19 problems

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 $P$ 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 $\lambda/\mu\subset [m]\times[n]$ 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…

Let $D$ and $E$ be shifted skew shapes. A frayed ribbon is a shifted near-ribbon containing two squares on the staircase, and let $D^a$ denote the antipodal reflection of $D$ acros…

Let $S$ be a skew shape, and let $k\geq 1$. For a filling of $S$, write $\operatorname{Tr}(S,\pi)$ for the number of transversal fillings avoiding the filling pattern $\pi$, and le…

Let $\mathcal{P}_N$ be the poset of skew shapes with $N$ cells, ordered by Schur-positivity, and let $P_{R,S}$ be the box diagonal diagram with $R$ rows, $S$ columns, and $N=R+S-1$…

Shifted skew-shape product conjecture. One has

Asymptotic constant conjecture. There is a constant $c=c(\overline\pi)$ such that

Krattenthaler–Panova conjecture. The number of tableaux is

Let $\delta_k$ denote the staircase shape, and let $\upsilon_k=\delta_{2k}/\delta_k$ be the thick ribbon. Write $n=|\upsilon_k|$. Thick-ribbon asymptotic conjecture. For some const…

Let $\sigma\subset\tau$, where $\tau=(\ell^k)$ is a rectangle, and let $\sigma^\ast$ be the $180$-degree rotation of $\sigma$. Form the skew shape $\lambda/\mu$ by attaching two co…

For skew shapes $A$ and $B$, let $A\leq B$ mean that $a_{A,\nu}\leq a_{B,\nu}$ for every partition $\nu$, equivalently that $G_B-G_A$ is $G$ positive. Transposition sends a skew sh…

Let $\delta_n=\langle n-1,n-2,\ldots,1\rangle$ be the staircase partition, and let $\mu$ be a diagram contained in $\delta_n$. For a skew shape $A$, write $A^T$ for its transpose.…

Let $A$ and $B$ be skew shapes of the same size. For each positive integer $k$, let $\mathrm{rows}_{k}(A)$ and $\mathrm{rows}_{k}(B)$ denote their row-overlap partitions, and let…

Let $N$ be fixed, let $\lambda\vdash N$, and consider the subposet of $\mathcal{P}_{N}$ consisting of ribbons $R$ whose row partition is $\operatorname{rows}(R)=\lambda$. For a rib…

Fix $N$ and the number of rows $l$, so that an equitable ribbon has $N-l+1$ columns, and restrict to equitable ribbons whose rows all have length at least $2$. Write…

Let $N$ be a positive integer, let $\mathcal{P}_{N}$ be the poset of skew shapes of size $N$ under Schur-positivity, and restrict to connected skew shapes. Two skew shapes are iden…

Let $M_n$ denote the $n$th Motzkin number. For each partition $\lambda$ with at most three parts, consider standard Young tableaux of skew shape $\lambda/(2,1)$ having $n$ entries.…