# Hands-free Telelocomotion of a Wheeled Humanoid toward Dynamic Mobile
Manipulation via Teleoperation

Amartya Purushottam1, Yeongtae Jung2, Kevin Murphy2, Donghoon Baek2, Joao
Ramos1,2 The authors are with the 1Department of Electrical and Computer
Engineering and the 2Department of Mechanical Science and Engineering at the
University of Illinois at Urbana-Champaign, USA.This work is supported by the
National Science Foundation via grant IIS-2024775.

###### Abstract

Robotic systems that can dynamically combine manipulation and locomotion could
facilitate dangerous or physically demanding labor. For instance, firefighter
humanoid robots could leverage their body by leaning against collapsed
building rubble to push it aside. Here we introduce a teleoperation system
that targets the realization of these tasks using human’s whole-body motor
skills. We describe a new wheeled humanoid platform, SATYRR, and a novel
hands-free teleoperation architecture using a whole-body Human Machine
Interface (HMI). This system enables telelocomotion of the humanoid robot
using the operator’s body motion, freeing their arms for manipulation tasks.
In this study we evaluate the efficacy of the proposed system on hardware, and
explore the control of SATYRR using two teleoperation mappings that map the
operators body pitch and twist to the robot’s velocity or acceleration.
Through experiments and user feedback we showcase our preliminary findings of
the pilot-system response. Results suggest that the HMI is capable of
effectively telelocomoting SATYRR, that pilot preferences should dictate the
appropriate motion mapping and gains, and finally that the pilot can better
learn to control the system over time. This study represents a fundamental
step towards the realization of combined manipulation and locomotion via
teleoperation.

###### Index Terms:

Biwheeled locomotion, telerobotics and teleoperation, human and humanoid
motion analysis and synthesis, human-in-the-loop

##  I INTRODUCTION

Humanoid robots have the potential to aid workers in physically demanding and
dangerous jobs, such as firefighting and disaster relief. However, to
adequately aid workers in these tasks, robots must be capable of Dynamic
Mobile Manipulation (DMM), which is defined as the ability to combine
locomotion and manipulation to amplify the forces applied to the environment.
These types of tasks are challenging for state-of-the-art robots because they
require whole-body coordination, intermittent contacts, large manipulation
forces, dynamic locomotion, and a physically capable machine.

Figure 1: The targeted vision of the teleoperated humanoid platform, SATYRR,
performing Dynamic Mobile Manipulation via teleoperation with the Human
Machine Interface.

Autonomous robots cannot yet match human capability to plan complex actions in
unpredictable scenarios inherent to emergency response. To address this, we
aim to leverage human sensorimotor skills to directly control the robot’s
whole-body motion via teleoperation [1, 2], as seen in Fig.1. Whole-body
humanoid teleoperation can be separated by manipulation and locomotion tasks.
In this work, we focus on the study of the latter, telelocomotion, using human
whole-body movement.

Existing work on robot locomotion utilizes joysticks, visual-inertial motion
camera systems, and human-machine-interfaces. In contrast to these works, our
goal is to create a hands-free strategy for telelocomotion as a fundamental
step towards DMM. We envision that the operator will use their arms to control
robot manipulation while intuitively regulating robot locomotion using the
pitch and twist of their body. Similar strategies have been used with self-
balancing personal transportation devices, such as the Segway [3].

Wheeled bipeds, such as SATYRR, Ascento [4], and Handle [5], are the perfect
platform to study DMM. Self-balancing wheeled humanoids combine the advantages
of legged robots with the agility of wheels. Existing machines have
demonstrated nimble navigation over challenging terrain with inclines and
curbs, while allowing the exploitation of dynamic upright instability to
facilitate DMM tasks. Their human-like design with high center of mass (CoM)
and small base of support allows them to exert large manipulation forces to
the environment by dynamically throwing their bodies [5].

However, dynamic control of wheeled humanoid robots remains a challenge. Due
to the highly nonlinear nature of their dynamics, their controls often employ
reduced-order models with simpler and more intuitive behavior. For instance,
the Linear Inverted Pendulum (LIP) is often used to capture the core dynamics
of bipedal locomotion for the planning and control of humanoid robots [6].
Similarly, in this work we leverage the Wheeled Inverted Pendulum (WIP) model
to describe the fundamental behavior of the self-balancing wheeled humanoid
SATYRR [7]. Due to this model’s inherent instability, a control framework
capable of balancing the robot is required. As seen in [4, 8], linear state-
space systems and their closed-loop feedback controllers have been effective
in stabilizing and enabling locomotion of wheeled bipedal robots.

The user’s body motion is captured by the HMI (Human-Machine-Interface) [9]
and mapped to a desired robot velocity or acceleration via a piece-wise
continuous mapping and the dynamics of the WIP, as similarly seen in [10]. We
designed force feedback to prevent the human from falling during teleoperation
and enable a larger range of motion. When the operator tilts forward, the
translation of the human CoM is mapped into a desired forward motion. When the
operator rotates their body, the rotation around the human CoM is mapped into
a desired turning rate for SATYRR. In this viability study, we provide three
key contributions:

  * •

The introduction of a bi-wheeled humanoid robot SATYRR and its integration
with a novel whole-body HMI, that enables hands-free telelocomotion.

  * •

The development of two human motion mappings (velocity and acceleration)
evaluated on two sets of experiments.

  * •

Discussions on the viability of the proposed approach and consolidation of
user feedback about their human-robot interaction.

The experiments conducted help evaluate the efficacy of the proposed system
and benchmark the performance. Preliminary results show that our first pilot
was only seconds slower when using the HMI compared to using a handheld
joystick. The subject also preferred acceleration mapping over velocity
mapping. Our second pilot demonstrated their ability to better control the
system over three days of trials. These findings suggest that the proposed
system is a capable platform for telelocomotion.

The remainder of this paper is structured as follows. In section II we discuss
the implementation of hardware for SATYRR and the HMI, along with the key
software integration components. In section III we discuss the reduced order
model and closed-loop feedback tracking controllers. In section IV the
velocity and acceleration based human motion mappings are introduced. Section
V outlines the experiment design. Section VI discusses the experiment results
and user feedback. Finally, section VII provides our conclusion.

##  II System Description

###  II-A SATYRR Hardware

SATYRR is an anthropomorphic robot with a pair of legs that employ active
wheels instead of feet, as shown in Fig. 2. Each leg consists of a one degree-
of-freedom (DoF) mechanism that couples links 1 and 2 to move the wheel
vertically in a perfect straight line. The actuator directly drives link 1 and
a pulley fixed to the torso couples the rotation of link 2, in respect to the
robot’s body via a 1:2 timing belt transmission. This implies that link 2 will
rotate two times faster than link 1, in respect to the robot’s torso,
resulting in a vertical motion of the wheel because links 1 and 2 have the
same length. SATYRR’s lower-body is equipped with four of the same actuators
utilized in the MIT Mini Cheetah [11, 12]. These have a built-in reduction
ratio of 6:1 and can deliver a peak torque of 24Nm and achieve speeds up to
30rad/s. The actuators are backdrivable and can indirectly control the output
torque via regulation of the current going through the windings [13]. The four
motors are daisy chained and communicate over CAN bus with the low-level
controller sbRIO 9626 from National Instruments. Each motor is equipped with a
12-bit encoder and an inertial measurement unit (IMU) VN-100 from VectorNav is
mounted to the torso. This IMU houses an on-board Kalman filter that outputs
the estimation of the robot’s torso orientation at a maximum frequency of 200
Hz. To power all the motors, computer, and IMU, a single Kobalt 24V and 4Ah
battery is used in tandem with two variable low-dropout regulators (LDOs). The
system weighs 6.8 kg and stands at a nominal hip height of 0.28 m. The robot
computer runs the controller in real-time frequency of 833 Hz and communicates
with the HMI via UDP.

Figure 2: (A) Design of SATYRR’s lower body, torso, and straight line
mechanism of the single degree-of-freedom leg. (B) WIP reduced order model and
states.

###  II-B HMI Hardware

The HMI introduced in [9] and [10] is composed of a high force haptic device,
a large force plate, and passive Motion Capture (MoCap) exoskeleton. The
haptic device consists of two backdrivable linear actuators that are capable
of generating up to 100N of force each to the operator’s torso near the CoM.
Each actuator is mounted on a a passive gimbal and measures the spatial
location of the operator’s torso. The actuators are utilized in our
experiments to create a spring-like force that allows the operator to tilt
their body forward and backward without falling. Future versions of this
device will employ four linear actuators (Fig. 1) and allow the generation of
forces and torques to the operator’s torso in multiple directions. The design
of the force plate is similar to that of a Stewart Platform [14] but with six
uniaxial load cells, instead of linear actuators. It measures the magnitude
and location of the net ground contact force applied by the operator’s feet.
Its point of application is known as the Center of Pressure (CoP).

The HMI is controlled by a real-time controller (cRIO-9082 from National
Instruments) that runs the at 1kHz. The HMI can be used for providing feedback
about disturbances to the robot’s dynamics to the user, which will
significantly improve teleoperation performance in the future. For example, if
a disturbance force is applied to the robot, the HMI applies a similar force
to the user, who uses their balancing reflexes for stabilizing locomotion.
User’s dynamics measured by the force plate and exoskeleton system can be used
to calculate the feedback force for the user comparing it to the robot’s
dynamics. This strategy has been employed in the author’s previous work on
whole-body teleoperation of humanoid robots [15, 16].

In this study, we introduce a simple unilateral control strategy for the
preliminary investigation of the human-robot control system. The position of
the user’s CoM is measured by the linear actuators and utilized for the
control of robot motion in its sagittal plane. The turn rate of the robot is
controlled by the twist angle of the human along the longitudinal $z$-axis as
shown in Fig. 3, and described in Section III.

Figure 3: (A) Full system layout and control flow work from pilot to the WIP
to the full robot. (B) Overview of the control architecture.

##  III Modeling & Controller Design

In section III-A we discuss the planar WIP model that reduces the
dimensionality of the system while still enabling driving and steering. In
section III-B we discuss the feedback controller that handles the dynamic
balance of the robot around the desired upright equilibrium. Section III-C
discusses additional controllers that regulate the robot height (leg length)
and turning rate (yaw) around the vertical $Z$ axis.

###  III-A The Wheeled Inverted Pendulum (WIP) Model

As seen in [4], [7], and [17], reduced order models (RoM) of self-balancing
wheeled systems have been effective for motion planning and dynamic
stabilization. We use the planar WIP model, composed of active wheels and the
lumped rigid-body that represents the robot’s torso, as shown in Fig. 3(B), to
develop the stabilization controller. Moreover, we assume no slip between the
wheels and the ground, which implies that the distance traveled, $x_{w}$, is
linearly proportional to the total number of wheel revolutions. The dynamic
equations of motion of the WIP are given by:

$\begin{gathered}\bigg{(}m_{o}\\!+\\!m_{w}\\!+\\!\frac{I_{w}}{r^{2}}\bigg{)}\ddot{x}_{w}\\!+\\!m_{o}Ls_{\theta_{R}}\dot{\theta}^{2}_{R}\\!-\\!m_{o}Lc_{\theta_{R}}\ddot{\theta}_{R}\\!=\\!u\\\
(m_{o}L^{2}+I_{o})\ddot{\theta}_{R}-m_{o}Lc_{\theta_{R}}\ddot{x}_{w}-m_{o}gLs_{\theta_{R}}\\!=\\!0,\end{gathered}\
$ (1)

where $m_{b}$ is the mass of the body, $m_{w}$ is the wheel mass, $L$ is the
length between the center of the wheel and the CoM of the body, $\theta_{b}$
is the pitch rotation of the body from vertical axis, $I_{w}$ is the inertia
of the wheel, $I_{b}$ is the inertia of the body evaluated at its CoM, $r$ is
the radius of the wheel, and $u$ is the control effort (force) exerted along
the $x-$axis on the wheel due to the motor torque.

