Measurements of charm production in pp collisions with ALICE

Tiantian Cheng1,2 * for the ALICE Collaboration

1 Central China Normal University, Wuhan, China

2 GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany

*<EMAIL_ADDRESS>
50th International Symposium on Multiparticle Dynamics
(ISMD2021)
12-16 July 2021 10.21468/SciPostPhysProc.?

## Abstract

The production of charm-hadrons at midrapidity in pp collisions at $\sqrt{s}$
= 5.02 and 13 TeV was recently measured by the ALICE Collaboration. The
results show that the baryon-to-meson yield ratios differ from predictions
where the hadronisation has been tuned to measurements obtained in $\rm
e^{+}e^{-}$ and ep collisions for different charm-baryon species. These
observations suggest that the charm fragmentation fractions are not universal
and that the baryon-to-meson ratio depends on the collision systems. The
results are compared to the predictions from Monte Carlo event generators and
theoretical calculations.

##  1 Introduction

Measurements of the production of heavy-flavour hadrons in high-energy
hadronic collisions provide important tests of Quantum Chromodynamics (QCD).
The production cross sections of heavy-flavour hadrons can be calculated with
the convolution of three terms: the parton distribution functions (PDFs) of
the incoming protons, the partonic cross section, and the fragmentation
functions describing the non-perturbative transition from charm quarks into
charm hadrons. It is generally assumed that the fragmentation fractions are
universal regardless of the collision system and energy. However, recent
measurements of charm-baryon production at midrapidity in pp collisions showed
an enhancement of the yield of $\rm\Lambda_{\rm c}^{0}$, $\rm\Xi_{\rm c}^{0}$,
$\rm\Sigma_{\rm c}^{0,++}$, $\rm BR\times\Omega_{\rm c}^{0}$ baryon to $\rm
D^{0}$ yield ratios with respect to predictions based on $\rm e^{+}e^{-}$ and
ep experiments, suggesting that the charm fragmentation fractions are not
universal among different collision systems.

##  2 Charm baryon-to-meson yield ratios in pp collisions

Recently, the ALICE collaboration reported measurements of the ground-state
charm hadrons, i.e. charm-meson ($\rm D^{0},D^{+},D_{s}^{+},D^{*+}$ [1]) and
charm-baryon ($\Lambda_{\rm c}^{+}$[2], $\Sigma_{\rm c}^{0,++}$ [3],
$\Xi^{0,+}_{\rm c}$[4, 5], and $\Omega_{\rm c}^{0}$). Both the prompt and non-
prompt meson-to-meson yield ratios are well described with the pQCD
calculations by using fragmentation functions measured in $\rm e^{+}e^{-}$
collisions. However, the calculations significantly underestimate all the
measured baryon-to-meson yield ratios. The top left panel of Fig. 1 shows the
$\Lambda_{\rm c}^{+}/{\rm D}^{0}$ ratio in pp collisions, where a clear
$p_{\rm T}$ dependence is observed. It is compared with model calculations.
The measurements of heavier charm baryons of $\Sigma_{\rm c}^{0,++}$,
$\Xi^{0,+}_{\rm c}$ and $\Omega_{\rm c}^{0}$ give further important
constraints on charm-quark hadronization models. The absolute decay branching
ratio (BR) of $\Omega_{\rm c}^{0}\rightarrow\Omega^{-}\pi^{+}$ is not
measured, hence only the BR multiplied to the cross section of $\Omega_{\rm
c}^{0}$ over cross section of $\rm D^{0}$ is reported. All the models
underestimate the $\rm\Xi^{0,+}_{\rm c}/D^{0}$ and $\rm BR\times\Omega_{\rm
c}^{0}/D^{0}$ ratios, except the Catania model [6], which implements a new
possible scenario for pp collisions allowing low-$p_{\rm T}$ charm quarks to
hadronize via coalescence also in addition to the fragmentation mechanism.

Figure 1: The results of charm baryon-to-meson ratios are compared with
different theoretical predictions. Top-left: The $\Lambda_{\rm c}^{+}/{\rm
D}^{0}$ ratio measured in pp collisions at $\sqrt{s}$ = 5.02 TeV [2]. Top-
right: $\Sigma_{\rm c}^{0,++,+}/{\rm D}^{0}$ in pp collisions at $\sqrt{s}$ =
13 TeV [3]. Bottom-left: $\Xi^{0}_{\rm c}/{\rm D}^{0}$ and $\Xi^{+}_{\rm
c}/{\rm D}^{0}$ ratios at a function of $p_{\rm T}$ in pp collisions at
$\sqrt{s}$ = 13 TeV [4]. Bottom-right: ${\rm BR}\times\Omega_{\rm c}^{0}/{\rm
D}^{0}$ ratios at a function of $p_{\rm T}$ in pp collisions at $\sqrt{s}$ =
13 TeV.

##  3 Charm production and fragmentation in pp collisions

The charm fragmentation fractions, $f(\rm c\rightarrow H_{\rm c})$ shown in
the left panel of Fig. 2, represent the probabilities of a charm quark to
hadronise into a given charm hadron. They were computed for the first time in
pp collisions at the LHC using measurements of charm baryons at midrapidity
and are observed to be different from the ones performed in $\rm e^{+}e^{-}$
collisions at LEP and B factories as well as in ep collisions, providing
evidence that the assumption of universality (collision-system independence)
of the parton-to-hadron fragmentation is not valid [7]. In the right panel of
Fig. 2, the $\rm c\overline{c}$ production cross section per unit of rapidity
at midrapidity in pp collisions is calculated by summing the $p_{\rm
T}$-integrated cross sections of all measured ground-state charm hadrons, and
is compared with FONLL [8] and NNLO [9, 10] predictions as a function of the
collision energy. The $\rm c\overline{c}$ cross sections measured at
midrapidity at the LHC lie at the upper edge of the theoretical pQCD
calculations.

Figure 2: Left: Charm-quark fragmentation fractions into charmed hadrons
measured in pp collisions at $\sqrt{s}$ = 5.02 TeV in comparison with
experimental measurements performed in $\rm e^{+}e^{-}$ collisions at LEP and
B factories, and in ep collisions at HERA. Right: Charm production cross
section at midrapidity per unit of rapidity as a function of the collisions
energy [7].

##  4 Conclusion

Recent measurements of charm-baryon production give stringent constraints to
theoretical calculations. The baryon-to-meson ratios are significantly higher
at low to intermediate $p_{\rm T}$ with respect to measurements done at the
electron-positron and electron-proton colliders. This observation may indicate
that the charm fragmentation functions are not universal for the different
collision systems.

