[Paper Review] Counting descent pairs with prescribed colors in the colored permutation groups

Eli Bagno, David Garber+1 more|ArXiv.org|Sep 17, 2007

Advanced Combinatorial Mathematics5 references4 citations

TL;DR

This paper introduces and enumerates new descent statistics—(c,d)-descents—on colored permutation groups $\mathbb{Z}_r \wr S_n$, generalizing classical descent statistics to colored permutations. Using generating functions, recurrences, and combinatorial manipulations, the authors derive explicit formulas for the number of elements with exactly $m$ $(c,d)$-descents for $c \leq d$ and $(c,c)$-descents, extending results from the hyperoctahedral group $B_n$ to the broader colored permutation setting.

ABSTRACT

We define new statistics, (c, d)-descents, on the colored permutation groups Z_r \wr S_n and compute the distribution of these statistics on the elements in these groups. We use some combinatorial approaches, recurrences, and generating functions manipulations to obtain our results.

Motivation & Objective

• To generalize classical descent statistics in symmetric and hyperoctahedral groups to the setting of colored permutation groups $\mathbb{Z}_r \wr S_n$.
• To define and study new descent statistics—(c,d)-descents—based on the colors of adjacent elements in a permutation.
• To compute the exact distribution of these statistics using generating functions and combinatorial recurrences.
• To extend previous results on signed permutations in $B_n$ to the full family of colored permutation groups $G_{r,n}$.
• To provide explicit closed-form formulas for the number of colored permutations with exactly $m$ $(c,d)$-descents for all $c \leq d$ and $(c,c)$-descents.

Proposed method

• Define $(c,d)$-descents in $G_{r,n}$ as positions $i$ where $\pi(i)$ has color $c$ and $\pi(i+1)$ has color $d$, with $c \leq d$.
• Use exponential generating functions $G_r(x,q)$, $G_r^+(x,q)$, and $G_r^-(x,q)$ to model the distribution of $(c,d)$-descents.
• Establish recurrence relations for $g_{r,n}(q)$, $g_{r,n}^+(q)$, and $g_{r,n}^-(q)$ based on the color of the first element.
• Apply generating function manipulations to derive closed-form expressions for the coefficients of $x^n q^m$.
• Use the generating function $A_r(x,q) = \frac{1}{1 - rx - (q-1)x^2}$ to model $(c,d)$-descents for $c < d$.
• Apply binomial expansions and coefficient extraction techniques to derive explicit formulas involving sums over indices $j$, $i$, $k$.

Experimental results

Research questions

• RQ1 How can classical descent statistics be generalized to colored permutation groups $\mathbb{Z}_r \wr S_n$ using color-based adjacency conditions?
• RQ2 What is the exact number of colored permutations in $G_{r,n}$ with exactly $m$ $(c,d)$-descents for $c \leq d$?
• RQ3 What is the distribution of $(c,c)$-descents in $G_{r,n}$, and how does it relate to the structure of the group?
• RQ4 Can the generating functions for these descent statistics be expressed in closed form, and what algebraic structure underlies them?
• RQ5 How do the results for $G_{r,n}$ specialize to known results in $B_n$ (e.g., for $r=2$) and $S_n$?

Key findings

• The number of colored permutations in $G_{r,n}$ with exactly $m$ $(c,d)$-descents for $c < d$ is given by $n! \sum_{j=0}^{n-2m} \binom{m+j}{j} \binom{j}{n-2m-j} r^{2j+2m-n} (-1)^{n-j}$.
• For $c = d$, the number of permutations in $G_{r,n}$ with exactly $m$ $(c,c)$-descents is $n! \sum_{j=0}^{n-m} \sum_{i=0}^{j} \sum_{k=0}^{i} \binom{j}{i} \binom{i}{k} \binom{n-j}{m} \frac{(-1)^{n+m+j}(1-r)^{i-k}k^{n-j+i}}{(n-j+i)!}$.
• The generating function for $(c,d)$-descents with $c < d$ is $A_r(x,q) = \frac{1}{1 - rx - (q-1)x^2}$, which yields the closed-form coefficient for $x^n q^m$.
• Substituting $r=2$ into the $(c,d)$-descent formula recovers the known result for $B_n$: the number of signed permutations with $m$ pn-descents is $n! \binom{n+1}{n-2m}$.
• The number of permutations with $m$ positive or negative descents in $B_n$ is given by a complex sum involving binomial coefficients and alternating signs, matching the general formula for $r=2$.
• The distribution of $(c,c)$-descents is invariant under color shift, so the count for any color $c$ equals that for color $0$, simplifying enumeration.