For the controller synthesis, we acquire the linear state-space model by
linearizing equation (1) around the upright equilibrium state $q_{0}$ at the
nominal center of mass height.

$\displaystyle\delta\dot{q}=A\delta q+B\delta u.$

Where $q=[x_{w}\quad\theta_{R}\quad\dot{x}_{w}\quad\dot{\theta}_{R}]^{\top}$
is the state vector; $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times
k}$ are the state-space matrices, $\delta q=q-q_{0}\in\mathbb{R}^{n}$ is the
deviation from the equilibrium state $q_{0}$ and $\delta
u=u-u_{0}\in\mathbb{R}^{k}$ is the deviation from the nominal control effort
$u_{0}$. Where $n=4$ and $k=1$.

###  III-B Stabilization Controller

The stabilization task can be simplified by leveraging the mechanical design
of our robot, which constrains the CoM to remain vertically aligned with the
center of the wheel. This feature enables decoupling of the posture control to
regulate height of the robot, the stabilization to keep SATYRR balanced, and
the steering controller [7], [8].

By discretizing the continuous state-space model with a sampling period
$T_{s}=1.0$ ms and assuming an infinite horizon, we can solve the discrete
time Algebraic Riccati Equation for the corresponding optimal gain matrix $K$
[18]. After extensive tuning of the cost matrices, the optimal gains generated
by the model were $K=[-180,-640,-120,-70]$.

To control the real robot using our planar model, we assumed that the legs
behave symmetrically. The left leg, right leg hip, and wheel angles are
averaged to determine robot height and wheel displacement. Final stage mapping
can be seen in Fig. 3.

###  III-C Height Adjustment & Yaw Controller

The height adjustment PD controller is given by:

$\tau_{Hip}=J_{c}^{\top}F_{g}+K_{p}(\theta_{1,des}-\theta_{1})+K_{d}(\dot{\theta}_{1,des}-\dot{\theta}_{1}),$
(2)

where $\theta_{1}$ represents the angle between the torso and link 1,
$F_{g}=m_{0}g$ represents the force due to gravity, and $J_{c}$ is the
Jacobian that relates the relative angular velocity $\dot{\theta}_{1}$ of link
1 with the vertical velocity of the foot. Tuned on hardware, $K_{p}=100$ and
$K_{d}=1$.

The robot’s heading PD controller is given by:

$\tau_{yaw}=K_{p,yaw}(\gamma_{des}-\gamma)+K_{d,yaw}(\dot{\gamma}_{des}-\dot{\gamma}).$
(3)

Where $\gamma$ represents the robot’s yaw, and $\theta_{w,R},\theta_{w,L}$
represent the left and right wheel encoder positions:

$\gamma=\frac{r_{w}}{r_{c}}(\theta_{w,R}-\theta_{w,L}).$ (4)

Here $r_{w}$ represents the wheel radius and $r_{c}$ represents the distance
between the two wheels. Hand tuned on hardware, $K_{p,yaw}=1$ and
$K_{d,yaw}=0.1$.

##  IV Locomotion Teleoperation Mappings

For intuitive hands-free locomotion control of SATYRR, we explore the efficacy
of two, hand tuned piece-wise linear mappings that map tilt to the robot WIP’s
velocity or acceleration.The two mappings differ in the type of control
authority they give the user [10]. Section IV-A discusses a proposed velocity
mapping. Section IV-B outlines how the user’s tilt is mapped to the robots
tilt and acceleration.

###  IV-A Human Motion Velocity Mapping

To control linear velocity of the robot, a joystick’s tilt or user’s CoM
displacement along the $x$ axis, $p_{x}$, is mapped to a desired translation
velocity of the robot’s wheel, $\dot{x}^{des}_{w}$. We employ a piece-wise
linear transformation with a small dead-band, $p^{db}$. This dead-band
accounts for natural human swaying or joystick noise while at home position.
The mappings, as seen in Fig. 3, enable the pilot to fine tune their desired
commands and robot response. The $x$-directional virtual spring rendered by
the HMI’s linear actuator allows the operator to lean further forward or
backwards without falling. The spring constant was set by user’s preference
for desired stiffness and maximum pitch.

The piece-wise linear mapping function is defined as:

$\dot{x}^{des}_{w}\\!\\!=\\!\\!\begin{cases}0&|p_{x}|<p^{db}_{x}\\\
sgn(p_{x})\alpha_{p_{1}}(|p_{x}|\\!-\\!p^{db}_{x})&p^{db}_{x}\\!\\!\leq\\!\\!|p_{x}|\\!\\!<\\!\\!p^{swp}_{x}\\\
sgn(p_{x})\alpha_{p_{2}}(|p_{x}|\\!-\\!p^{swp}_{x})\\!+\\!c^{swp}&p^{swp}_{x}\\!\\!\leq\\!\\!|p_{x}|\\!\\!<\\!\\!p^{max}_{x}\\\
sgn(p_{x})\dot{x}^{max}_{w}&otherwise\end{cases}$ (5)

where $\alpha_{p_{1}},\alpha_{p_{2}}$ are the slopes of the individual piece-
wise linear sections. $p^{swp}$ represents the switching coordinate, and
$c^{swp}$ represents the intersection of the two piece-wise sections that
ensures continuity. The $sgn()$ function returns a binary value $[-1\quad 1]$
depending on the sign of the input.

The turning rate of the robot is analogously controlled by the rotational
angle of the human along $z$-axis, $\gamma_{h}$. The angle is mapped to a
desired yaw rate, $\dot{\gamma}_{des}$, using a similar piece-wise linear
function but with different coefficients. By controlling driving velocity and
turning rate using their whole-body, the operator’s hands are left free for
future telemanipulation tasks.

###  IV-B Human Motion Acceleration Mapping

The acceleration mapping (6) maps the joystick tilt or human tilt,
$\theta_{H}$, to the robot’s desired WIP tilt, $\theta_{R}^{des}$. Since the
WIP’s tilt is proportional to its CoM acceleration, this mapping can be
understood as commanding acceleration. Under the small-angle assumption, the
human’s CoM displacement along the y axis is approximated as their tilt,
$p_{y}\approx\theta_{H}$. Accordingly, the piece-wise mapping function is
given by:

$\theta_{R}^{des}\\!\\!=\\!\\!\begin{cases}0&|\theta_{H}|<\theta_{H}^{db}\\\
sgn(\theta_{H})\alpha_{\theta_{H}}(|\theta_{H}|\\!-\\!\theta_{H}^{db})&\theta^{db}_{H}\\!\\!\leq\\!\\!|\theta_{H}|\\!\\!<\\!\\!\theta^{max}_{H}\\\
sgn(\theta_{H})\theta^{max}_{R}&else\end{cases}$ (6)

where $\theta_{H}^{db}$ represents a dead-band around zero,
$\alpha_{\theta_{H}}$ is the section slope, and $\theta_{R}^{max}$ represents
the maximum commanded tilt. Differentiating this input we find
$\ddot{\theta}_{R}^{des}$. Under the small angle approximation
($sin(\theta_{R})\approx\theta_{R};\,\cos(\theta)\approx 1$), and assuming
small pendulum velocities ($\dot{\theta_{R}^{2}}\approx 0$), we rearrange (1)
to:

$\ddot{x}_{w}^{des}=-\frac{(m_{0}L^{2}+I_{0})}{m_{0}L}\ddot{\theta_{R}^{des}}+g\theta_{R}^{des}$
(7)

Integrating (7) we find the the desired states, $\dot{x}_{w}^{des}$ and
$x_{w}^{des}$, for our stabilization controller.

##  V Experimental Design

The goal of the experiments is to demonstrate the integration of SATYRR with
the HMI, and perform benchmark evaluations to determine the viability of
different human-robot-motion mappings for telelocomotion. Two sets of human
subject experiments were developed: 1) System and mapping efficacy experiments
and 2) Pilot-system response experiments. Inspired by sports-based agility
tests [19, 20], two test courses were designed: A) Straight-line speed course
and B) 3-cone drill course as seen in Fig. 4. These tests focus on
demonstrating rapid linear motion and confined navigation around obstacles,
respectfully. Each experiment (1 or 2) required the completion of these two
courses (A+B). The experiment layout can be seen in Fig. 4.

Figure 4: (A) Obstacle avoidance experiment test course. The linear motion
test was setup in the same area without the cones. (B) The two experimental
designs and layout for benchmarking.

The goal of both tests is to reach the end destination in the shortest time
possible. The second test introduced obstacles (cones) that the user is also
required to avoid colliding with by going through the cones in a weave-like
pattern. The successful completion of a single run is defined as the
translation from start to goal line following the design of the test.
Instability caused by collisions while steering, venturing outside the
predefined region, and slipping/falling due to user error are qualified as
failed runs. Moreover, the user is required to retain stable control of the
robot upon passing of the goal line. Experiment 1 required Pilot A to test all
eight combinations of tests (e.g., joystick-velocity-mapping-cone test is a
single combination), as can be seen in in Fig. 4. The user was allowed to
practice with each combination until satisfaction but only allowed to change
their mapping and controller gains for different controller-mechanism-mapping
schemes (e.g., the gains using the joystick for velocity mapping were fixed
for its straight-line and cone drill test). This enforces that the user choose
gains capable of telelocomoting the robot along open straight paths and around
obstacles since a priori knowledge of the terrain would not be known in
practicality. After tuning and practicing, the pilot was required to complete
10 successful completion runs for each of the 8 combinations. Each combination
took between 3-3.5 hours to tune, practice, and complete.

Experiment 2 required Pilot B to choose a preferred motion mapping for their
tests. After an initial trial preference period, pilot B chose velocity
mapping. Their mapping gains were adjusted prior to day 1 and set as their
fixed gain set – unmodifiable over the next 3 days. The variable gain set
could be adjusted and tested daily prior to starting. Over the 3 days, Pilot B
was required to complete 5 successful runs with their fixed gains and variable
gains for both the straight-line test and 3-cone agility test.

For these experiments the robot CoM height is fixed at 0.28m. The pilot is
given a first-person view of the robot during the experiments through DJI FPV
googles and a single monocular camera on the robot. The camera orientation is
fixed to align with the center of the robot and pitched as per the user’s
preference prior to the start of the tests. For both cone and straight-line
tests, the robot is started from the same position 4m away from the finish
line. The human’s task is to teleoperate the robot after a 3 second countdown.

Due to the lengthy duration of the experiments and because this work
represents a viability study only two subjects of age 32 years – pilot A – and
21 years - pilot B – were recruited. Neither pilot had competitive athletic
involvement beyond high school. These experiments were conducted after the
University’s Institutional Review Board reviewed and approved this research
study.

##  VI Experimental Results and Discussion

Sections A, and B, discuss the results of the system efficacy experiments with
pilot A. Section C discusses the key points from the second set of experiments
with pilot B. Section D consolidates feedback from both pilots.