## References

  * [1] S. Acharya _et al._ ,  _Measurement of beauty and charm production in pp collisions at $\sqrt{s}$ = 5.02 TeV via non-prompt and prompt D mesons_,  JHEP 2021(5) (2021),  10.1007/jhep05(2021)220.
  * [2] S. Acharya _et al._ ,  _$\mathrm{\Lambda_{c}^{+}}$ production in pp and in p-Pb collisions at $\sqrt{s_{\rm{NN}}}=5.02$ TeV_ (2020), http://arxiv.org/abs/2011.06079.
  * [3] S. Acharya _et al._ ,  _Measurement of prompt D 0, $\Lambda_{c}^{+}$, and $\Sigma_{c}^{0,++}$(2455) production in pp collisions at $\sqrt{s}=13$ TeV,_ (2021), http://arxiv.org/abs/2106.08278.
  * [4] S. Acharya _et al._ ,  _Measurement of the cross sections of $\Xi^{0}_{\rm c}$ and $\Xi^{+}_{\rm c}$ baryons and branching-fraction ratio BR($\Xi^{0}_{\rm c}\rightarrow\Xi^{-}{\rm e}^{+}\nu_{\rm e}$)/BR($\Xi^{0}_{\rm c}\rightarrow\Xi^{-}\pi^{+}$) in pp collisions at 13 TeV,_ (2021), http://arxiv.org/abs/2105.05187.
  * [5] S. Acharya _et al._ ,  _Measurement of the production cross section of prompt $\Xi^{0}_{\rm c}$ baryons at midrapidity in pp collisions at $\sqrt{s}$ = 5.02 TeV_,  JHEP 2021(10) (2021),  10.1007/JHEP10(2021)159.
  * [6] V. Minissale, S. Plumari and V. Greco,  _Charm hadrons in pp collisions at lhc energy within a coalescence plus fragmentation approach_ ,  Physics Letters B 821, 136622 (2021),  10.1016/j.physletb.2021.136622.
  * [7] S. Acharya _et al._ ,  _Charm-quark fragmentation fractions and production cross section at midrapidity in pp collisions at the LHC,_ (2021), http://arxiv.org/abs/2105.06335.
  * [8] M. Cacciari, S. Frixione, N. Houdeau, M. L. Mangano, P. Nason and G. Ridolfi,  _Theoretical predictions for charm and bottom production at the lhc_ ,  Journal of High Energy Physics 2012(10) (2012),  10.1007/jhep10(2012)137.
  * [9] D. d’Enterria and A. M. Snigirev,  _Triple parton scatterings in proton-nucleus collisions at high energies_ (2016), 1612.08112.
  * [10] M. Czakon, P. Fiedler and A. Mitov,  _Total top-quark pair-production cross section at hadron colliders through $o(\alpha_{s}^{4})$_,  Physical Review Letters 110(25) (2013),  10.1103/physrevlett.110.252004.


$t_{i}\leq_{\mathbf{c}}t_{i+1}$ whenever $k_{i}=k_{i+1}$.

Such sequences are in bijection with $\mathrm{NC}^{(k)}(W,\mathbf{c})$ by
“halving the colors,” i.e., by the map that sends
$\big{(}(t_{1},2k_{1}),(t_{2},2k_{2}),\ldots,(t_{r},2k_{r})\big{)}\mapsto\big{(}(t_{1},k_{1}),(t_{2},k_{2}),\ldots,(t_{r},k_{r})\big{)}$.
∎

###  8.3. Cambrian rotation

Recall that $\mathbf{c}=(s_{1},s_{2},\dots,s_{r})$. Let
$\mathbf{c}^{\prime}=(s_{2},\ldots,s_{r},s_{1})$. The _Cambrian rotation_ is a
bijection
$\mathrm{Sort}^{(k)}(W,\mathbf{c})\xrightarrow{\sim}\mathrm{Sort}^{(k)}(W,\mathbf{c}^{\prime})$
(equivalently,
$\mathrm{NC}^{(k)}(W,\mathbf{c})\xrightarrow{\sim}\mathrm{NC}^{(k)}(W,\mathbf{c}^{\prime})$
or
$\mathrm{Clus}^{(k)}(W,\mathbf{c})\xrightarrow{\sim}\mathrm{Clus}^{(k)}(W,\mathbf{c}^{\prime})$)
satisfying certain properties that enable inductive arguments; see [STW15] for
background. Cambrian rotation is a distinguishing feature of noncrossing
families [AST13]. The goal of this subsection is to develop an analogous
bijection for maximal $\mathbf{c}^{kh+1}$-Deograms.

######  Lemma 8.9.

Let $\mathbf{w}=(s_{1},s_{2},\dots,s_{m})$ be a word and let
$\mathbf{w}^{\prime}\coloneqq(s_{2},\dots,s_{m},s_{1})$. Then there is a
bijection

(8.1)
$\mathcal{D}_{e,\mathbf{w}}\xrightarrow{\sim}\mathcal{D}_{e,\mathbf{w}^{\prime}}.$

preserving the statistics $\operatorname{d}(\cdot)$,
$\operatorname{e}(\cdot)$. In particular, it restricts to a bijection
$\mathcal{M}_{e,\mathbf{w}}\xrightarrow{\sim}\mathcal{M}_{e,\mathbf{w}^{\prime}}$.

###### Proof.

We describe the bijection. Choose a word
$\mathbf{u}=(u_{1},u_{2},\dots,u_{m})\in\mathcal{D}_{e,\mathbf{w}}$. Let
$s=s_{1}$ and $\mathbf{u}^{\prime}\coloneqq(u_{2},\dots,u_{m},u_{1})$.

Assume that $u_{1}=e$. Then we claim that $\mathbf{u}^{\prime}$ is
distinguished, i.e., that
$\mathbf{u}^{\prime}\in\mathcal{D}_{e,\mathbf{w}^{\prime}}$. Indeed, we have
$u^{\prime}_{(i)}=u_{(i+1)}$ for all $0\leq i\leq m-1$, and moreover
$u^{\prime}_{(m-1)}=u^{\prime}_{(m)}=e$, which implies
$\mathbf{u}^{\prime}\in\mathcal{D}_{e,\mathbf{w}^{\prime}}$.

Now assume that $u_{1}=s$. We first treat the case where $W=\mathfrak{S}_{2}$,
where we have $s_{i}=s$ for all $i\in[m]$. Since $u_{1}=s$, we have $u_{2}=s$
because $\mathbf{u}$ is distinguished. Thus,
$\mathbf{u}=(s,s,u_{3},\dots,u_{m})$, which we send to
$\mathbf{u}^{\prime\prime}\coloneqq(u_{3},\dots,u_{m},s,s)$. The map
$\mathbf{u}\mapsto\mathbf{u}^{\prime\prime}$ is the desired bijection (8.1).

