Distribution of sets of descent tops and descent bottoms on restricted permutations

TL;DR

This work investigates how descent-top and descent-bottom sets distribute over pattern-avoiding permutations, focusing on refined equivalences beyond classical Wilf equivalence. It develops Destop/Desbot statistics, proves Destop-Wilf equivalences among the length-3 patterns $132$, $231$, $312$, and constructs bijections that preserve these statistics, including a joint $(\mathrm{Destop},\mathrm{Desbot})$-Wilf bijection for $\mathrm{Av}(312)$ and $\mathrm{Av}(231)$. A Vincular-pattern framework is introduced via a Françon–Viennot–style bijection and a signature function $\mathrm{sg}_{n,(T,B)}$ to obtain a closed product formula for the joint distribution of $(2\underline{31},\underline{31}2)$ on fixed $(\mathrm{Destop},\mathrm{Desbot})$, yielding equidistribution results between $231$- and $312$-avoiders. For length-4 patterns, the authors propose a shape-Wilf extension and conjecture several nontrivial Destop/Desbot equivalences, with recent multi-statistic Wilf-equivalence results by Zhou, Zang, and Yan (2024) supporting these refinements and connecting to Ferrers-board analogues. Overall, the paper advances a refined understanding of how descent-related statistics distribute across pattern-avoiding permutations and frames several conjectures now partially settled by recent work.

Abstract

In this note, we prove some and conjecture other results regarding the distribution of descent top and descent bottom sets on some pattern-avoiding permutations. In particular, for 3-letter patterns, we show bijectively that the set of descent tops and the set of descent bottoms are jointly equidistributed on the avoiders of 231 and 312. We also conjecture similar equidistributions for 4-letter patterns, in particular, that the set of descent tops and the set of descent bottoms are jointly equidistributed on the avoiders of 3142, 3241, 4132. This conjecture and several others made in this paper have now been proved by Zhou, Zang, and Yan (2024).

Figures (3)

• Figure 1: Direct sum $\sigma\oplus\tau$ and skew-sum $\sigma\ominus\tau$ of permutations $\sigma$ and $\tau$.
• Figure 2: Bijections $\Phi$ and $\Psi$ of Theorem \ref{['thm:destop-132-231-312']}. Only the case $\sigma',\sigma"\ne\emptyset$ is shown for $\Phi$. For the remaining cases, see Figure \ref{['fig:sum-skewsum']}.
• Figure 3: An instance of $(\mathop{\mathrm{Destop}}\nolimits,\mathop{\mathrm{Desbot}}\nolimits)$-preserving bijection from Theorem \ref{['thm:231-312-destop-desbot']} between $\mathop{\mathrm{Av}}\nolimits(312)$ and $\mathop{\mathrm{Av}}\nolimits(231)$ with descent matching $(T,B)=(\{2,5,8,9\},\{1,2,3,7\})$ and size $n=9$.

Theorems & Definitions (26)

• Theorem 1
• proof
• Definition 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• proof : of Theorem \ref{['thm:231-312-destop-desbot']}
• Example 6