###  VI-A HMI Control Efficacy

To test the practicality of using the HMI with feedback force for hands-free
tele-locomotion, we use the conventional joystick as a gold-standard for
comparison. Across the same mappings we notice that pilot A’s performance was
superior using the joystick. The fastest time across the obstacle test was
completed using the joystick at 9.2 seconds. However, the user had the fastest
performance along the straight-line test using the HMI in 5.9 seconds. The
average total distance traveled along the joystick-cone test was 5.03m and
5.18m for the HMI-cone test – a difference of 150 cm. Fig. 6 highlights the
paths taken by the robot using the joystick and HMI for all 10 successful run.
Note that the robot’s path was reconstructed using wheel odometry, which is
prone to error accumulation due to wheel slip.

We hypothesize three reasons the user may have been slower with the HMI: 1)
There may exist a difference in experience between operating the two systems
as most people are likely more familiar with the usage of a handheld joystick.
2) The pilot noted that during the HMI experiments, first-person view
dizziness was augmented by mismatch between their body motion and the robot’s
direction of motion. To mitigate dizziness the pilot may have unintentionally
modulated their control with more conservative body movements and velocities.
This pilot also felt less dizzy when standing stationary using the joystick.
3) Finger motion has an inherently faster bandwidth in comparison to the
movement of the operator’s whole-body. This results in a slower rise time of
the desired velocities commanded by the pilot and subsequently a slower
response from the robot.

These preliminary results indicate that the HMI is only seconds slower than
the completion time when using the joystick, and the difference in average
path lengths differs by centimeters. This supports the assumption that the
operator can use their body to control the robot with almost the same
dexterity as with their hands.

Figure 5: Experiment 1 average completion time for pilot A on 10 successful
test runs. Error bars show max and min values for each of 8 combinations.

Figure 6: Reconstructed odometry data for all 8 combination test runs for the
system efficacy experiments with pilot A. Each combination shows the first 10
successful runs. Figure 7: HMI single-run 3-Cone obstacle avoidance agility
test: actual and desired states, applied torques, and pilot CoM position for
velocity and acceleration mapping. (A) and (B) show characteristic behavior of
non-minmum phase systems e.g pendulums - the robot moves backward initially
before moving forward. Legend guide: VM - velocity mapping. AM - acceleration
mapping

###  VI-B Human Motion Mapping Preference

Pilot A performed the experiments more quickly using the acceleration mapping
as 3 of the 4 test setups resulted in lower time, as seen in Fig. 5. The
fastest completion time recorded for any test was using the HMI-acceleration
mapping at $5.9$ seconds. This strategy enables the user to move faster than
they could have with the velocity mapping, but in turn places responsibility
on them to stabilize the robot – continuous acceleration causes the robot to
pitch too far and fall. We hypothesize 2 reasons that the velocity mapping may
have resulted in slower times: 1) The user’s commanded inputs were perturbed
by small oscillations of the robot making velocity control more difficult
since directly commanding yawing and forward motion are coupled dynamics.
These oscillations ($\approx\pm 0.075$m in amplitude) around the desired wheel
position occur at slower speeds and while the robot is trying to stay
stationary. They are caused by a combination of actuation delay and friction
in the wheels. 2) Pilot A’s natural control preference leans towards control
of acceleration rather than velocity.

Our observations suggest a strategy for constructing these piece-wise mapping
functions. For both mappings, the subject desired slower and more precise
movements at lower angles of tilt for the obstacle avoidance test but more
aggressive gains – i.e. higher achievable velocities or tilt – for the
straight line test where the user throttled maximum input. For the velocity
mapping this dictates a moderate gain within the first section of the piece-
wise function and higher gains for the second section. Note, creating a
function with widely different gains can result in jerkiness in the robot’s
response and thus requires careful tuning with user feedback. For the
acceleration mapping, a simple linear function with small gains can be used –
pilot A preferred mapping their maximum pitch to a desired max $1.5^{o}$ tilt
of the robot. We believe there are 2 reasons for this small gain preference:
1) The robot’s acceleration is a function of the pilot’s pitch and pitch
acceleration shown in (7). This acceleration is significant as the user is
never truly standing still in the HMI. 2) Feedback from the user also revealed
the key to their acceleration mapping control strategy – accelerating to a
desired constant velocity, then commanding zero acceleration, as seen by the
virual spring force in Fig.7E, and using quick yawing motions to correct for
the robot’s position. Smaller acceleration gains would result in a slower
velocity increase and prevent the user from overshooting their desired robot
speed. Note that with this mapping, the robot velocity is not limited for the
straight-line test.

The velocity mapping had 7 failed runs with the joystick, and 11 failed runs
with the HMI before the completion of the 10 successful runs – a total of 18
failed runs. The acceleration mapping had 3 failed runs with the joystick and
4 failed runs with the HMI – a total of 7 failed runs. The completion times
and consistency of the acceleration mapping serve as early indicators that our
pilot preferred acceleration mapping. The testing conducted on hardware
indicates small timing differences – seconds apart – between the different
mappings. This supports our claim that both mappings present viable approaches
to control of SATYYR, and pilots may indeed have preferences in control
mapping schemes.

###  VI-C Does the Pilot improve with time?

Implicitly assumed in our goal towards whole-body teleoperation is the idea
that the pilot can learn to operate the robot more effectively through
adaptation in their control. Through our preliminary findings we show here
that this assumption is well founded and that pilots can improve their course
telelocomotion performance by practicing and altering their body motion.

Fig. 8(A) illustrates comparisons between two sets of gains – fixed gains and
varied gains – over time. The user-preferred fixed gain maximum velocity was
1.0 m/s. Over the duration of the experiments, the varied mappings gains were
increased by 10%, 25% and 35% by user preference. The average completion time
of D1-FGCT (Day 1 Fixed-Gain-Cone-Test) was 12.65 seconds, and 11.41 seconds
for D3-FGCT – a decrease of 9.8% over 15 trials. The average completion time
of D3-VGCT was 8.79 seconds. There was no significant difference in the
completion times of the straight-line tests. From these results we see that
the user was able to complete the 3-cone agility test more quickly over the 3
days and consistently showed better performance with the higher varied gains.
We believe that for the straight-line test, which requires less movement and
skill, the pilot was limited by the achievable maximum velocity of the robot –
1.4 m/s.

For velocity mapping (chosen by the pilot) we define human- control
sensitivity, $S$ as:

$S=\frac{\dot{x}^{max}_{w}}{\theta^{max}_{H}}$

The maximum pitch, $\theta^{max}_{H}$, is determined by the user’s comfort
prior to the start of the experiments and is fixed. The preferred maximum
velocity is user set and seen in Eqn. (5). In the pilot’s attempt to complete
the course faster, they desired a greater maximum velocity. This leads to an
increase in the sensitivity of the system. This pilot adjusted for this
increased sensitivity by using smaller, more acute body motions as seen by the
reduction in the used human motion area in Fig. 8(B)-(C). We noticed this
emerging behavior with other pilots who operated the system as well. This may
suggest that with long-term training, velocity mapping users may converge to a
high-gain-small-tilt strategy [10]. To address our initial question and
benchmark the systems development over time, we conclude that the pilot can
adapt and learn to better control this system. Future studies look to better
understand the limitations of this human-robot learning and generalize these
preliminary result for larger sample sizes.

Figure 8: (A) Performance over time data. Fixed-Gain-Cone-Test (FGCT).
Variable-Gain-Cone-Test (VGCT). (B) User human motion area for Day 1. (C) User
human motion area for Day 3.

###  VI-D Human-Robot Interaction

Both pilots reported that the usage of the joystick and HMI for teleoperation
resulted in some disorientation. As discussed above, we believe this problem
was a byproduct of human-robot motion mismatch while using the HMI and robot
pendulum-like swaying motion. Also, without a 3rd person view of the robot,
both pilots expressed concern about knowing where the wheels of the robot
relative to their view and were sometimes unaware they had hit the cones. This
exposes a potential need for a more interactive and user-friendly interface to
perform more dynamic tests in the future.

With larger LQR tracking gains, the swaying motions were exasperated as the
robot was more aggressive in its response to the pilot’s command. Sudden stops
and turns resulted in some overshoot, that led to a quicker and more
aggressive pendulum like motion response from the robot. As such, both pilots
requested that the wheel position tracking gains of the stabilization
controller be reduced significantly. These findings suggest that our
stabilization controller with a human in the loop may benefit from preference
based control designs [21].

These adjusted controller gains, user field-of-view, and swaying dizziness may
have led to sub-optimal robot paths, seen in Fig. 6. While increased yawing
sensitivity was crucial for the obstacle avoidance test, the linear motion
test would require less aggressive gains for yawing, suggesting that this
mapping should depend on the expected path and desired forward velocity.
Finally, both subjects did report feeling more comfortable with the viewpoint
and control of the robot with increased practice and test runs. This provides
a positive indicator that further training of the operators with the HMI could
result in increased performance.

###  VI-E Limitations of the Study & Future Work

The experiments conducted suggest the viability of the hands-free approach for
telelocomotion and were not designed to search for optimal performance.
Different motion mappings could lead to improved results. In addition, due to
the complexity of the system and the extensive testing required from the
pilots, the results for only two operators are shown in this work. Moreover,
due to physical limitations of the robot – its wheel size and maximum motor
speed – we could not achieve speeds beyond 1.4 m/s. Moving forward we look to
employ head tracking to reduce user motion dizziness and a pair of arms for
manipulation. We believe the added payload can be compensated by the balancing
controller to minimally degrade human performance, [5]. Finally, we look
forward to integrating bilateral feedback [1] into our design as to better
achieve dynamic similarity between the user and robot.

##  VII Conclusions

The contributions presented here are: 1) Introduction of a bi-wheeled
humanoid, SATYRR, and its integration with the HMI towards whole-body
teleoperation. 2) Completion of benchmark experiments to test robot agility
and telelocomotion by using body pitch. For contribution 1) we show the
hardware setup, reduced order model, tracking controllers, and motion mappings
that enabled hands-free teleoperation of SATYRR. For contribution 2) we ran 2
experiments. In experiment 1 we used completion time (CT), average path length
traveled, and number of failed runs to address the efficacy of the proposed
system with different mapping types, and subject preferences. In experiment 2
we fix the mapping and look at used motion area and CT over 3 days to
determine if the pilot can adjust and learn to operate better. The results
suggest that the HMI is a capable platform for controlling SATYRR, that motion
mapping preferences are dictated by the pilot, and that users can show
improved performance by practicing and adapting their motion behaviors over
time. Finally, we discuss feedback from the pilots to better understand the
limitations of the system. These preliminary results suggest viability and
support future developments of a hands-free telelocomotion approach.