Finally, we treat the general case. Let

$\displaystyle J(\mathbf{u},s)$ $\displaystyle\coloneqq\\{j\in[m]\mid
s_{j}^{u_{(j)}}=s\\}$ $\displaystyle\text{and}\qquad J(\mathbf{u}^{\prime},s)$
$\displaystyle\coloneqq\\{j\in[m]\mid s_{j}^{u^{\prime}_{(j)}}=s\\}.$

For $j\in J(\mathbf{u},s)$, let $k_{j}$ be the color of the corresponding
reflection in $\mathrm{inv}(\mathbf{u})$ defined by (4.1). For $j\in
J(\mathbf{u}^{\prime},s)$, we similarly write $k^{\prime}_{j}$ for the
corresponding color.

Note that $1\in J(\mathbf{u},s)$ and $m\in J(\mathbf{u}^{\prime},s)$. We thus
have $J(\mathbf{u},s)=\\{1=j_{1}<j_{2}<\cdots<j_{\bar{m}}\\}$, and we claim
that similarly,
$J(\mathbf{u}^{\prime},s)=\\{j^{\prime}_{1}<\cdots<j^{\prime}_{\bar{m}}=m\\}$,
where $j^{\prime}_{i}=j_{i+1}-1$ for $i\in[\bar{m}-1]$. Indeed, this holds
because the reflections in $\mathrm{inv}(\mathbf{u}^{\prime})$ are obtained
from those in $\mathrm{inv}(\mathbf{u})$ by conjugation by $s$. Under such
conjugation, the color changes if and only if the reflection itself was equal
to $s$, in which case the color decreases by one. In other words, we have
$k^{\prime}_{j_{i}-1}=k_{j_{i}}-1$ for $i\in[2,\bar{m}]$.

Consider the word $\bar{\mathbf{w}}\coloneqq(s,s,\dots,s)$, where $s$ occurs
$\bar{m}$ times, and the subword
$\bar{\mathbf{u}}\coloneqq(\bar{u}_{1},\dots,\bar{u}_{\bar{m}})$ given by
$\bar{u}_{i}=e$ if $u_{j_{i}}=e$ and $\bar{u}_{i}=s$ otherwise. Let
$\bar{\mathbf{u}}^{\prime\prime}\coloneqq(\bar{u}^{\prime\prime}_{1},\dots,\bar{u}^{\prime\prime}_{\bar{m}})$
be the result of applying the above bijection for $\mathfrak{S}_{2}$ to
$\bar{\mathbf{u}}$.

Let
$\mathbf{u}^{\prime\prime}=(u^{\prime\prime}_{1},u^{\prime\prime}_{2},\dots,u^{\prime\prime}_{m})$
be the subword of $\mathbf{w}$ defined as follows. For $j\notin
J(\mathbf{u}^{\prime},s)$, set $u^{\prime\prime}_{j}\coloneqq u^{\prime}_{j}$.
For $j=j^{\prime}_{i}\in J(\mathbf{u}^{\prime},s)$, let
$u^{\prime\prime}_{j}\coloneqq e$ if $\bar{u}^{\prime\prime}_{i}=e$ and
$u^{\prime\prime}_{j}\coloneqq w^{\prime}_{j}$ otherwise. Once again, one
checks that the map $\mathbf{u}\mapsto\mathbf{u}^{\prime\prime}$ gives the
desired bijection. ∎

Let $\mathbf{c}^{\prime}\coloneqq(s_{2},\dots,s_{r},s_{1})$. Then the lemma
above gives a bijection
$\mathcal{M}_{e,\mathbf{c}^{kh+1}}\xrightarrow{\sim}\mathcal{M}_{e,(\mathbf{c}^{\prime})^{kh+1}}$.
This bijection has the following property: It sends an element
$\mathbf{u}=(u_{1},u_{2},\dots,u_{m})\in\mathcal{M}_{e,\mathbf{c}^{kh+1}}$ to
an element
$\mathbf{u}^{\prime}=(u_{1}^{\prime},u_{2}^{\prime},\dots,u_{m}^{\prime})\in\mathcal{M}_{e,(\mathbf{c}^{\prime})^{kh+1}}$
satisfying $u^{\prime}_{m}=u_{1}$.

###  8.4. Cambrian and Deodhar recurrences

Our next goal is to show that the subset of
$\mathbf{u}\in\mathcal{M}_{e,\mathbf{c}^{kh+1}}$ satisfying $u_{1}=e$ is in
bijection with
$\mathcal{M}_{e,(\mathbf{c}^{\prime\prime})^{kh^{\prime\prime}+1}}$, where
$\mathbf{c}^{\prime\prime}=(s_{2},\ldots,s_{r})$ is a Coxeter word for the
parabolic subgroup $W_{\langle s_{1}\rangle}$ of $W$ generated by
$S\setminus\\{s_{1}\\}$. This will match the Cambrian recurrence on
noncrossing families described in [STW15, Section 4].

Let $c^{\prime\prime}\coloneqq s_{1}c$ be the associated Coxeter element of
$W_{\langle s_{1}\rangle}$, and let $h^{\prime\prime}$ be the Coxeter number
of $W_{\langle s_{1}\rangle}$. Suppose that
$\mathbf{u}\in\mathcal{M}_{e,\mathbf{c}^{kh+1}}$ starts with $u_{1}=e$. Let
$\big{(}(t_{1},k_{1}),(t_{2},k_{2}),\ldots,(t_{r},k_{r})\big{)}\in\mathrm{NC}^{(k)}(W,\mathbf{c})$
be the $k$-noncrossing partition assigned to $\mathbf{u}$ under the bijection
of Theorem 8.8. Then we have $(t_{1},k_{1})=(s_{1},0)$. By [STW15, Proposition
4.3], the subset of $\mathrm{NC}^{(k)}(W,\mathbf{c})$ that satisfies
$(t_{1},k_{1})=(s_{1},0)$ is in bijection with the set
$\mathrm{NC}^{(k)}(W_{\langle s_{1}\rangle},\mathbf{c}^{\prime\prime})$, which
is itself in bijection with
$\mathcal{M}_{e,(\mathbf{c}^{\prime\prime})^{kh^{\prime\prime}+1}}$ by Theorem
8.8.

The _Cambrian recurrence_ on $\mathcal{M}_{e,\mathbf{c}^{kh+1}}$ is the
modification of the Cambrian recurrence that performs the map from
$\mathcal{M}_{e,\mathbf{c}^{kh+1}}$ to
$\mathcal{M}_{e,(\mathbf{c}^{\prime})^{kh+1}}$ from (8.1) when a subword
$\mathbf{u}\in\mathcal{M}_{e,\mathbf{c}^{kh+1}}$ does not start with
$u_{1}=e$, and performs the map above from $\mathcal{M}_{e,\mathbf{c}^{kh+1}}$
to $\mathcal{M}_{e,(\mathbf{c}^{\prime\prime})^{kh^{\prime\prime}+1}}$ when
$\mathbf{u}$ starts with $u_{1}=e$.

