17 problems

Let $P$ be a finite poset on $n$ elements, and let $\mathcal{O}(P)$ and $\mathcal{C}(P)$ denote its order polytope and chain polytope, respectively. For each $k=0,\dots,n-1$, write…

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…

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…

Lee–Vindas-Meléndez–Wang conjecture. The Ehrhart $h^*$-polynomial $h^*(\mathcal{O}(P),t)$ is real-rooted. This conjecture concerns the root structure of Ehrhart $h^*$-polynomials f…

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 $P$ be a finite poset and let $\mathcal{O}(P)$ denote its order polytope. Its Ehrhart $h^{*}$-polynomial is the numerator of the Ehrhart series of $\mathcal{O}(P)$. Neggers' co…

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 $S_n$ 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…

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…

Let $\mathbf{w} in \mathcal{V}$, and let $\mathcal{O}(Q_{\mathbf{w}})$ be the order polytope associated with the generalized snake poset. Its secondary polytope has vertices corres…

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.

Lower-dimensional Ehrhart realization conjecture. For any order $d$-polytope $O_{\Pi}$ different from the $d$-cube $C_d$, there is a lattice polytope $P_{\Pi}$ of dimension $d-2$ s…

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

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