## References

  * [1] J. Ramos and S. Kim, “Dynamic bilateral teleoperation of the cart-pole: A study toward the synchronization of human operator and legged robot,” _IEEE Robotics and Automation Letters_ , vol. 3, no. 4, pp. 3293–3299, 2018\.
  * [2] J. Ramos and S. Kim, “Humanoid dynamic synchronization through whole-body bilateral feedback teleoperation,” _IEEE Transactions on Robotics_ , vol. 34, no. 4, pp. 953–965, 2018.
  * [3] “Segway, howpublished = https://www.segway.com/, note = Accessed: 2021-10-01.”
  * [4] V. Klemm, A. Morra, C. Salzmann, F. Tschopp, K. Bodie, L. Gulich, N. Küng, D. Mannhart, C. Pfister, M. Vierneisel, F. Weber, R. Deuber, and R. Siegwart, “Ascento: A two-wheeled jumping robot,” in _2019 International Conference on Robotics and Automation (ICRA)_ , 2019, pp. 7515–7521.
  * [5] E. Guizzo, “Boston dynamics enters warehouse robots market, acquires kinema systems,” Jun 2021. [Online]. Available: https://spectrum.ieee.org/boston-dynamics-warehouse-robots-acquires-kinema-systems
  * [6] S. Kajita, F. Kanehiro, K. Kaneko, K. Yokoi, and H. Hirukawa, “The 3d linear inverted pendulum mode: a simple modeling for a biped walking pattern generation,” in _Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180)_ , vol. 1, 2001, pp. 239–246 vol.1.
  * [7] H. Chen, B. Wang, Z. Hong, C. Shen, P. M. Wensing, and W. Zhang, “Underactuated motion planning and control for jumping with wheeled-bipedal robots,” 2020.
  * [8] H. Zhou, X. Li, H. Feng, J. Li, S. Zhang, and Y. Fu, “Model decoupling and control of the wheeled humanoid robot moving in sagittal plane,” in _2019 IEEE-RAS 19th International Conference on Humanoid Robots (Humanoids)_ , 2019, pp. 1–6.
  * [9] S. Wang, K. Murphy, D. Kenney, and J. Ramos, “A comparison between joint space and task space mappings for dynamic teleoperation of an anthropomorphic robotic arm in reaction tests,” in _2021 IEEE International Conference on Robotics and Automation (ICRA)_ , 2021.
  * [10] S. Wang and J. Ramos, “Dynamic locomotion teleoperation of a wheeled humanoid robot reduced model with a whole-body human-machine interface,” in _arXiv preprint arXiv:2109.03906_ , 2021.
  * [11] B. Katz, “A low cost modular actuator for dynamic robots,” 2018, master’s Thesis, MIT.
  * [12] B. Katz, J. D. Carlo, and S. Kim, “Mini cheetah: A platform for pushing the limits of dynamic quadruped control,” in _2019 International Conference on Robotics and Automation (ICRA)_ , 2019, pp. 6295–6301.
  * [13] P. M. Wensing, A. Wang, S. Seok, D. Otten, J. Lang, and S. Kim, “Proprioceptive actuator design in the mit cheetah: Impact mitigation and high-bandwidth physical interaction for dynamic legged robots,” _IEEE Transactions on Robotics_ , vol. 33, no. 3, pp. 509–522, 2017.
  * [14] D. Stewart, “A platform with six degrees of freedom,” in _Proceedings of the institution of mechanical engineers_ , vol. 180, no. 1, 1965, pp. 371–386.
  * [15] A. Wang, J. Ramos, J. Mayo, W. Ubellacker, J. Cheung, and S. Kim, “The hermes humanoid system: A platform for full-body teleoperation with balance feedback,” in _2015 IEEE-RAS 15th International Conference on Humanoid Robots (Humanoids)_ , 2015, pp. 730–737.
  * [16] J. Ramos and S. Kim, “Dynamic locomotion synchronization of bipedal robot and human operator via bilateral feedback teleoperation,” _Science Robotics_ , vol. 4, no. 35, p. eaav4282, 2019. [Online]. Available: https://www.science.org/doi/abs/10.1126/scirobotics.aav4282
  * [17] J. Ramos and S. Kim, “Dynamic bilateral teleoperation of the cart-pole: A study toward the synchronization of human operator and legged robot,” _IEEE Robotics and Automation Letters_ , vol. 3, no. 4, pp. 3293–3299, 2018\.
  * [18] G. F. Franklin, J. D. Powell, M. L. Workman _et al._ , _Digital control of dynamic systems_. Addison-wesley Reading, MA, 1998, vol. 3.
  * [19] J. Dawes and M. Roozen, _Developing Agility and Quickness_. Human Kinetics, IL, 2012, vol. 1.
  * [20] P. F. Stewart, A. N. Turner, and S. C. Miller, “Reliability, factorial validity, and interrelationships of five commonly used change of direction speed tests,” _Scandinavian journal of medicine & science in sports_, vol. 24, no. 3, pp. 500–506, 2014.
  * [21] M. Tucker, N. Csomay-Shanklin, W.-L. Ma, and A. D. Ames, “Preference-based learning for user-guided hzd gait generation on bipedal walking robots,” in _2021 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2021, pp. 2804–2810.


# Sign-balance of excedances over mod-k-alternating permutations and gamma-
positivity

Hiranya Kishore Dey Department of Mathematics, Indian Institute of Science,
Bangalore 560012, India. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Iswar Mahato Department of Mathematics, Indian Institute of Technology Bombay,
Mumbai 400076, India. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A permutation is called mod-k-alternating if its entries are restricted to
having the same remainder as the index, modulo some integer $k\geq 1.$ In this
paper, we find the sign-balance for mod-k-alternating permutations with
respect to the statistic excedance. Moreover, we study the sign-balance for
excedances over mod-k-alternating derangements. The results are obtained by
constructing suitable matrices and connecting their determinants with the
signed excedance enumeration of mod-k-alternating permutations. As an
application of the signed excedance enumeration, we prove that when $n\equiv
k\pmod{2k}$, the excedance enumerating polynomials over the even and odd mod-
k-alternating permutations, starting with a fixed remainder, are gamma-
positive.

Keywords: Mod-k-alternating permutations, excedance, derangement, gamma-
positive.

AMS Subject Classification (2020): 05A15, 15A15, 05E99.

##  1 Introduction

Enumerating permutations by different statistics is an interesting and well-
studied area in combinatorics. The most famous statistics on a permutation are
descent, excedance, inversion and major-index, and their enumerators are
classical results in combinatorics (see [8], [18]). One can look at the course
notes of Foata and Han [9] and the book of Petersen [22] for a self-contained
introduction to this area.

For a positive integer $n$, let $\mathfrak{S}_{n}$ be the symmetric group on
the set $[n]=\\{1,2,\ldots,n\\}$. For a permutation $\pi$, written in one-line
notation as $\pi=\pi_{1}\pi_{2}\dots\pi_{n}$, let
$\mathrm{Des}(\pi)=\\{i\in[n-1]:\pi_{i}>\pi_{i+1}\\}$ be its set of descents
and let $\mathrm{des}(\pi)=|\mathrm{Des}(\pi)|.$ The classical Eulerian
number, denoted by $A_{n,k}$, is the number of permutations in
$\mathfrak{S}_{n}$ with $k$ descents and
$A_{n}(t)=\sum_{\pi\in\mathfrak{S}_{n}}t^{\mathrm{des}(\pi)}$ is the Eulerian
polynomial. Define $\mathrm{Exc}(\pi)=\\{i\in[n-1]:\pi_{i}>i\\}$ as its set of
excedances and let $\mathrm{exc}(\pi)=|\mathrm{Exc}(\pi)|.$ It is well known
(see, [18]) that excedances and descents are equidistributed when summed over
the elements of $\mathfrak{S}_{n},$ that is,
$\sum_{\pi\in\mathfrak{S}_{n}}t^{\mathrm{des}(\pi)}=\sum_{\pi\in\mathfrak{S}_{n}}t^{\mathrm{exc}(\pi)}.$

For $\pi\in\mathfrak{S}_{n},$ the number of inversions of $\pi$ is defined as
$\mathrm{inv}(\pi)=|\\{(i,j):i<j,\pi_{i}>\pi_{j}\\}|$ and the sign of a
permutation is defined as $\mathrm{sgn}(\pi)=(-1)^{\mathrm{inv}(\pi)}.$ It is
well-known that the sign of a permutation $\pi\in\mathfrak{S}_{n}$ is $1$ if
and only if $\pi$ is even, that is, it belongs to the alternating group
$\mathcal{A}_{n}$. The sign-balance of a set $T_{n}\subseteq\mathfrak{S}_{n},$
respecting a statistic $\mathrm{stat}$, is given by

$\sum_{\pi\in T_{n}}(-1)^{\mathrm{inv}(\pi)}t^{\mathrm{stat}(\pi)}.$

The sign-balance of $\mathfrak{S}_{n}$ with respect to the statistic
$\mathrm{des}$ was first considered by Loday in [15]. He defined
$\mathrm{SgnDes}(n,t)=\sum_{\pi\in\mathfrak{S}_{n}}(-1)^{\mathrm{inv}(\pi)}t^{\mathrm{des}(\pi)}$
as the signed descent enumerator and conjectured the following recurrence
relation, which was proved by Désarménien and Foata in [5, Theorem 1].

######  Theorem 1.1 (Désarménien-Foata).

For positive integers $n$, we have

$\displaystyle\mathrm{SgnDes}_{n}(t)$ $\displaystyle=$
$\displaystyle\begin{cases}(1-t)^{m}A_{m}(t)&\text{if $n=2m$},\\\
(1-t)^{m}A_{m+1}(t)&\text{if $n=2m+1$.}\end{cases}$

Later, Wachs in [28] gave a sign-reversing involution on $\mathfrak{S}_{n}$,
thereby giving an alternate, bijective proof of Theorem 1.1. Gessel and Simion
in [28, Corollary 2] gave an elegant factorial-type product formula for the
refined sign-balance of $\mathfrak{S}_{n}$ respecting major indices.

There have been many works on the sign-balance of some restricted subsets of
the symmetric group $\mathfrak{S}_{n}$. For example, Simion and Schmidt [24]
determined the sign-balance for the 321-avoiding permutations in
$\mathfrak{S}_{n}$ with respect to various statistics. Adin and Roichman [1]
gave a refinement, respecting the position of the last descent. Reifegerste
[23] proved another refinement, respecting the length of the longest
increasing subsequence. The sign-balance of $\mathfrak{S}_{n}$ and its various
subsets, respecting the statistic $\mathrm{exc}$ have been investigated by
Mantaci in [19]. He proved the following remarkable result on signed excedance
enumeration.

######  Theorem 1.2 (Mantaci).

For positive integers $n$, we have

$\sum_{\pi\in\mathfrak{S}_{n}}(-1)^{\mathrm{inv}(\pi)}t^{\mathrm{exc}(\pi)}=(1-t)^{n-1}.$

Later, Mantaci in [20] gave a combinatorial proof of this result. In [25],
Sivasubramanian gave an alternate proof of Mantaci’s result by evaluating the
determinant of an appropriately defined $n\times n$ matrix. Motivated by these
works, in this article, we study the sign-balance of the statistic
$\mathrm{exc}$ over mod-k-alternating permutations. The results of this
article can be considered as an application of linear algebraic techniques in
the area of combinatorics, which are obtained by constructing suitable
matrices and connecting their determinants with the signed excedance
enumeration of mod-k-alternating permutations.

