25 problems

Guo–Zeng conjecture. For every $n\in\mathbb{N}$, there exist coefficients $a_{n,i}\in\mathbb{N}$ such that

Summed edge-deletion nonnegativity conjecture. The polynomial $Z_G(t)$ has nonnegative coefficients. This is proposed as a stronger statement whose gamma-nonnegativity would imply…

Let $G$ be a 2-connected graph with edge set $E$, and let $P_G$ denote its symmetric edge polytope. For an edge $e\in E$, let $P_{G\setminus e}$ be the symmetric edge polytope of t…

Let $W$ be a Coxeter group and let $u<v$ be elements of $W$. The Coxeter Chow polynomial $\mathrm{H}_{uv}$ is the polynomial associated with the Bruhat interval $[u,v]$; it is…

Let $P=[m]$, let $Q$ be a subposet of $P\times[n]$, and let $C^{Q,\operatorname{id}}(x)$ denote the dissonant canon polynomial associated with $Q$. Gamma-positivity conjecture. The…

For $n\geq1$ and $S\subseteq[n]$, let $\operatorname{Pin}(\pi)$ be the set of values at peak positions of $\pi\in\mathfrak{S}_n$, and define … A polynomial is $\gamma$-positive wit…

For $n\geq1$ and $0\leq k\leq\lfloor(n-1)/2\rfloor$, define … A symmetric polynomial with center of symmetry $(n+1)/2$ is $\gamma$-positive when its expansion in the corresponding…

Let $\Delta$ be a flag homology ball, and suppose that its boundary $\partial\Delta$ is an induced subcomplex of $\Delta$. Let $\theta(\Delta,x)$ denote the polynomial introduced f…

For $m1$, define the BorosMoll polynomial $M_m(x)$ and its transform … A polynomial is symmetric if its coefficients read the same forwards and backwards, unimodal if its coefficie…

Ohsugi–Tsuchiya conjecture. For every $i\geq 0$,

For a multiset $\mathcal{M}$, let $\overline{\mathcal{Q}}_{\mathcal{M}}$ denote the set of quasi-Stirling permutations of $\mathcal{M}$, and define … where $asc(\pi)$, $des(\pi)$,…

Let $\mathfrak{S}_n$ be the symmetric group, and define the two-sided alternating Eulerian polynomial by … Two-sided alternating gamma-expansion conjecture. For every $n\geq 1$, ……

Let $\mathfrak{S}_n$ be the symmetric group, and let $\operatorname{altdes}(\pi)$ and $\operatorname{altmaj}(\pi)$ denote the alternating descent number and alternating major index…

Let $n\geq 1$ and $0\leq i\leq n$. Write … for the trivariate extension of $A_{n,i}(y)$. A trivariate polynomial $p(x,y,z)=\sum_i s_i(x,y)z^i$ is partial $\gamma$-positive when eve…

Let ${\mathcal A}_G$ be a symmetric edge polytope of type A, and let $h^*({\mathcal A}_G,x)$ denote its $h^*$-polynomial. Type A symmetric edge-polytope conjecture. The $h^*$-polyn…

Let ${\mathcal P}\subset {\mathbb R}^d$ be a locally anti-blocking reflexive polytope, and let $h^*({\mathcal P},x)$ denote its $h^*$-polynomial. Locally anti-blocking reflexive po…

For every positive integer $n$, let $[r]$ denote a chain with $r$ elements and let $\mathcal{N}_{[2]\times[n]\times[n+1]}(x)$ be the antichain generating polynomial of the product…

For $n\equiv 0,1\pmod 4$, let $\mathcal{A}_n$ be the alternating subgroup of $\mathfrak{S}_n$, and define … For even $n$, define the type-$B$ polynomials $\mathrm{TSB}_n^+(s,t)$ an…

Let $P$ be a naturally labeled, graded poset with nonnegative rank function $\rho$, and set ${\boldsymbol s}=\rho+1$. Let $O(P,{\boldsymbol s})$ denote the corresponding…

Let $W$ be a finite crystallographic Coxeter group, and let $\operatorname{Cox}(W)$ be its Coxeter complex. Equivariant Gal conjecture. The pair $(\operatorname{Cox}(W),W)$ exhibit…

Let $V$ be an $n$-element set, let $2^V$ denote the simplex on $V$, and let $\Gamma$ be a flag triangulation of $2^V$. Let $ell_V(\Gamma,x)$ denote its local $h$-polynomial. Athanas…

For each positive integer $n$, let $I_n(x)$ denote the polynomial defined by the paper through the descent enumerator of standard Young tableaux with $n$ squares. Guo–Zeng's conjec…

Let $W$ be a finite Coxeter group of rank $n$, and let … be its two-sided Eulerian polynomial. Generalized Gessel's conjecture. There exist nonnegative integers $\gamma_{a,b}^W$ su…

Let $D4AE_n$ denote the symmetric group on $n$ letters. Let $D4AE_n(231)$ be the set of permutations avoiding the pattern $231$, and let $S_n(t)$ be its descent polynomial. Write ……

Blanco–Petersen's conjecture. There exist polynomials $\gamma_{n,k}(q)$ with nonnegative integer coefficients such that