14 problems

Hibi–Li conjecture.

Let $P$ be a finite poset, let $F$ be uniformly distributed in its order polytope, let $H(x)=\mathbb E F(x)$, and let $\operatorname{Win}(x)=\mathbb E[R(x)-Q(x)]$, where…

Let $\mathbf{w}$ be a generalized snake word of length $n+1$. Let $h^*(\mathbf{w};z)$ denote its $h^*$-polynomial, and let $L(\mathcal{O}(P(\mathbf{w}));t)$ denote the Ehrhart poly…

Let $P$ be a finite poset with $|P|=d>1$. Write $f_i(Q)$ for the number of $i$-dimensional faces of a polytope $Q$, and let ${\mathcal O}(P)$ and ${\mathcal C}(P)$ denote the order…

Let $S_n$ denote the poset used in the paper, and let $\mathcal{O}(S_n)$ be its order polytope. Let $\operatorname{Cat}(m)$ denote the $m$th Catalan number. Odd-Catalan enumeration…

Let $\mathbf{w} in \mathcal{V}$, and let $\mathcal{O}(Q_{\mathbf{w}})$ be the associated order polytope. A triangulation is regular if it arises from a lifting function, equivalent…

Assume $J(Q_{\mathbf{w}})=\widehat{P}(\mathbf{w})$, where $\mathbf{w}=\varepsilon L R^{n-2}$ and $ngeq 3$. Consider triangulations of $\mathcal{O}(Q_{\mathbf{w}})$ and their dual gr…

Let $Q_{\mathbf{w}}$ be a poset with Jordan--Hölder poset $J(Q_{\mathbf{w}})=\widehat{P}(\mathbf{w})$, and let a turn mean the specified local turn in the generalized snake poset.…

Let $\mathbf{w} in \mathcal{V}$, and consider the flip graph whose vertices are the regular triangulations of the order polytope $\mathcal{O}(Q_{\mathbf{w}})$. Let $k$ denote the d…

Antichain obstruction conjecture. If $\operatorname{\mathcal{O}}(\operatorname{\mathscr{P}})$ is projectively unique, then $\operatorname{\mathscr{P}}$ has no antichain of size $3$…

Unranked graphic order polytope conjecture. The order polytope $\operatorname{\mathcal{O}}(\operatorname{\mathscr{P}})$ is graphic.

Stapledon decomposition conjecture. The polynomial $p_{\Pi}(z)$ can be decomposed as

Stapledon decomposition conjecture. The polynomial $h_G(z)$ can be decomposed as

Let $P$ and $Q$ be finite posets with $|P|=|Q|=d$. For the toric ideal $I_{\Omega(P,-Q)}$ associated with the twinned order polytope $\Omega(P,-Q)$, quadratic generation conjecture…