A permutation is called a parity alternating permutation, if its entries
assume odd and even integers alternately. In 2010, Tanimoto [26] introduced
the notion of parity alternating permutation and studied its various
combinatorial properties. In [27], the author determined the sign-balance of
parity alternating permutation, respecting the statistic descent, and
described the signed Eulerian numbers by the parity alternating permutations.
Recently, Kebede and Rakotondrajao [14] studied the parity alternating
permutations starting with an odd integer. Very recently, Alexandersson et.
al. [2] generalized the notion of parity alternating permutations to mod-k-
alternating-permutations, which is defined as follows:

######  Definition 1.1.

For any positive integer $k$, a mod-k-alternating permutation of size $n$ is a
permutation $\pi\in\mathfrak{S}_{n}$ such that $\pi_{i}\equiv i\pmod{k}$ for
all $i=1,2,\dots,n.$

Let $\mathrm{MP}_{n}^{k}$ be the set of such mod-k-alternating permutations in
$\mathfrak{S}_{n}$ and let $\mathrm{mp}_{n}^{k}$ denote the cardinality of the
set $\mathrm{MP}_{n}^{k}$. It is easy to see that $\mathrm{MP}_{n}^{k}$ is a
subgroup of $\mathfrak{S}_{n}$. Note that

$\mathrm{mp}_{n}^{k}=\Big{(}\Big{\lceil}\frac{n}{k}\Big{\rceil}!\Big{)}^{j}\Big{(}\Big{\lfloor}\frac{n}{k}\Big{\rfloor}!\Big{)}^{k-j},$

where $j$ is the remainder when $n$ is divided by $k$. For example,
$\mathrm{MP}_{5}^{3}$ has $(2!)^{2}\cdot(1!)^{1}=4$ elements such as
$12345,15342,42315$ and $45312$.

Alexandersson et. al. [2] also introduced the notion of mod-k-alternating
permutations starting with the remainder $r\pmod{k}$ as follows:

######  Definition 1.2.

Let $\mathrm{MP}_{n,r}^{k}$ denote the set of permutations of length $n$ that
satisfy $\pi_{i}-i\equiv r-1\pmod{k}$ for $1\leq i\leq n$. Clearly, when
$r=1$, we have $\mathrm{MP}_{n,1}^{k}=\mathrm{MP}_{n}^{k}$. Moreover, let
$\mathrm{mp}_{n,r}^{k}=|\mathrm{MP}_{n,r}^{k}|.$

In this article, our aim is to study the signed excedance enumeration over
$\mathrm{MP}_{n,r}^{k}$. We define the polynomials

$\mathrm{MPE}_{n,r}^{k}(t)=\sum_{\pi\in\mathrm{MP}_{n,r}^{k}}t^{\mathrm{exc}(\pi)}\hskip
8.53581pt\mbox{ and }\hskip
8.53581pt\mathrm{SgnMPE}_{n,r}^{k}(t)=\sum_{\pi\in\mathrm{MP}_{n,r}^{k}}(-1)^{\mathrm{inv}(\pi)}t^{\mathrm{exc}(\pi)},$

and prove the following result, which generalizes Theorem 1.2 proved by
Mantaci in [19].

######  Theorem 1.3.

Let $n,k$ be positive integers. If $n$ is not divisible by $k$, say $n=mk+j$
with $1\leq j\leq k-1$, we have

$\mathrm{SgnMPE}_{n}^{k}(t)=\mathrm{SgnMPE}_{n,1}^{k}(t)=(1-t)^{n-k}.$

If $n$ is divisible by $k$, say $n=mk$, we have

$\displaystyle\mathrm{SgnMPE}_{n,r}^{k}(t)$ $\displaystyle=$
$\displaystyle\begin{cases}(1-t)^{n-k}&\text{if $r=1$},\\\
(-1)^{(mr-1)(k-r+1)}t^{k-r+1}(1-t)^{n-k}&\text{if $2\leq r\leq
k$}.\end{cases}$

For a permutation $\pi=\pi_{1}\pi_{2}\dots\pi_{n}\in\mathfrak{S}_{n}$, we say
that $i\in[n]$ is a fixed point of $\pi$ if $\pi_{i}=i.$ If $\pi$ has no fixed
point, then $\pi$ is called a derangement and the set of all derangements in
$\mathfrak{S}_{n}$ is denoted by $\mathcal{D}_{n}$. Recall that for a natural
number $n$, its $q$-analogue is defined as

$[n]_{q}=\frac{1-q^{n}}{1-q}=1+q+q^{2}+\ldots+q^{n-1}.$

In [21], Mantaci and Rakotondrajao determined the signed excedance enumerator
for derangements and obtained the following result.

######  Theorem 1.4 (Mantaci-Rakotondrajao).

For $n\geq 2$, let
$\mathrm{DSgnExc}_{n}(t)=\sum_{\pi\in\mathcal{D}_{n}}(-1)^{\mathrm{inv}(\pi)}t^{\mathrm{exc}(\pi)}$
be the signed excedance enumerator over derangements. Then
$\mathrm{DSgnExc}_{n}(t)=(-1)^{n-1}t[n-1]_{t}$.

Recently, Alexandersson and Getachew [3] proved some multivariate
generalizations of a formula for enumerating signed excedances in
derangements. Kebede and Rakotondrajao [14] studied the parity alternating
derangements which are the derangements that also are parity alternating
permutations starting with odd integers. Motivated by these works, in this
article, we study the signed excedance enumeration over mod-k-alternating
derangements starting with the remainder $r\pmod{k}$.

Let $\mathrm{MPD}_{n,r}^{k}$ be the set of mod-k-alternating permutations
starting with remainder $r\pmod{k}$, which are also derangements and
$\mathrm{mpd}_{n,r}^{k}=|\mathrm{MPD}_{n,r}^{k}|$. Define

$\mathrm{SgnMPDE}_{n,r}^{k}(t)=\sum_{\pi\in\mathrm{MPD}_{n,r}^{k}}(-1)^{\mathrm{inv}(\pi)}t^{\mathrm{exc}(\pi)}.$

We prove the following result about the signed excedance enumeration over mod-
k-alternating derangements.

######  Theorem 1.5.

Let $n,k$ be positive integers. If $n=mk+j$ where $0\leq j\leq k-1$, we have

$\displaystyle\mathrm{SgnMPDE}_{n,r}^{k}(t)=$
$\displaystyle\begin{cases}(-1)^{n}(-t)^{k}\big{(}[m]_{t}\big{)}^{j}\big{(}[m-1]_{t}\big{)}^{k-j}&\text{if
$r=1$},\\\ \mathrm{SgnMPE}_{n,r}^{k}(t)&\text{if $r\geq 2$.}\end{cases}$

As an application of the signed excedance enumeration, we also prove that, for
suitable choices of $n$ and $k$, the excedance enumerating polynomials over
the even and odd mod-k-alternating permutations starting with a fixed
remainder $r\pmod{k}$, are gamma-positive. To avoid repetition, we will define
gamma-positivity in details in Section 4. Define

$\mathrm{MPE}_{n,r}^{k,+}(t)=\sum_{\pi\in\mathrm{MP}_{n,r}^{k}\cap\mathcal{A}_{n}}t^{\mathrm{exc}(\pi)},\hskip
8.53581pt\mbox{ and }\hskip
8.53581pt\mathrm{MPE}_{n,r}^{k,-}(t)=\sum_{\pi\in\mathrm{MP}_{n,r}^{k}\cap(\mathfrak{S}_{n}\setminus\mathcal{A}_{n})}t^{\mathrm{exc}(\pi)}.$

In this context, we prove the following result.

######  Theorem 1.6.

For positive integers $n\equiv k\pmod{2k}$ with $n\geq 5k$, the polynomials
$\mathrm{MPE}_{n,1}^{k,+}(t)$ and $\mathrm{MPE}_{n,1}^{k,-}(t)$ are gamma-
positive with center of symmetry $(n-k)/2.$ Moreover, for $2\leq r\leq k$, the
polynomials $\mathrm{MPE}_{n,r}^{k,+}(t)$ and $\mathrm{MPE}_{n,r}^{k,-}(t)$
are gamma-positive with center of symmetry $(n+1-r)/2.$

The article is organized as follows. In Section $2$, we give the proof of
Theorem 1.3 by constructing suitable matrices and connecting their
determinants with the signed excedance enumeration of mod-k-alternating
permutations. We also establish a relationship between the un-signed excedance
enumerating polynomials and the Eulerian numbers. We discuss the proof of
Theorem 1.5 and Theorem 1.6 in Section $3$ and $4$, respectively.

##  2 Excedances and permanents, signed excedances and determinants

The main goal of this section is to prove Theorem 1.3. This will be achieved
by constructing appropriate matrices and connecting their determinants with
the signed excedance enumerating polynomials over $\mathrm{MP}_{n,r}^{k}$. As
a by-product of our method, we will also obtain an explicit expression for the
un-signed excedance enumerating polynomials. For that, first let us recall the
definition of the permanent of a matrix.

######  Definition 2.1.

For a square matrix $M$ of size $n\times n$, the permanent of $M$, denoted by
$\mathrm{perm}(M)$, is defined as

$\mathrm{perm}(M)=\displaystyle\sum_{\pi\in\mathfrak{S}_{n}}\prod_{i=1}^{n}m_{i,\pi_{i}}.$

Now, we state an important lemma which will be used in the subsequent proofs.

######  Lemma 2.1 ([2, Lemma 9]).

We have $\mathrm{mp}_{n,r}^{k}=0$ unless $r=1$ or $n=mk$ for some positive
integer $k$.

In the following two propositions, we establish a relation among the
permanents and the determinants of suitably constructed matrices with the un-
signed and signed excedance enumerating polynomials over
$\mathrm{MP}_{n,r}^{k}$, respectively.

######  Proposition 2.2.

Let $n$ be a positive integer. Consider the $n\times n$ matrix
$M_{n,1}=(m^{1}_{i,j})$ defined by

$\displaystyle m^{1}_{i,j}$ $\displaystyle=$
$\displaystyle\begin{cases}0&\text{if $|i-j|$ is not divisible by $k$},\\\
1&\text{if $|i-j|$ is divisible by $k$ and $i\geq j$},\\\ t&\text{if $|i-j|$
is divisible by $k$ and $i<j$}.\end{cases}$

Then, we have

$\mathrm{perm}(M_{n,1})=\mathrm{MPE}_{n,1}^{k}(t),\hskip 8.53581pt\mbox{ and
}\hskip 8.53581pt\det(M_{n,1})=\mathrm{SgnMPE}_{n,1}^{k}(t).$

###### Proof.

By the definition, it follows that

$\displaystyle\mathrm{perm}(M_{n,1})$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\pi\in\mathfrak{S}_{n}}\prod_{i=1}^{n}m^{1}_{i,\pi_{i}}=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,1}^{k}}\prod_{i=1}^{n}m^{1}_{i,\pi_{i}}+\displaystyle\sum_{\pi\in(\mathfrak{S}_{n}-\mathrm{MP}_{n,1}^{k})}\prod_{i=1}^{n}m^{1}_{i,\pi_{i}},$

and

$\displaystyle\det(M_{n,1})$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\pi\in\mathfrak{S}_{n}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}m^{1}_{i,\pi_{i}}$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{\pi\in\mathrm{MP}_{n,1}^{k}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}m^{1}_{i,\pi_{i}}+\displaystyle\sum_{\pi\in(\mathfrak{S}_{n}-\mathrm{MP}_{n,1}^{k})}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}m^{1}_{i,\pi_{i}}.$

For a permutation $\pi\in(\mathfrak{S}_{n}-\mathrm{MP}_{n,1}^{k})$, it is
necessary that $|\pi_{i}-i|$ is not divisible by $k$ for some $1\leq i\leq n$
and hence $m^{1}_{i,\pi_{i}}=0$. Thus, we have

$\mathrm{perm}(M_{n,1})=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,1}^{k}}\prod_{i=1}^{n}m^{1}_{i,\pi_{i}}\hskip
8.53581pt\mbox{ and }\hskip
8.53581pt\det(M_{n,1})=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,1}^{k}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}m^{1}_{i,\pi_{i}}.$