######  Remark 8.10.

To perform the Cambrian recurrence on $\mathcal{M}_{e,\mathbf{w}}$, we split
the set of all $\mathbf{u}\in\mathcal{M}_{e,(\mathbf{c})^{kh+1}}$ into subsets
satisfying $u_{1}=e$ and $u_{1}=s_{1}$. This same split—without the descent to
the parabolic subgroup made possible by the reflection-factorization
properties of a Coxeter element—occurs in the Deodhar recurrence (Proposition
4.11). Thus the Deodhar recurrence may be seen as a generalization of the
Cambrian rotation from collections of Fuss–Catalan objects to the more general
sets $\mathcal{D}_{u,\mathbf{w}}^{k}$.

$\begin{array}[]{|cccccccc|cc|}\lx@intercol\hfil\mathcal{M}_{e,\mathbf{c}^{h+1}}\hfil\lx@intercol&\lx@intercol\hfil\mathrm{NC}^{(1)}(W,\mathbf{c})\hfil\lx@intercol\\\
\hline\cr&&&&&&&&&\\\\[-13.00005pt]
s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&\ddot{(1\kern 1.00006pt2)}&\ddot{(2\kern
1.00006pt3)}&\big{(}\dot{(1\kern 1.00006pt2)},&\dot{(2\kern
1.00006pt3)}\big{)}\\\ s_{1}&s_{2}&(2\kern
1.00006pt3)&s_{2}&s_{1}&s_{2}&\ddot{(1\kern 1.00006pt3)}&s_{2}&\big{(}(2\kern
1.00006pt3),&\dot{(1\kern 1.00006pt3)}\big{)}\\\ s_{1}&(1\kern
1.00006pt3)&s_{1}&s_{2}&s_{1}&\ddot{(1\kern
1.00006pt2)}&s_{1}&s_{2}&\big{(}(1\kern 1.00006pt3),&\dot{(1\kern
1.00006pt2)}\big{)}\\\ (1\kern
1.00006pt2)&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&\ddot{(2\kern
1.00006pt3)}&\big{(}(1\kern 1.00006pt2),&\dot{(2\kern 1.00006pt3)}\big{)}\\\
(1\kern 1.00006pt2)&(2\kern
1.00006pt3)&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&\big{(}(1\kern
1.00006pt2),&(2\kern 1.00006pt3)\big{)}\\\ \hline\cr\end{array}$ Figure 10.
For $W=\mathfrak{S}_{3}$ and $\mathbf{c}=(s_{1},s_{2})$, the bijection between
$\mathcal{M}_{e,\mathbf{c}^{h+1}}$ and $\mathrm{NC}^{(1)}(W,\mathbf{c})$. As
usual, we replace the positions $i$ where $u_{i}=e$ with the corresponding
colored reflection in $\mathrm{inv}_{e}(\mathbf{u})$.
$\begin{array}[]{|cccccccccccccc|}\lx@intercol\hfil\mathcal{M}_{e,\mathbf{c}^{2h+1}}\hfil\lx@intercol\\\
\hline\cr
s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&\ddddot{(1\kern
1.00006pt2)}&\ddddot{(2\kern 1.00006pt3)}\\\
s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&\ddot{(2\kern
1.00006pt3)}&s_{2}&s_{1}&s_{2}&\ddddot{(1\kern 1.00006pt3)}&s_{2}\\\
s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&\ddot{(1\kern
1.00006pt3)}&s_{1}&s_{2}&s_{1}&\ddddot{(1\kern 1.00006pt2)}&s_{1}&s_{2}\\\
s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&\ddot{(1\kern
1.00006pt2)}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&\ddddot{(2\kern
1.00006pt3)}\\\ s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&\ddot{(1\kern
1.00006pt2)}&\ddot{(2\kern 1.00006pt3)}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}\\\
s_{1}&s_{2}&(2\kern
1.00006pt3)&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&\ddddot{(1\kern
1.00006pt3)}&s_{2}\\\ s_{1}&s_{2}&(2\kern
1.00006pt3)&s_{2}&s_{1}&s_{2}&\ddot{(1\kern
1.00006pt3)}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}\\\ s_{1}&(1\kern
1.00006pt3)&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&\ddddot{(1\kern
1.00006pt2)}&s_{1}&s_{2}\\\ s_{1}&(1\kern
1.00006pt3)&s_{1}&s_{2}&s_{1}&\ddot{(1\kern
1.00006pt2)}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}\\\ (1\kern
1.00006pt2)&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&\ddddot{(2\kern
1.00006pt3)}\\\ (1\kern
1.00006pt2)&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&\ddot{(2\kern
1.00006pt3)}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}\\\ (1\kern
1.00006pt2)&(2\kern
1.00006pt3)&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}&s_{1}&s_{2}\\\
\hline\cr\end{array}$ Figure 11. For $W=\mathfrak{S}_{3}$ and
$\mathbf{c}=(s_{1},s_{2})$, the 12 elements of
$\mathcal{M}_{e,\mathbf{c}^{2h+1}}$. As usual, we replace the positions $i$
where $u_{i}=e$ with the corresponding colored reflection in
$\mathrm{inv}_{e}(\mathbf{u})$.

###  8.5. Rational Noncrossing Parking Functions

In the first four sections of [Ede80], Edelman proposed a $k$-generalization
of type $A$ noncrossing partitions, which was subsequently generalized to all
Coxeter groups by Armstrong [Arm09]. In [Ede80, Section 5], Edelman proposed a
definition he called “noncrossing $2$-partitions.” Inspired by a related
construction of _nonnesting_ parking functions, Armstrong, Reiner, and Rhoades
independently proposed a generalized version for all Coxeter groups in
[ARR15]. Rhoades gave a Fuss generalization in [Rho14].

Because of the $v$-twisting (cf. Definition 1.7), we find it convenient to
pass from our canonical factorization definition of
$\mathrm{NC}^{(k)}(W,\mathbf{c})$ back to $k$-multichain language. We refer
the reader back to Section 8.1.2 for a discussion of this equivalence. Given
$\pi=\big{(}\pi_{1}\leq_{T}\pi_{2}\leq_{T}\cdots\leq_{T}\pi_{k}\big{)}$ with
$\pi_{i}\in\mathrm{NC}(W,c)$, let $W_{\langle\pi_{1}\rangle}\coloneqq\langle
t:t\leq_{T}\pi_{1}\rangle$ be the reflection subgroup of $W$ generated by the
reflections below $\pi_{1}$ in absolute order. By [Dye90, Corollary 3.4(ii)]
(see Lemma 8.12 below), every coset $vW_{\langle\pi_{1}\rangle}$ has a unique
element $z$ of minimal length, characterized by the property that
$\mathrm{inv}(z^{-1})$ has no reflections belonging to
$W_{\langle\pi_{1}\rangle}$. Here, for $u\in W$, we set

$\displaystyle\mathrm{inv}(u)\coloneqq\left\\{s_{1}^{u_{(1)}},s_{2}^{u_{(2)}},\dots,s_{m}^{u_{(m)}}\right\\},$

where $m=\ell(u)$ and $\mathbf{u}=(u_{1},u_{2},\dots,u_{m})$ is any reduced
word for $u$. In other words, $\mathrm{inv}(u)$ is obtained from
$\mathrm{inv}(\mathbf{u})$ by forgetting the colors and the order of the
reflections. We write $W^{\langle\pi_{1}\rangle}$ to denote the set of minimal
coset representatives of $W/W_{\langle\pi_{1}\rangle}$.

