30 problems

Let $\tau$ be a preorder of size $n$, and let $h(\tau,t)=\sum_{i=0}^n h_i(\tau)t^i$ be its $h$-polynomial. Gamma-positivity conjecture. There are nonnegative integers…

Gessel's conjecture. The polynomial $A_n(s,t)$ has the symmetry property of being gamma-positive: it can be expressed as a non-negative integer linear combination of the bivariate…

Ohsugi–Tsuchiya conjecture. The Ehrhart $h^*$-polynomials of symmetric edge polytopes are $\gamma$-positive.

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…

A classical symmetric edge polytope is the symmetric edge polytope associated with a graph, and its $h^\ast$-polynomial is the Ehrhart $h^\ast$-polynomial of that polytope. Gamma-p…

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 $V$ be a finite 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. Athanasiadi…

Peterson's generalized Gessel conjecture. The generating polynomials of the $W$-Eulerian numbers $N_{ij}$ are $\gamma$-positive.

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…

Gamma-positivity conjecture. Let $p>1$ and $q\geqslant 1$ be two given real numbers. Then the two polynomials in the symmetric decomposition of…

Autonomous-subposet recurrence.

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…

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…

Let $\mathsf{M}$ be a matroid of rank $k$, and write its Z-polynomial in the form … where $\gamma_{\mathsf{M}}(t)$ is the associated gamma-polynomial. Gamma-positivity conjecture.…

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…

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…

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…

Sun's conjecture. For every integer $n\geq 1$, all coefficients $c_{n,j}$ are positive integers.

Dilks's conjecture.

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…