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Web permutations, Seidel triangle and normalized $γ$-coefficients

Yao Dong, Zhicong Lin, Qiongqiong Pan

TL;DR

The paper addresses the problem of giving a combinatorial interpretation for Seidel triangle entries in the context of web permutations and relates the normalized $\gamma$-coefficients of the $(\alpha,t)$-Eulerian polynomials to cycle-up-down statistics. It develops a chord-diagram framework that translates smoothing/switching processes into diagram expansions and uses Nakamigawa’s results to match these counts with Seidel-triangle numbers, culminating in a full combinatorial proof of the Hwang–Jang–Oh conjecture. It also introduces two commuting group actions (Foata–Strehl and block-Foata–Strehl) to prove a $\gamma$-coefficient identity and constructs a min-max-tree–based bijection $\Lambda$ between $Web_n$ and $\Delta_n$, linking drop with a new mix statistic. Together, these results unify web-permutation theory, chord-diagram expansions, and refined Eulerian polynomial statistics, providing explicit combinatorial interpretations and equidistribution results that advance understanding in algebraic combinatorics.

Abstract

The web permutations were introduced by Hwang, Jang and Oh to interpret the entries of the transition matrix between the Specht and $\mathrm{SL}_2$-web bases of the irreducible $§_{2n}$-representation indexed by $(n,n)$. They conjectured that certain classes of web permutations are enumerated by the Seidel triangle. Using generating functions, Xu and Zeng showed that enumerating web permutations by the number of drops, fixed points and cycles gives rise to the normalized $γ$-coefficients of the $(α,t)$-Eulerian polynomials. They posed the problems to prove their result combinatorially and to find an interpretation of the normalized $γ$-coefficients in terms of cycle-up-down permutations. In this work, we prove the enumerative conjecture of Hwang-Jang-Oh and answer the two open problems proposed by Xu and Zeng.

Web permutations, Seidel triangle and normalized $γ$-coefficients

TL;DR

The paper addresses the problem of giving a combinatorial interpretation for Seidel triangle entries in the context of web permutations and relates the normalized -coefficients of the -Eulerian polynomials to cycle-up-down statistics. It develops a chord-diagram framework that translates smoothing/switching processes into diagram expansions and uses Nakamigawa’s results to match these counts with Seidel-triangle numbers, culminating in a full combinatorial proof of the Hwang–Jang–Oh conjecture. It also introduces two commuting group actions (Foata–Strehl and block-Foata–Strehl) to prove a -coefficient identity and constructs a min-max-tree–based bijection between and , linking drop with a new mix statistic. Together, these results unify web-permutation theory, chord-diagram expansions, and refined Eulerian polynomial statistics, providing explicit combinatorial interpretations and equidistribution results that advance understanding in algebraic combinatorics.

Abstract

The web permutations were introduced by Hwang, Jang and Oh to interpret the entries of the transition matrix between the Specht and -web bases of the irreducible -representation indexed by . They conjectured that certain classes of web permutations are enumerated by the Seidel triangle. Using generating functions, Xu and Zeng showed that enumerating web permutations by the number of drops, fixed points and cycles gives rise to the normalized -coefficients of the -Eulerian polynomials. They posed the problems to prove their result combinatorially and to find an interpretation of the normalized -coefficients in terms of cycle-up-down permutations. In this work, we prove the enumerative conjecture of Hwang-Jang-Oh and answer the two open problems proposed by Xu and Zeng.

Paper Structure

This paper contains 7 sections, 13 theorems, 51 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Expansions of chord diagrams and proof of conjecture \ref{['conj:hwang']}
  3. Web permutations
  4. Chord diagrams
  5. Proof of conjecture \ref{['conj:hwang']}
  6. A group action proof of \ref{['divi:gam:at']}
  7. The bijection $\Lambda$ and proof of Theorem \ref{['thm:pro2xz']}

Key Result

Theorem 1.1

A permutation $\sigma\in\mathfrak{S}_n$ is a web permutation iff each cycle of $\sigma$ is an André cycle.

Figures (12)

  • Figure 1: The Seidel triangle $s_{i,j}$ for $i\leq 9$ and $j\leq 5$.
  • Figure 2: $(a)$, $(b)$ and $(c)$ represent the grid configurations $G(1324,\emptyset)$, $G(1324,\{(1,4)\})$ and $G(3124,\{(1,4)\})$, respectively.
  • Figure 3: A crossing to an elbow.
  • Figure 4: The process of resolving maximal crossings when $n=3$
  • Figure 5: The matching $M(1324)=\{\{0,1\},\{2,5\}, \{3,4\},\{6,7\}\}$.
  • ...and 7 more figures

Theorems & Definitions (21)

  • Theorem 1.1: Hwang, Jang and Oh
  • Conjecture 1.2: Hwang, Jang and Oh HJO
  • Theorem 1.3: Xu and Zeng
  • Theorem 1.6
  • Remark 1.7
  • Definition 1
  • Proposition 2.1: See HJO
  • Lemma 2.3: See TN16
  • Lemma 2.4: See TN
  • Theorem 2.5: Nakamigawa TN
  • ...and 11 more