######  Definition 8.11 ([ARR15, Rho14]).

The _$(W,k)$ -noncrossing parking functions_ are

$\displaystyle\mathrm{Park}^{(k)}(W,c)\coloneqq\left\\{\big{(}v,\big{(}\pi_{1}\leq_{T}\pi_{2}\leq_{T}\cdots\leq_{T}\pi_{k}\big{)}\big{)}\
\middle|\ \pi_{i}\in\mathrm{NC}(W,c),v\in W^{\langle\pi_{1}\rangle}\right\\}.$

For $W=\mathfrak{S}_{3}$, the sixteen $(W,1)$-noncrossing parking functions
$(v,\pi_{1})$ in $\mathrm{Park}^{(1)}(\mathfrak{S}_{3},s_{1}s_{2})$ are
illustrated in the right column of Figure 12.

$\begin{array}[]{|c||cc!{\graytablecolor\vrule}cc!{\graytablecolor\vrule}cc!{\graytablecolor\vrule}cc||c|}\hline\cr
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\hline\cr&&\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}&&\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}&&\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>s_{2}s_{1}&s_{1}&s_{2}\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}&(2\kern
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s_{1}s_{2}&s_{1}&s_{2}\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}&s_{1}&s_{2}\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}&\ddot{(1\kern
1.00006pt3)}&\ddot{(1\kern
1.00006pt2)}\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>\hline\cr&&\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}&&\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}&&\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>s_{1}s_{2}s_{1}&s_{1}&s_{2}\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}&s_{1}&\ddot{(1\kern
1.00006pt2)}\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color@gray@fill{0.6}&\ddot{(2\kern
1.00006pt3)}&s_{2}\hfil\lx@intercol\color[rgb]{0.6,0.6,0.6}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.6,0.6}\pgfsys@color@gray@stroke{0.6}\pgfsys@color<EMAIL_ADDRESS>\hline\cr\end{array}$ Figure 12. The 16 elements of
${\mathcal{P}_{e,\mathbf{c}^{p}}(W)}$ for $W=\mathfrak{S}_{3}$ and
$\mathbf{c}=(s_{1},s_{2})$ and $p=4$, shown together with the corresponding
parking functions of $\mathrm{Park}(W,c)$. For the wiring diagram
representations, see Figure 4.

In the rest of this subsection, we show that the noncrossing parking functions
of Definition 8.11 are in uniform bijection with the parking objects of
Definition 1.7.

######  Lemma 8.12 ([Dye90, Corollary 3.4(ii)]).

Let $v\in W$ and $\pi\in\mathrm{NC}(W,c)$. Then $v\in W^{\langle\pi\rangle}$
if and only if we have $t\not\in\mathrm{inv}(v^{-1})$ for all $t\leq_{T}\pi$.

######  Example 8.13.

For $W=\mathfrak{S}_{4}$ and $\pi=(1\kern 1.00006pt2)(3\kern 1.00006pt4)$, the
minimal coset representatives in $W^{\langle\pi\rangle}$ are $e$, $s_{2}$,
$s_{1}s_{2}$, $s_{3}s_{2}$, $s_{1}s_{3}s_{2}$, $s_{2}s_{1}s_{3}s_{2}$. The
inverses of these permutations have inversion sets $\emptyset$, $\\{(2\kern
1.00006pt3)\\}$, $\\{(1\kern 1.00006pt3),(2\kern 1.00006pt3)\\}$, $\\{(2\kern
1.00006pt4),(2\kern 1.00006pt3)\\}$, $\\{(1\kern 1.00006pt3),(2\kern
1.00006pt4),(2\kern 1.00006pt3)\\}$, $\\{(1\kern 1.00006pt4),(1\kern
1.00006pt3),(2\kern 1.00006pt4),(2\kern 1.00006pt3)\\}$. The reflections $t\in
T$ satisfying $t\leq_{T}\pi$ are $(1\kern 1.00006pt2)$ and $(3\kern
1.00006pt4)$, which are precisely the reflections that never appear in the
inversion sets above.

It is natural to use $v\in W$ to twist the color of the colored reflections in
$\mathrm{inv}(\mathbf{u})$ while preserving the reflection itself,
generalizing Section 2.3. We explain this construction in more detail. Given a
subword $\mathbf{u}=(u_{1},u_{2},\ldots,u_{m})$, recall the colored
reflections $t_{j}(\mathbf{u})=(s_{j}^{u_{(j)}},k_{j})$ for $1\leq j\leq m$
that we defined in (4.1). For $1\leq j\leq m$, set

$\displaystyle
t^{(v)}_{j}(\mathbf{u})\coloneqq(s_{j}^{u_{(j)}},k_{j}^{\prime}),\quad\text{where}\quad
k_{j}^{\prime}\coloneqq k_{j}+\begin{cases}1&\text{if
}s_{j}^{u_{(j)}}\in\mathrm{inv}(v^{-1}),\\\ 0&\text{ otherwise.}\end{cases}$

Write
$\mathrm{inv}^{(v)}(\mathbf{u})\coloneqq\left(t^{(v)}_{1}(\mathbf{u}),t^{(v)}_{2}(\mathbf{u}),\dots,t^{(v)}_{m}(\mathbf{u})\right)$
and $\mathrm{inv}_{e}^{(v)}(\mathbf{u})$ for the restriction of
$\mathrm{inv}^{(v)}(\mathbf{u})$ to the indices $j$ for which $u_{j}=e$.

In order to state the bijection between
${\mathcal{P}_{e,\mathbf{c}^{kh+1}}(W)}$ and
$\mathrm{Park}^{(k)}(W,\mathbf{c})$, we need to understand the behavior of the
twisted colored reflections of a subword. This will allow us to go between
chains of noncrossing partitions using only even colors, and certain chains of
noncrossing partitions using both even and odd colors.

