10 problems

Gessel–Zhuang's conjecture. Every shuffle-compatible statistic is a descent statistic.

Let $(W,S)$ be a Coxeter system with length function $\ell:W\to\mathbb N$. For $w\in W$, let $T(w)=\{t\in T\mid \ell(tw)<\ell(t)\}$ be its inversion set, where…

For a permutation $\tau$, let $\operatorname{Destop}(\tau)$ and $\operatorname{Desbot}(\tau)$ denote its descent top and descent bottom sets. Patterns are…

Let $S_4$ be the set of permutations of length $4$. For a permutation $\tau$, define its descent bottom set by … Two patterns are Desbot-Wilf equivalent if their avoidance classes…

Let $dd(I;n)$ denote the number of permutations in the symmetric group on $n$ letters whose double descent set is $I$. Let $I,J\subset\mathbb{Z}_{\geq2}$ be finite sets such that…

Let $dd(I;n)$ denote the number of permutations in the symmetric group on $n$ letters whose double descent set is $I$. For fixed $i,j\in\mathbb{Z}_{\geq2}$, the positive ratio-limi…

Let $dd(I;n)$ denote the number of permutations in the symmetric group on $n$ letters whose double descent set is $I$. For $n,i\in\mathbb{N}$ with…

Let $dd(I;n)$ denote the number of permutations in the symmetric group on $n$ letters whose double descent set is $I$. Given a fixed $n\in\mathbb{N}$, the down-up-down-up conjectur…

Let $dd(I;n)$ denote the number of permutations in the symmetric group on $n$ letters whose double descent set is $I$. For fixed $0<\alpha<\beta<1$, the asymptotic equidistribution…

Let a permutation statistic be a statistic on permutations, and call it shuffle-compatible if the multiset of its values on shuffles of two permutations depends only on the two inp…