6 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…

Let $E$ be an $n$-element set and let $\tau$ be an arbor on $E$. Its $h$-polynomial is $h(\tau,t)=\sum_{i=0}^n h_i(\tau)t^i$, where $h_i(\tau)$ counts the lattice points of the arb…

Let $P$ be a finite connected poset with an autonomous subposet that is a chain $C_n$, and let $A_1,\ldots,A_n$ be antichains replacing the elements of that chain. Nguyen's conject…

Let $A^{(2)}_n(z)$ denote the $1/2$-Eulerian polynomial, and let $\Delta(\mathcal{Q}^+(v))$ be the positive signed sum system associated to a generic weight $v$. Tolosa Villareal's…

Let $V$ be a finite set, let $2^V$ denote the simplex on $V$, and let $\Gamma$ be a triangulation of $2^V$. The local $h$-polynomial of $\Gamma$, denoted $\ell_V(\Gamma,x)$, is the…

Let $\Phi$ be an irreducible root system of type $A_n$, and let $\Gamma(\Phi)$ be its cluster subdivision. Its local $h$-polynomial is … where $I$ indexes the simple roots,…