######  Lemma 8.14.

Fix $v\in W$. Any $\pi\in\mathrm{NC}(W,c)$ can be uniquely factored as
$\pi=\pi_{v}\cdot\pi^{v}$, such that

  * •

$\pi_{v},\pi^{v}\in\mathrm{NC}(W,c)$,

  * •

$\ell_{T}(\pi_{v})+\ell_{T}(\pi^{v})=\ell_{T}(\pi)$, and

  * •

if $t\leq_{T}\pi_{v}$, then $t\in\mathrm{inv}(v^{-1})$, whereas

  * •

if $t\leq_{T}\pi^{v}$, then $t\not\in\mathrm{inv}(v^{-1})$.

###### Proof.

First, assume that $\pi=c$. We argue using the theory of Reading’s Cambrian
lattices [Rea06, Rea07a]. These Cambrian lattices are quotients of the weak
order induced by the projections $\pi_{\downarrow}^{c}:W\to\mathrm{Sort}(W,c)$
and $\pi^{\uparrow}_{c}$ from $W$ to the $c$-antisortable elements of $W$.
(Recall that $w$ is called _$c$ -antisortable_ if $ww_{\circ}$ is
$c^{-1}$-sortable.) More precisely, for $v\in W$, we define
$\pi_{\downarrow}^{c}(v)$ is as the largest $c$-sortable element less than $v$
in weak order, while $\pi^{\uparrow}_{c}(v)$ is the smallest $c$-antisortable
element larger than $v$ in weak order.

Then any $v\in W$ is contained in a unique interval
$[\pi_{\downarrow}^{c}(v),\pi^{\uparrow}_{c}(v)]$. The product of the lower
cover reflections for $\pi_{\downarrow}^{c^{-1}}(v^{-1})$ in the order
$\leq_{c}$ defines a noncrossing partition $c_{v}\in\mathrm{NC}(W,c)$, while
the product of the upper cover reflections for $\pi^{\uparrow}_{c^{-1}}(v)$ in
the order $\leq_{c}$ defines the Kreweras complement $c^{v}=c_{v}^{-1}c$.
(Note that here, we use the $c^{-1}$-Cambrian lattice projections
$\pi_{\downarrow}^{c^{-1}}$ and $\pi^{\uparrow}_{c^{-1}}$ instead of the
$c$-Cambrian lattice projections $\pi_{\downarrow}^{c}$ and
$\pi^{\uparrow}_{c}$, to ensure that the order of the factors in the product
$c=c_{v}\cdot c^{v}$ use the lower cover reflections _before_ the upper cover
reflections. Using $\pi_{\downarrow}^{c}$ and $\pi^{\uparrow}_{c}$ would
instead result in a factorization where the product of the lower cover
reflections appear _after_ the product of the upper cover reflections.) These
$c_{v}$ and $c^{v}$ are the desired factors: Uniqueness follows from
uniqueness of the interval, and the last two properties follow from the fact
that the Cambrian lattices form stronger partial orders than the noncrossing
partition lattices [Rea11, Proposition 8.11].

Now let $\pi$ be a general noncrossing partition with canonical EL-
factorization $\pi=t_{1}t_{2}\cdots t_{a}$, where
$t_{1}<_{c}t_{2}<_{c}\cdots<_{c}t_{a}$. Consider the parabolic subgroup
$W_{\langle\pi\rangle}$ of $W$ in which $t_{1},\ldots,t_{a}$ are the simple
reflections. (We can treat them as simple reflections because they are
precisely the lower cover reflections of some sortable element $w\in W$.) Each
element $v\in W$ appears in some coset of $W_{\langle\pi\rangle}$, and each
such coset contains a minimal representative in weak order by [GP00,
Proposition 2.1.1]. After translating by this representative, we can identify
the coset with the Coxeter group $W_{\langle\pi\rangle}$. In this way, we can
identify $v$ with an element of $W_{\langle\pi\rangle}$. The inversions for
the latter are obtained from the inversions for $v$ by restricting to the
reflections in $W_{\langle\pi\rangle}$. If we build the Cambrian lattices on
the cosets of $W_{\langle\pi\rangle}$, using $t_{1},\ldots,t_{a}$ as the
simple reflections and $t_{1}t_{2}\cdots t_{a}$ as the Coxeter element, then
we will have reduced to the previous case. ∎

######  Example 8.15.

Continuing Example 8.13, take $W=\mathfrak{S}_{4}$ and
$c=s_{1}s_{2}s_{3}=(1\kern 1.00006pt2\kern 1.00006pt3\kern 1.00006pt4)$ and
$\pi=c$. This is the Coxeter-element case of Lemma 8.14, which requires the
full Cambrian lattice.

  * •

For each of $v\in\\{s_{2}s_{1}s_{3}s_{2},s_{1}s_{2},s_{1}s_{3}s_{2}\\}$, we
have

$\displaystyle\pi_{\downarrow}^{c^{-1}}(v^{-1})$
$\displaystyle=\text{$s_{2}s_{1}$ with lower cover reflections $\\{(1\kern
1.00006pt3)\\}$}$
$\displaystyle\text{and}\qquad\pi^{\uparrow}_{c^{-1}}(v^{-1})$
$\displaystyle=\text{$s_{2}s_{1}s_{3}s_{2}$ with upper cover reflections
$\\{(1\kern 1.00006pt2),(3\kern 1.00006pt4)\\}$.}$

giving the factorization $(1\kern 1.00006pt2\kern 1.00006pt3\kern
1.00006pt4)=(1\kern 1.00006pt3)\cdot(1\kern 1.00006pt2)(3\kern 1.00006pt4)$.

  * •

For $v\in\\{s_{3}s_{2},s_{2}\\}$, we have

$\displaystyle\pi_{\downarrow}^{c^{-1}}(v^{-1})$ $\displaystyle=\text{$s_{2}$
with lower cover reflections $\\{(2\kern 1.00006pt3)\\}$}$
$\displaystyle\text{and}\qquad\pi^{\uparrow}_{c^{-1}}(v^{-1})$
$\displaystyle=\text{$s_{2}s_{3}$ with cover reflections $\\{(1\kern
1.00006pt3),(3\kern 1.00006pt4)\\}$.}$

giving the factorization $(1\kern 1.00006pt2\kern 1.00006pt3\kern
1.00006pt4)=(2\kern 1.00006pt3)\cdot(1\kern 1.00006pt3\kern 1.00006pt4)$.

  * •

For $v=e$, we get the factorization $(1\kern 1.00006pt2\kern 1.00006pt3\kern
1.00006pt4)=e\cdot(1\kern 1.00006pt2\kern 1.00006pt3\kern 1.00006pt4)$, since

$\displaystyle\pi_{\downarrow}^{c^{-1}}(e)=\pi^{\uparrow}_{c^{-1}}(e)=e$

with no lower cover reflections and with the upper cover reflections
$\\{(1\kern 1.00006pt2),(2\kern 1.00006pt3),(3\kern 1.00006pt4)\\}$.

