13 problems

Let $P$ be a poset on $n$ elements, and let $\Omega(P;t)$ be its order polynomial. Kahn–Saks conjecture. The map … is weakly decreasing for positive integer values of $t$. The conj…

Let $\lambda/\mu$ be a skew shape with $\lambda_1\geq\ell(\lambda)$, and let $P_{\lambda/\mu}$ be its cell poset. For $S=\{0,\ldots,\lambda_1\}$, write $M[S]$ for the class of poly…

Let $W$ be a finite Coxeter group, let $w\in W$, and let $\Omega_W(w;t)$ denote the order polynomial associated with $W$ and $w$. Assume that $t_W$ is a linear function depending o…

Let $P$ be a circular fence poset of size $n$, let $S_n(P)$ denote the relevant set of permutations, and let $\widehat{bl}_P(\sigma)$ be the block statistic introduced for circular…

Let $P$ be a skew shape poset of size $n$, let $S_n(P)$ denote the set of $P$-partitions of the labeling set represented by permutations, and let $\widetilde{bl}_P(\sigma)$ be the…

Let $\lambda$ and $\mu$ be strict partitions such that $\lambda/\mu$ is a shifted skew shape, and let $Q_{\lambda/\mu}$ be its underlying cell poset. Write…

Let $P_{\lambda/\mu/d}$ be the cell poset associated with a cylindric skew shape $\lambda/\mu$ of period parameter $d$, and let $\Omega(P_{\lambda/\mu/d};t)$ be its order polynomia…

Let $\lambda/\mu\subset [m]\times[n]$ be a skew shape encoding a width-two poset $R_{\lambda/\mu}$, with $m+n$ elements. Width-two positivity conjecture. … Thus the normalized orde…

Let $\Omega(P;t)$ denote the order polynomial of a finite poset $P$. The classes under consideration are posets of width two, cell posets of cylindric skew shapes, and cell posets…

A poset $P$ has order polynomial $\Omega(P;t)$, whose coefficients are considered after normalization by $|P|!$. A family of posets is stable under ideals when it is closed under t…

Let $P=(X,\prec)$ be a finite poset, and let $t\geq k\geq 1$ be positive integers. Chan–Panova inductive conjecture. There exists $x\in X$ such that … This conjecture was proposed…

Let $P=(X,\prec)$ be a finite poset, let $x\in X$, and let $a,t\in\mathbb N_{\geq 1}$ with $1<a<t$. Write $\Omega(P,t;x,a)$ for the number of order-preserving maps $g:X\to[t]$ sati…

Let $P=(X,\prec)$ be a finite poset, and let $t\geq k\geq 1$ be positive integers. Kahn–Saks–FKG conjecture. There exists $x\in X$ such that … This conjecture is motivated by an in…