Now, for $\pi\in\mathrm{MP}_{n,1}^{k}$, let
$T_{\pi}=\prod_{i=1}^{n}m^{1}_{i,\pi_{i}}$ be the term occuring in the
permanent (respectively, determinant) expansion corresponding to $\pi$. Since
$m^{1}_{i,j}=t$ if $i<j$ and $m^{1}_{i,j}=1$ otherwise, we have
$T_{\pi}=t^{\mathrm{exc}(\pi)}$. Hence,

$\mathrm{perm}(M_{n,1})=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,1}^{k}}\prod_{i=1}^{n}t^{\mathrm{exc}(\pi)}=\mathrm{MPE}_{n,1}^{k}(t),$

and

$\det(M_{n,1})=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,1}^{k}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}t^{\mathrm{exc}(\pi)}=\mathrm{SgnMPE}_{n,1}^{k}(t).$

∎

######  Proposition 2.3.

Let $n$ be a positive integer divisible by $k$. Consider the $n\times n$
matrix $M_{n,r}=(m^{r}_{i,j})$ defined by

$\displaystyle m_{i,j}^{r}$ $\displaystyle=$
$\displaystyle\begin{cases}0&\text{if \quad$j-i\not\equiv r-1\pmod{k}$},\\\
1&\text{if \quad$j-i\equiv r-1\pmod{k}$ and $i\geq j$},\\\ t&\text{if
\quad$j-i\equiv r-1\pmod{k}$ and $i<j$}.\end{cases}$

Then, we have

$\mathrm{perm}(M_{n,r})=\mathrm{MPE}_{n,r}^{k}(t),\hskip 8.53581pt\mbox{ and
}\hskip 8.53581pt\det(M_{n,r})=\mathrm{SgnMPE}_{n,r}^{k}(t).$

###### Proof.

The proof follows via the similar steps as the proof of Proposition 2.2. Yet,
we give it for the sake of completeness. Note that

$\displaystyle\mathrm{perm}(M_{n,r})$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\pi\in\mathfrak{S}_{n}}\prod_{i=1}^{n}m^{r}_{i,\pi_{i}}=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,r}^{k}}\prod_{i=1}^{n}m^{r}_{i,\pi_{i}}+\displaystyle\sum_{\pi\in(\mathfrak{S}_{n}-\mathrm{MP}_{n,r}^{k})}\prod_{i=1}^{n}m^{r}_{i,\pi_{i}},$

and

$\displaystyle\det(M_{n,r})$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\pi\in\mathfrak{S}_{n}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}m^{r}_{i,\pi_{i}}$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{\pi\in\mathrm{MP}_{n,r}^{k}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}m^{r}_{i,\pi_{i}}+\displaystyle\sum_{\pi\in(\mathfrak{S}_{n}-\mathrm{MP}_{n,r}^{k})}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}m^{r}_{i,\pi_{i}}.$

Observe that for a permutation
$\pi\in(\mathfrak{S}_{n}-\mathrm{MP}_{n,r}^{k})$, it is necessary that
$\pi_{i}-i\not\equiv r-1\pmod{k}$ for some $1\leq i\leq n$, and hence
$m^{r}_{i,\pi_{i}}=0$. Thus, we have

$\mathrm{perm}(M_{n,r})=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,r}^{k}}\prod_{i=1}^{n}m^{r}_{i,\pi_{i}}\hskip
8.53581pt\mbox{ and }\hskip
8.53581pt\det(M_{n,r})=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,r}^{k}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}m^{r}_{i,\pi_{i}}.$

Now, for $\pi\in\mathrm{MP}_{n,r}^{k}$, let
$T_{\pi}=\prod_{i=1}^{n}m^{r}_{i,\pi_{i}}$ be the term occuring in the
permanent (respectively, determinant) expansion corresponding to $\pi$. For
$\pi\in\mathrm{MP}_{n,r}^{k}$, we have $m^{r}_{i,j}=t$ if $i<j$ and
$m^{r}_{i,j}=1$ if $i>j$. Hence, $T_{\pi}=t^{\mathrm{exc}(\pi)}$. Thus,

$\mathrm{perm}(M_{n,r})=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,r}^{k}}\prod_{i=1}^{n}t^{\mathrm{exc}(\pi)}=\mathrm{MPE}_{n,r}^{k}(t),$

and

$\det(M_{n,r})=\displaystyle\sum_{\pi\in\mathrm{MP}_{n,r}^{k}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}t^{\mathrm{exc}(\pi)}=\mathrm{SgnMPE}_{n,r}^{k}(t).$

This completes the proof. ∎

Recall that the Kronecker product of two matrices $A=(a_{ij})_{m\times n}$ and
$B=(b_{ij})_{p\times q}$, denoted by $A\otimes B$, is defined to be the
$mp\times nq$ block matrix $[a_{ij}B]$. If $A$ and $B$ are $n\times n$ and
$p\times p$ matrices, respectively, then $\det(A\otimes B)=(\det A)^{p}(\det
B)^{n}$. We are now in a position to prove the first of our main results.

###### Proof of Theorem 1.3.

We divide the proof into the following two cases:
Case 1. Let $n$ be not divisible by $k$. By Lemma 2.1, we have
$\mathrm{mp}_{n,r}^{k}=0$ for $r\geq 2$. Hence,
$\mathrm{SgnMPE}_{n}^{k}(t)=\mathrm{SgnMPE}_{n,1}^{k}(t)$. Now, consider the
$n\times n$ matrix $M_{n,1}=(m^{1}_{i,j})$ defined in Proposition 2.2, for
which we have $\mathrm{SgnMPE}_{n,1}^{k}(t)=\det(M_{n,1})$. So, it is enough
to compute the determinant of $M_{n,1}.$ By performing suitable row and column
operations, we transform $M_{n,1}$ into a block diagonal matrix and compute
its determinant as follows:

Let $n=mk+j$, where $1\leq j\leq k-1.$ We can write any positive integer
$q\leq n$ as $q=ak+b$, where $0\leq a\leq m$ and $0\leq b\leq k-1$. If
$0<b\leq j$, we apply the operations $C_{ak+b}\rightarrow C_{a+1+(b-1)(m+1)}$
and $R_{ak+b}\rightarrow R_{a+1+(b-1)(m+1)}$; if $j<b\leq k-1$, we apply the
operations $C_{ak+b}\rightarrow C_{j(m+1)+a+1+(b-j-1)m}$ and
$R_{ak+b}\rightarrow R_{j(m+1)+a+1+(b-j-1)m}$; and if $b=0$, we apply the
operations $C_{ak}\rightarrow C_{n-(m-a)}$ and $R_{ak}\rightarrow
R_{n-(m-a)}$. This gives

$\displaystyle\det(M_{n,1})=\det\left(\begin{array}[]{cc}A_{m+1}\otimes
I_{j}&\bf{0}\\\ \bf{0}&{A_{m}}\otimes
I_{k-j}\end{array}\right),~{}~{}\text{where}~{}~{}A_{k}=\left(\begin{array}[]{cccc}1&t&\cdots&t\\\
1&1&\ddots&\vdots\\\ \vdots&\ddots&\ddots&t\\\ 1&\cdots&1&1\\\
\end{array}\right)_{k\times k}.$

Again, by applying the operations $C_{i}^{\prime}=C_{i}-C_{i+1}$ for $1\leq
i\leq k-1$ on $A_{k}$, we have

$\displaystyle\det(A_{k})=\det\left(\begin{array}[]{ccccc}1-t&0&\cdots&0&t\\\
0&1-t&\ddots&\vdots&t\\\ \vdots&\ddots&\ddots&0&\vdots\\\ 0&\cdots&0&1-t&t\\\
0&\cdots&0&0&1\\\ \end{array}\right)=(1-t)^{k-1}.$ (6)

Hence,

$\displaystyle\det(M_{n,1})$
$\displaystyle=(\det(A_{m+1}))^{j}(\det(A_{m}))^{k-j}=(1-t)^{mj}(1-t)^{(m-1)(k-j)}=(1-t)^{n-k}.$

Case 2. Let $n$ be divisible by $k$, say $n=mk$. For $r=1$, the proof will be
similar to Case 1 and we have $\det(M_{n,1})=(1-t)^{n-k}$. For $2\leq r\leq
k$, consider the $n\times n$ matrix $M_{n,r}=(m^{r}_{i,j})$ defined in
Proposition 2.3. Again, by Proposition 2.3, we have
$\mathrm{SgnMPE}_{n,r}^{k}(t)=\det(M_{n,r})$. So, it is sufficient to compute
the determinant of $M_{n,r}$.

Let $n=mk$. We can write any positive integer $q\leq n$ as $q=ak+b$, where
$0\leq a\leq m$ and $0\leq b\leq k-1$. If $b\neq 0$, we apply the operations
$C_{ak+b}\rightarrow C_{a+1+m(b-1)}$; $R_{ak+b}\rightarrow R_{a+1+m(b-1)}$ and
if $b=0$, we apply the operations $C_{ak}\rightarrow C_{a+m(k-1)}$;
$R_{ak}\rightarrow R_{a+m(k-1)}$. This gives

$\displaystyle\det(M_{n,r})=\det\left(\begin{array}[]{cc}\bf{0}&B_{m}\otimes
I_{k-r+1}\\\ A_{m}\otimes I_{r-1}&\bf{0}\end{array}\right),$

where $A_{m}$ and $B_{m}$ are $m\times m$ matrices with

$\displaystyle A_{m}=\left(\begin{array}[]{cccc}1&t&\cdots&t\\\
1&1&\ddots&\vdots\\\ \vdots&\ddots&\ddots&t\\\ 1&\cdots&1&1\\\
\end{array}\right)\text{and}~{}~{}B_{m}=\left(\begin{array}[]{cccc}t&t&\cdots&t\\\
1&t&\ddots&\vdots\\\ \vdots&\ddots&\ddots&t\\\ 1&\cdots&1&t\\\
\end{array}\right).$

From (6), we have $\det(A_{m})=(1-t)^{m-1}$. Again, by applying the operations
$R_{i}^{\prime}=R_{i}-R_{i+1}$ for $1\leq i\leq m-1$ on $B_{m}$, we have

$\det(B_{m})=\det\left(\begin{array}[]{ccccc}t-1&0&\cdots&0&0\\\
0&t-1&\ddots&\vdots&0\\\ \vdots&\ddots&\ddots&0&\vdots\\\ 0&\cdots&0&t-1&0\\\
1&\cdots&1&1&t\\\ \end{array}\right)=(-1)^{m-1}t(1-t)^{m-1}.$

Thus,

$\displaystyle\det(M_{n,r})$
$\displaystyle=(-1)^{m^{2}(r-1)(k-r+1)}(\det(A_{m}))^{r-1}(\det(B_{m}))^{k-r+1}$
$\displaystyle=(-1)^{m(r-1)(k-r+1)+(m-1)(k-r+1)}t^{k-r+1}(1-t)^{(r-1)(m-1)+(m-1)(k-r+1)}$
$\displaystyle=(-1)^{(mr-1)(k-r+1)}t^{k-r+1}(1-t)^{n-k}.$

This completes the proof. ∎

######  Remark 2.1.

Note that by putting $k=1$ in Theorem 1.3, we get Theorem 1.2 as an immediate
corollary.

We get the following result as a by-product of the method of the proof of
Theorem 1.3.