######  Example 8.16.

Take $W=\mathfrak{S}_{4}$ and $c=(1\kern 1.00006pt2\kern 1.00006pt3\kern
1.00006pt4)$ and $\pi=(1\kern 1.00006pt3\kern 1.00006pt4)$, a more generic
case of Lemma 8.14. We have $W_{\langle\pi\rangle}=\langle(1\kern
1.00006pt3),(3\kern 1.00006pt4)\rangle\simeq\mathfrak{S}_{3}$. The simple
reflections of $W_{\langle\pi\rangle}$ are $(1\kern 1.00006pt3)$ and $(3\kern
1.00006pt4)$, and the reflections of $W_{\langle\pi\rangle}$ are $\\{(1\kern
1.00006pt3),(1\kern 1.00006pt4),(3\kern 1.00006pt4)\\}$.

  * •

For $v\in\\{e,s_{2},s_{3}s_{2}\\}$, we have $\mathrm{inv}(v^{-1})\cap
W_{\langle\pi\rangle}=\emptyset$, so

$\displaystyle\pi_{\downarrow}^{(1\kern 0.70004pt4\kern
0.70004pt3)}(v|_{W_{\langle\pi\rangle}})=\pi^{\uparrow}_{(1\kern
0.70004pt4\kern 0.70004pt3)}(v|_{W_{\langle\pi\rangle}})=e,$

giving the factorization $(1\kern 1.00006pt3\kern 1.00006pt4)=e\cdot(1\kern
1.00006pt3\kern 1.00006pt4)$.

  * •

For $v\in\\{s_{1}s_{2},s_{1}s_{3}s_{2}\\}$, we have $\mathrm{inv}(v^{-1})\cap
W_{\langle\pi\rangle}=\\{(1\kern 1.00006pt3)\\}$, so

$\displaystyle\pi_{\downarrow}^{(1\kern 0.70004pt4\kern
0.70004pt3)}(v|_{W_{\langle\pi\rangle}})$ $\displaystyle=\text{$(1\kern
1.00006pt3)$ with lower cover reflection $(1\kern 1.00006pt3)$}$
$\displaystyle\text{and}\qquad\pi^{\uparrow}_{(1\kern 0.70004pt4\kern
0.70004pt3)}(v|_{W_{\langle\pi\rangle}})$ $\displaystyle=\text{$(1\kern
1.00006pt3)(3\kern 1.00006pt4)$ with upper cover reflection $(3\kern
1.00006pt4)$},$

giving the factorization $(1\kern 1.00006pt3\kern 1.00006pt4)=(1\kern
1.00006pt3)\cdot(3\kern 1.00006pt4)$.

  * •

For $v=s_{2}s_{1}s_{3}s_{2}$, we have $\mathrm{inv}(v^{-1})\cap
W_{\langle\pi\rangle}=\\{(1\kern 1.00006pt4),(1\kern 1.00006pt3)\\}$, so
again,

$\displaystyle\pi_{\downarrow}^{(1\kern 0.70004pt4\kern
0.70004pt3)}(v|_{W_{\langle\pi\rangle}})$ $\displaystyle=(1\kern 1.00006pt3)$
$\displaystyle\text{and}\qquad\pi^{\uparrow}_{(1\kern 0.70004pt4\kern
0.70004pt3)}(v|_{W_{\langle\pi\rangle}})$ $\displaystyle=(1\kern
1.00006pt3)(3\kern 1.00006pt4),$

giving the same factorization $(1\kern 1.00006pt3\kern 1.00006pt4)=(1\kern
1.00006pt3)\cdot(3\kern 1.00006pt4)$ as in the previous case.

######  Theorem 8.17.

Fix $k\in\mathbb{N}$. Then there is a uniform bijection

$\displaystyle{\mathcal{P}_{e,\mathbf{c}^{kh+1}}(W)}$
$\displaystyle\simeq\mathrm{Park}^{(k)}(W,\mathbf{c}).$

###### Proof.

Any element of ${\mathcal{P}_{e,\mathbf{c}^{kh+1}}(W)}=\bigsqcup_{v\in
W}\mathcal{M}_{e,\mathbf{c}^{kh+1}}^{(v)}$ belongs to
$\mathcal{M}_{e,\mathbf{c}^{kh+1}}^{(v)}$ for some $v$. Now,
$\mathcal{M}_{e,\mathbf{c}^{kh+1}}^{(v)}$ is exactly the subset of
$\mathrm{Clus}^{(2k)}(W,\mathbf{c})$ of those subwords $\mathbf{u}$ for which
each colored reflection in $\mathrm{inv}_{e}^{(v)}(\mathbf{u})$ has even
color. Since applying $v$ to a colored reflection $(t,k)$ increases $k$ by one
if and only if $t\in\mathrm{inv}(v)$, we deduce that
$\mathcal{M}_{e,\mathbf{c}^{kh+1}}^{(v)}$ is in bijection with $(2k+1)$-tuples
of noncrossing partitions
$e\leq_{T}\pi_{0}\leq_{T}\pi_{1}\leq_{T}\cdots\leq_{T}\pi_{2k}=c$ such that:

  * •

if $t\leq_{T}\pi_{i-1}^{-1}\pi_{i}$ and $i$ is odd, then
$t\in\mathrm{inv}(v^{-1})$, whereas

  * •

if $t\leq_{T}\pi_{i-1}^{-1}\pi_{i}$ and $i$ is even, then
$t\not\in\mathrm{inv}(v^{-1})$.

We wish to place such tuples in bijection with the noncrossing parking
functions of the form
$\big{(}v,\big{(}\sigma_{1}\leq_{T}\sigma_{2}\leq_{T}\cdots\leq_{T}\sigma_{k}\big{)}\big{)}$
with $v\in W^{\langle\sigma_{1}\rangle}$, by setting

$\displaystyle\sigma_{i}\coloneqq\pi_{2(i-1)}\qquad\text{for $1\leq i\leq
k$.}$

Note that if $t\leq_{T}\pi_{0}$, then $t\not\in\mathrm{inv}(v^{-1})$, so $v$
is a minimal coset representative of
$W_{\langle\sigma_{1}\rangle}=W_{\langle\pi_{0}\rangle}$ by Lemma 8.12. Thus,
the map from tuples to noncrossing parking functions is well-defined.

To see that the map is a bijection, apply Lemma 8.14 to each factor
$\sigma_{i}^{-1}\sigma_{i+1}$ in succession. Working on colors
$i=1,2,\ldots,m$, we use the reflections $t\leq_{T}\sigma_{i}$ that are also
in $\mathrm{inv}(v^{-1})$ to split each noncrossing partition
$\sigma_{i}^{-1}\sigma_{i+1}$ into a noncrossing partition in the odd color
$2i-3$ and one in the even color $2i-2$, which become the factors of the
element in $\mathcal{M}_{e,\mathbf{c}^{kh+1}}^{(v)}$. By further factoring the
noncrossing partitions $c_{v}$ and $c^{v}$ uniquely into $<_{c}$-increasing
products of reflections, as in Section 8.1.2, we construct the
$\mathrm{inv}_{e}^{v}$ sequence for the subword in
$\mathcal{M}_{e,\mathbf{c}^{h+1}}^{(v)}$ with no reflections of color $0$. So
the product of the reflections with twisted color $2$ is $c$ itself. As we
have now identified the colored reflections in $\mathrm{inv}_{e}^{v}$, we can
reconstruct the subword itself using Remark 4.6. ∎

######  Example 8.18.

Continuing Example 8.15, we illustrate Theorem 8.17 by using Lemma 8.14 to
reconstruct the subwords in $\mathcal{M}_{e,\mathbf{c}^{h+1}}^{(v)}$
corresponding to the noncrossing parking functions
$\big{(}v,\big{(}e\big{)}\big{)}$.

  * •

For $v\in\\{s_{2}s_{1}s_{3}s_{2},s_{1}s_{2},s_{1}s_{3}s_{2}\\}$, we need to
put $(1\kern 1.00006pt3)$ in color $1$, whereas we need to put $(1\kern
1.00006pt2),(3\kern 1.00006pt4)$ in color $2$ so that after twisting, all
reflections have color $2$. Using Remark 4.6, we obtain

$\displaystyle(s_{1},s_{2},s_{3},s_{1},s_{2},s_{3},s_{1},\dot{(1\kern
1.00006pt3)},\ddot{(1\kern
1.00006pt2)},s_{1},s_{2},s_{3},s_{1},s_{2},\ddot{(3\kern 1.00006pt4)}).$

  * •

For $v\in\\{s_{3}s_{2},s_{2}\\}$, we need to put $(2\kern 1.00006pt3)$ in
color 1 and $(1\kern 1.00006pt3),(3\kern 1.00006pt4)$ in color $2$. We obtain

$\displaystyle(s_{1},s_{2},s_{3},s_{1},s_{2},s_{3},s_{1},s_{2},\dot{(2\kern
1.00006pt3)},s_{1},s_{2},s_{3},\ddot{(1\kern 1.00006pt3)},s_{2},\ddot{(3\kern
1.00006pt4)}).$

  * •

For $v=e$, we put all reflections $(1\kern 1.00006pt2)$, $(2\kern
1.00006pt3)$, and $(3\kern 1.00006pt4)$ in color $2$, so that

$\displaystyle(s_{1},s_{2},s_{3},s_{1},s_{2},s_{3},s_{1},s_{2},s_{3},s_{1},s_{2},s_{3},\ddot{(1\kern
1.00006pt2)},\ddot{(2\kern 1.00006pt3)},\ddot{(3\kern 1.00006pt4)}).$

######  Example 8.19.

For a larger illustration of Theorem 8.17, take $W=\mathfrak{S}_{6}$ and
$c=(1\kern 1.00006pt2\kern 1.00006pt3\kern 1.00006pt4\kern 1.00006pt5\kern
1.00006pt6)$ and $m=1$. Fix $v=s_{5}s_{2}s_{3}s_{4}s_{2}s_{3}\in\in
W^{\langle(1\kern 0.70004pt3\kern 0.70004pt6)\rangle}$, and consider the
noncrossing parking function $\big{(}v,\big{(}(1\kern 1.00006pt3\kern
1.00006pt6)\big{)}\big{)}$.

First, $\mathrm{inv}(v^{-1})=\\{(3\kern 1.00006pt4),(2\kern
1.00006pt4),(3\kern 1.00006pt5),(2\kern 1.00006pt5),(4\kern
1.00006pt5),(3\kern 1.00006pt6)\\}$. The noncrossing partition $(1\kern
1.00006pt3\kern 1.00006pt6)$ corresponds to the factorization into noncrossing
partitions $c=\pi_{0}\cdot\pi_{1}$, where $\pi_{0}=(1\kern 1.00006pt3\kern
1.00006pt6)$ and $\pi_{1}=(1\kern 1.00006pt2)(3\kern 1.00006pt4\kern
1.00006pt5)$. To each of $\pi_{0}$ and $\pi_{1}$, we apply Lemma 8.14:

  * •

Since $v$ is a minimal coset representative of $W^{\langle(1\kern
0.70004pt3\kern 0.70004pt6)\rangle}$, we have $\mathrm{inv}(v^{-1})\cap
W_{\langle\pi_{0}\rangle}=\emptyset$. Thus we put $(1\kern 1.00006pt3),(3\kern
1.00006pt6)$ in color $0$.

  * •

Since $\mathrm{inv}(v^{-1})\cap W_{\langle\pi_{1}\rangle}=\\{(3\kern
1.00006pt4),(3\kern 1.00006pt5),(4\kern 1.00006pt5)\\}$, we have

$\displaystyle\pi_{\downarrow}^{(1\kern 0.70004pt2)(3\kern 0.70004pt5\kern
0.70004pt4)}(v|_{W_{\langle\pi_{1}\rangle}})=\pi^{\uparrow}_{(1\kern
0.70004pt2)(3\kern 0.70004pt5\kern
0.70004pt4)}(v|_{W_{\langle\pi_{1}\rangle}})=(3\kern 1.00006pt4)(4\kern
1.00006pt5)(3\kern 1.00006pt4)$

in $W_{\langle\pi_{1}\rangle}$, with lower cover reflections $(3\kern
1.00006pt4),(4\kern 1.00006pt5)$ and upper cover reflection $(1\kern
1.00006pt2)$, giving the factorization $(1\kern 1.00006pt2)(3\kern
1.00006pt4\kern 1.00006pt5)=(3\kern 1.00006pt4\kern 1.00006pt5)\cdot(1\kern
1.00006pt2)$. Thus we put $(3\kern 1.00006pt4),(4\kern 1.00006pt5)$ in color
$1$ and $(1\kern 1.00006pt2)$ in color $2$.

We can finally reconstruct the corresponding subword of $\mathbf{c}^{7}$ using
Remark 4.6:

$\displaystyle\begin{array}[]{ccccc}(s_{1},&(1\kern
1.00006pt3),&s_{3},&s_{4},&(3\kern 1.00006pt6),\\\
s_{1},&s_{2},&s_{3},&s_{4},&s_{5},\\\
s_{1},&s_{2},&s_{3},&s_{4},&\ddot{(1\kern 1.00006pt2)},\\\ s_{1},&\dot{(3\kern
1.00006pt4)},&s_{3},&s_{4},&s_{5},\\\ \dot{(4\kern
1.00006pt5)},&s_{2},&s_{3},&s_{4},&s_{5},\\\
s_{1},&s_{2},&s_{3},&s_{4},&s_{5},\\\
s_{1},&s_{2},&s_{3},&s_{4},&s_{5}).\end{array}$

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