######  Theorem 2.1.

Let $n,k$ be positive integers. If $n$ is not divisible by $k$, say $n=mk+j$
with $1\leq j\leq k-1$, then

$\mathrm{MPE}_{n}^{k}(t)=\mathrm{MPE}_{n,1}^{k}(t)=A_{m+1}(t)^{j}A_{m}(t)^{k-j}.$

If $n$ is divisible by $k$, say $n=mk$, then

$\displaystyle\mathrm{MPE}_{n,r}^{k}(t)$ $\displaystyle=$
$\displaystyle\begin{cases}A_{m}(t)^{k}&\text{if $r=1$},\\\
t^{k+1-r}A_{m}(t)^{k}&\text{if $2\leq r\leq k$}.\end{cases}$

###### Proof.

First, let us consider the case when $n$ is not divisible by $k$. By Lemma
2.1, it follows that $\mathrm{MPE}_{n}^{k}(t)=\mathrm{MPE}_{n,1}^{k}(t)$. Now,
consider the $n\times n$ matrix $M_{n,1}=(m^{1}_{i,j})$ defined in Proposition
2.2, for which we have $\mathrm{MPE}_{n,1}^{k}(t)=\mathrm{perm}(M_{n,1})$. So,
it is enough to compute the permanent of $M_{n,1}.$ Following the proof of
Theorem 1.3, we have

$\mathrm{perm}(M_{n,1})=(\mathrm{perm}(A_{m+1}))^{j}(\mathrm{perm}(A_{m}))^{k-j},$

where $A_{m}$ is the same matrix as defined in the proof of Theorem 1.3. Since
$\mathrm{perm}(A_{m})$ is same as the Eulerian polynomial $A_{m}(t)$, we have

$\mathrm{MPE}_{n}^{k}(t)=\mathrm{MPE}_{n,1}^{k}(t)=(\mathrm{perm}(A_{m+1}))^{j}(\mathrm{perm}(A_{m}))^{k-j}=A_{m+1}(t)^{j}A_{m}(t)^{k-j}.$

Next, we consider the case when $n$ is divisible by $k$, say $n=mk$. For
$r=1$, it is easy to see that

$\mathrm{MPE}_{n,1}^{k}(t)=A_{m}(t)^{k}.$

For $2\leq r\leq k$, from the proof of Theorem 1.3 Case 2, we have

$\displaystyle\mathrm{MPE}_{n,r}^{k}(t)=\mathrm{perm}(M_{n,r})$
$\displaystyle=(\mathrm{perm}(A_{m}))^{r-1}(\mathrm{perm}(B_{m}))^{k-r+1}$
$\displaystyle=A_{m}(t)^{r-1}(tA_{m}(t))^{k-r+1}$
$\displaystyle=t^{k-r+1}A_{m}(t)^{k}.$

This completes the proof. ∎

##  3 Mod-k-alternating derangements

In this section, we study the sign-balance for excedances over mod-k-
alternating derangements. The following lemma will be useful to get an
explicit expression for the polynomial $\mathrm{SgnMPDE}_{n,r}^{k}(t)$.

######  Lemma 3.1 ([7, Theorem 2.1]).

Let

$\displaystyle
f_{n}=\det\left(\begin{array}[]{cccccc}c_{1}&a_{1}&0&\cdots&\cdots&0\\\
b_{2}&c_{2}&a_{2}&\ddots&&\vdots\\\ 0&b_{3}&c_{3}&\ddots&0&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0\\\ \vdots&&0&\ddots&\ddots&a_{n-1}\\\
0&\cdots&\cdots&0&b_{n}&c_{n}\end{array}\right).$

Then $f_{n}$ satisfies the following recurrence relation:

$f_{n}=c_{n}f_{n-1}-b_{n}a_{n-1}f_{n-2},$

where the initial values for $f_{n}$ are $f_{0}=1$ and $f_{-1}=0$.

###### Proof of Theorem 1.5.

Note that any mod-k-alternating permutation starting with the remainder
$r\pmod{k}$ is a derangement when $r\geq 2$. So, we only need to consider the
case when $r=1$. Let $D_{n}=(d_{i,j})$ be an $n\times n$ matrix with

$\displaystyle d_{i,j}$ $\displaystyle=$ $\displaystyle\begin{cases}0&\text{if
$|i-j|$ is not divisible by $k$ or $i=j$},\\\ 1&\text{if $|i-j|$ is divisible
by $k$ and $i>j$},\\\ t&\text{if $|i-j|$ is divisible by $k$ and
$i<j$}.\end{cases}$

Note that

$\displaystyle\det(D_{n})$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\pi\in\mathfrak{S}_{n}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}d_{i,\pi_{i}}$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{\pi\in\mathrm{MPD}_{n,1}^{k}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}d_{i,\pi_{i}}+\displaystyle\sum_{\pi\in(\mathfrak{S}_{n}-\mathrm{MPD}_{n,1}^{k})}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}d_{i,\pi_{i}}.$

For a permutation $\pi\not\in\mathrm{MPD}_{n,1}^{k}$, we must have
$|\pi_{i}-i|$ is not divisible by $k$ for some $1\leq i\leq n$, or $\pi_{i}=i$
for some $1\leq i\leq n$. Hence, if $\pi\not\in\mathrm{MPD}_{n,1}^{k}$, we
must have $d_{i,\pi_{i}}=0$. Thus, we have

$\det(D_{n})=\sum_{\pi\in\mathrm{MPD}_{n,1}^{k}}(-1)^{\mathrm{inv}(\pi)}\prod_{i=1}^{n}d_{i,\pi_{i}}.$

Now, let $T_{\pi}=\prod_{i=1}^{n}d_{i,\pi_{i}}$ be the term occurring in the
determinant expansion corresponding to $\pi$. From the definition of
$d_{i,\pi_{i}}$, we have $T_{\pi}=t^{\mathrm{exc}(\pi)}$. Hence,
$\mathrm{SgnMPDE}_{n,1}^{k}(t)=\det(D_{n}).$ Now, to find the determinant of
$D_{n}$, we transform $D_{n}$ into a block diagonal matrix by applying
suitable row and column operations.

Let $n=mk+j$, where $1\leq j\leq k-1.$ We can write any positive integer
$q\leq n$ as $q=ak+b$ where $0\leq a\leq m$ and $0\leq b\leq k-1$. If $0<b\leq
j$, we apply the operations $C_{ak+b}\rightarrow C_{a+1+(b-1)(m+1)}$ and
$R_{ak+b}\rightarrow R_{a+1+(b-1)(m+1)}$; if $j<b\leq k-1$, we apply the
operations $C_{ak+b}\rightarrow C_{j(m+1)+(b-j-1)m+a+1}$ and
$R_{ak+b}\rightarrow R_{j(m+1)+(b-j-1)m+a+1}$; and if $b=0$, we apply the
operations $C_{ak}\rightarrow C_{n-(m-a)}$ and $R_{ak}\rightarrow
R_{n-(m-a)}$. This gives

$\displaystyle\det(D_{n})=\det\left(\begin{array}[]{cc}A_{m+1}\otimes
I_{j}&\bf{0}\\\ \bf{0}&{A_{m}}\otimes
I_{k-j}\end{array}\right),~{}~{}\text{where}~{}~{}A_{k}=\left(\begin{array}[]{cccc}0&t&\cdots&t\\\
1&0&\ddots&\vdots\\\ \vdots&\ddots&\ddots&t\\\ 1&\cdots&1&0\\\
\end{array}\right)_{k\times k}.$

Again, by applying the operations $C_{i}^{\prime}=C_{i}-C_{i+1}$ and
$R_{i}^{\prime}=R_{i}-R_{i+1}$ for $1\leq i\leq k-1$ on $A_{k}$, we have

$\displaystyle\det(A_{k})=\det\left(\begin{array}[]{cccccc}-1-t&t&0&\cdots&0&0\\\
1&-1-t&t&\ddots&\vdots&\vdots\\\ 0&1&-1-t&\ddots&0&0\\\
\vdots&\ddots&\ddots&\ddots&t&0\\\ 0&\cdots&0&1&-1-t&t\\\ 0&\cdots&0&0&1&0\\\
\end{array}\right).$

Now, expanding the determinant with respect to the last row, we have

$\displaystyle\det(A_{k})=(-1)^{2k-1}\det\left(\begin{array}[]{cccccc}-1-t&t&0&\cdots&0&0\\\
1&-1-t&t&\ddots&\vdots&\vdots\\\ 0&1&-1-t&\ddots&0&0\\\
\vdots&\ddots&\ddots&\ddots&t&0\\\ 0&\cdots&0&1&-1-t&0\\\ 0&\cdots&0&0&1&t\\\
\end{array}\right)_{(k-1)\times(k-1)}.$

Again, expanding with respect to the last column, we have

$\displaystyle\det(A_{k})=(-1)^{2k-1}t\det\left(\begin{array}[]{ccccc}-1-t&t&0&\cdots&0\\\
1&-1-t&t&\ddots&\vdots\\\ 0&1&-1-t&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&t\\\ 0&\cdots&0&1&-1-t\\\
\end{array}\right)_{(k-2)\times(k-2)}$

Now, by Lemma 3.1, we have
$\det(A_{k})=(-1)^{2k-1}t(-1)^{k-4}(1+t+t^{2}+\ldots+t^{k-2})=(-1)^{k-1}t[k-1]_{t}$.
Thus,

$\displaystyle\det(D_{n})$
$\displaystyle=(\det(A_{m+1}))^{j}(\det(A_{m}))^{k-j}$
$\displaystyle=(-1)^{mj}t^{j}\big{(}[m]_{t}\big{)}^{j}(-1)^{(m-1)(k-j)}t^{k-j}\big{(}[m-1]_{t}\big{)}^{k-j}$
$\displaystyle=(-1)^{n}(-t)^{k}\big{(}[m]_{t}\big{)}^{j}\big{(}[m-1]_{t}\big{)}^{k-j}.$

This completes the proof for $r=1$ and hence the proof of the theorem. ∎

##  4 Gamma-positivity of even and odd mod-k-alternating permutations

Let $f(t)=\sum_{i=0}^{n}a_{i}t^{i}\in\mathbb{Q}[t]$ be a univariate polynomial
of degree $n$, where $a_{i}\in\mathbb{Q}$ with $a_{n}\not=0$. Let $r$ be the
least non-negative integer such that $a_{r}\not=0$. The polynomial $f(t)$ is
said to be palindromic if $a_{r+i}=a_{n-i}$ for $0\leq
i\leq\left\lfloor(n-r)/2\right\rfloor$. Define the center of symmetry of
$f(t)$ to be $(n+r)/2$. Note that for a palindromic polynomial $f(t)$, its
center of symmetry could be half integral.

Let $\mathrm{SymPoly}_{(n+r)/2,r}(t)$ denote the set of palindromic univariate
polynomials $f(t)$ with minimum nonzero exponent of $t$ being at least $r$ and
having center of symmetry $(n+r)/2$. Let
$\Gamma=\\{t^{r+i}(1+t)^{n-r-2i}:0\leq i\leq\lfloor(n-r)/2\rfloor\\}$. It is
easy to see that if $f(t)\in\mathrm{SymPoly}_{(n+r)/2,r}(t)$, then we can
write
$f(t)=\sum_{i=0}^{\left\lfloor(n-r)/2\right\rfloor}\gamma_{n,i}t^{r+i}(1+t)^{n-r-2i}$.
The polynomial $f(t)$ is said to be gamma-positive if $\gamma_{n,i}\geq 0$ for
all $i$.

The study of gamma-positivity is an interesting and important topic in
enumerative combinatorics. It appears widely in finite geometries,
combinatorics and number theory. Foata and Schützenberger [10] showed gamma-
positivity of the Eulerian polynomials $A_{n}(t)$ that enumerate descents in
the symmetric group $\mathfrak{S}_{n}.$ Subsequently, Foata and Strehl [11]
used a group action based proof which has been rediscovered and termed as
“valley hopping” by Shapiro, Woan, and Getu. More recent interest in gamma-
positivity was sparked by Gal [12] in 2005 when he showed that some questions
in topology could be resolved by demonstrating the gamma-positivity of their
combinatorial invariants. For more details on the gamma-positivity, one can
look at [4, 13, 16, 17] and the references therein.

In this section, we are interested in the gamma-positivity of the excedance
enumerating polynomials over mod-k-alternating permutations. First, we give
some preliminary results which will be used to prove the main result.

######  Lemma 4.1 ([6, Lemma 1]).

Let $f_{1}$ and $f_{2}$ be two gamma-positive polynomials with centers of
symmetry $a_{1}$ and $a_{2}$, respectively. Then, their product $f_{1}f_{2}$
is a gamma-positive polynomial with center of symmetry $a_{1}+a_{2}$.

######  Lemma 4.2 ([6, Theorem 4]).

For odd positive integers $n\geq 5$, the polynomials
$\sum_{\pi\in\mathcal{A}_{n}}t^{\mathrm{exc}(\pi)}$ and
$\sum_{\pi\in(\mathfrak{S}_{n}-\mathcal{A}_{n})}t^{\mathrm{exc}(\pi)}$ are
gamma-positive and both of them have center of symmetry $(n-1)/2.$

######  Lemma 4.3.

Let $f,g$ be palindromic polynomials with same center of symmetry such that
$f+g$ and $f-g$ are gamma-positive with same center of symmetry $a$. Then, for
all positive integers $r$, the polynomials $f^{r}+g^{r}$ and $f^{r}-g^{r}$ are
gamma-positive with center of symmetry $ra$.

###### Proof.

We will prove this by induction on $r$. When $r=1$, the statement is true as
$f+g$ and $f-g$ are gamma-positive with the same center of symmetry. Let us
assume that for all positive integers $r\leq k$, the polynomials $f^{r}+g^{r}$
and $f^{r}-g^{r}$ are gamma-positive with center of symmetry $ra$. Now,
consider $r=k+1$. It is easy to check that

$f^{k+1}+g^{k+1}=\frac{1}{2}\big{[}(f+g)(f^{k}+g^{k})+(f-g)(f^{k}-g^{k})\big{]},$

and

$f^{k+1}-g^{k+1}=\frac{1}{2}\big{[}(f+g)(f^{k}-g^{k})+(f-g)(f^{k}+g^{k})\big{]}.$

By Lemma 4.1, it follows that $(f+g)(f^{k}+g^{k})$ is gamma-positive with
center of symmetry $ka+a=(k+1)a$ and the same holds for $(f-g)(f^{k}-g^{k}).$
Hence, $f^{r}+g^{r}$ and $f^{r}-g^{r}$ are gamma-positive with center of
symmetry $ra$ for $r=k+1$. This completes the proof. ∎

######  Remark 4.1.

Using Theorem 2.1 and Lemma 4.1, it is immediate that the polynomials
$\mathrm{MPE}_{n,r}^{k}(t)$ are gamma-positive.

######  Remark 4.2.

From Theorem 1.3 and Theorem 2.1, we can see that the polynomials
$\mathrm{MPE}_{n,r}^{k,+}(t)$ and $\mathrm{MPE}_{n,r}^{k,-}(t)$ are not
palindromic unless $n$ is divisible by $k$. Therefore, there is no question of
gamma-positivity unless $n$ is divisible by $k$.

Now, by using the above results, we give the proof of Theorem 1.6. We use the
notation $\mathrm{MPE}_{n,r}^{k,\pm}(t)$ to refer to both the polynomials
$\mathrm{MPE}_{n,r}^{k,+}(t)$ and $\mathrm{MPE}_{n,r}^{k,-}(t)$.

###### Proof of Theorem 1.6.

Let $n\equiv k\pmod{2k}$ with $n\geq 5k$. Suppose $n=2km+k$, where $m$ is a
positive integer with $m\geq 2$. Then, we have

$\mathrm{MPE}_{2mk+k,r}^{k,\pm}(t)=\frac{1}{2}\left[\sum_{\pi\in\mathrm{MPE}_{2mk+k,r}^{k}}t^{\mathrm{exc}(\pi)}\pm\sum_{\pi\in\mathrm{MPE}_{2mk+k,r}^{k}}\mathrm{sgn}(\pi)t^{\mathrm{exc}(\pi)}\right].$

By Theorem 2.1 and Theorem 1.3, it follows that

$\mathrm{MPE}_{2mk+k,1}^{k,\pm}(t)=\frac{1}{2}\left[A_{2m+1}(t)^{k}\pm(1-t)^{2mk}\right]~{}\text{for}~{}r=1,$

and for $r\geq 2$

$\mathrm{MPE}_{2mk+k,r}^{k,\pm}(t)=\frac{1}{2}t^{k+1-r}\left[A_{2m+1}(t)^{k}\pm(-1)^{(m^{2}r-1)(k+1-r)}(1-t)^{2mk}\right].$

Now, by Lemma 4.2, the polynomials $A_{2m+1}(t)+(1-t)^{2m}$ and
$A_{2m+1}(t)-(1-t)^{2m}$ are gamma-positive with centers of symmetry $m$.
Therefore, by Lemma 4.3 it follows that the polynomials
$\mathrm{MPE}_{2mk+k,1}^{\pm,k}(t)$ are gamma-positive with the center of
symmetry $km$. Similarly, for $2\leq r\leq k$, the polynomials
$\mathrm{MPE}_{n,r}^{k,+}(t)$ and $\mathrm{MPE}_{n,r}^{k,-}(t)$ are gamma-
positive with center of symmetry $(k+1-r+2km)/2.$ ∎

## Declaration of competing interest

The authors declare that they have no conflict of interest to this work.

## Data availability

No data was used for the research described in the article.

## Acknowledgements

The authors would like to thank Prof. Sivaramakrishnan Sivasubramanian for
valuable comments and suggestions during the preparation of the manuscript.
The first author acknowledges a NBHM Post-Doctoral Fellowship (File No.
0204/10(10)/2023/R&D-II/2781) during the preparation of this work and
profusely thanks National Board of Higher Mathematics, India for this funding.
The first author also acknowledges excellent working conditions in the
Department of Mathematics, Indian Institute of Science. The second author
thanks Indian Institute of Technology Bombay, India for financial support
through the Institute Post-Doctoral Fellowship.

## References

  * [1] R. M. Adin, Y. Roichman, Equidistribution and sign-balance on 321-avoiding permutations, Sém. Lothar. Combin. 51 (2004) Art. B51d.
  * [2] P. Alexandersson, F. G. Kebede, S. A. Fufa, D. Qiu, Pattern-Avoidance and Fuss-Catalan Numbers, J. Integer Seq. 26 (2023) Art. 23.4.2.
  * [3] P. Alexandersson, F. Getachew, An Involution on Derangements Preserving Excedances and Right-to-Left Minima, Sém. Lothar. Combin. 86B (2022) Art. 14.
  * [4] C. A. Athanasiadis, Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin. 77 (2018), Art. B77i.
  * [5] J. Désarménien, D. Foata, The signed Eulerian numbers, Discrete Math. 99 (1992) 49-58.
  * [6] H. K. Dey, S. Sivasubramanian, Gamma positivity of the Excedance-Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups, Ann. Comb. 24 (2020) 711-738.
  * [7] M. E. A. El-Mikkawy, On the inverse of a general tridiagonal matrix, Appl. Math. Comput. 150 (2004) 669-679.
  * [8] D. Foata, On the Netto inversion number of a sequence, Proc. Amer. Math. Soc. 19(1) (1968) 236–240.
  * [9] D. Foata, G.-N. Han, The q-series in combinatorics; Permutation statistics (Preliminary version), available at http://www-irma.u-strasbg.fr/ foata/ qseries.html (2004).
  * [10] D. Foata, M.-P. Schützenberger, Théorie géométrique des polynômes Eulériens, Lecture Notes in Mathematics, 138, Berlin, Springer-Verlag, available at http://www.mat.univie.ac.at/$\sim$slc/books/, 1970.
  * [11] D. Foata, V. Strehl, Euler numbers and variations of permutations, Colloquio Internazionale sulle Teorie Combinatoire (Roma 1973) Tomo I, Atti dei Convegni Lincei, No 17, Accad. Naz. Lincei, Rome, 119-131, 1976.
  * [12] S. R. Gal, Real Root Conjecture Fails for Five- and Higher-Dimensional Spheres, Discrete Comput. Geom. 34 (2005) 269-284.
  * [13] B. Han, gamma positivity in enumerative combinatorics, PhD Thesis, Université Claude Bernard-Lyon 1 (2019).
  * [14] F. G. Kebede, F. Rakotondrajao, Parity alternating permutations starting with an odd integer, Enumerative Combin. Appl. 1 (2021) Art. S2R16.
  * [15] J. L. Loday, Opérations sur ĺhomologie cyclique des algébres commutatives, Invent. Math. 96 (1989) 205-230.
  * [16] S.-M. Ma, J. Ma, Y.-N. Yeh, $\gamma$-positivity and partial $\gamma$-positivity of descent-type polynomials, J. Combin. Theory Ser. A 167 (2019) 257-293.
  * [17] S.-M. Ma, J. Ma, J. Yeh, Y.-N. Yeh, Excedance-type polynomials, gamma positivity and alternatingly increasing property, European J. Combin. 118 (2024) 103869.
  * [18] P. MacMahon, The indices of permutations the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects, Amer. J. Math. 35 (1913) 314-321.
  * [19] R. Mantaci. Statistiques Eulériennes sur les Groupes de Permutation, PhD thesis, Université Paris (1991).
  * [20] R. Mantaci, Binomial coefficients and anti-excedances of even permutations: A combinatorial proof, J. Combin. Theory Ser. A 63 (1993) 330-337.
  * [21] R. Mantaci, F. Rakotondrajao, Exceedingly deranging!, Adv. Appl. Math. 30 (2003) 177–188.
  * [22] T. K. Petersen, Eulerian Numbers, 1st ed. Birkhäauser (2015).
  * [23] A. Reifegerste, Refined sign-balance on 321-avoiding permutations, European J. Combin. 26 (2005) 1009–1018.
  * [24] R. Simion, F. W. Schmidt, Restricted permutations, European J. Combin. 6 (1985) 383–406.
  * [25] S. Sivasubramanian, signed excedance enumeration via determinants, Adv. Appl. Math. 47 (2011) 783–794.
  * [26] S. Tanimoto, Parity Alternating Permutations and signed Eulerian numbers, Ann. Comb. 14 (2010) 355-366.
  * [27] S. Tanimoto, Combinatorics of the group of parity alternating permutations, Adv. Appl. Math. 44 (2010) 225-230.
  * [28] M. L. Wachs, An involution for signed Eulerian numbers, Discrete Math. 99 (1992) 59-62.