15 problems

Let $EN_{s,t}(\sigma)(q)$ denote the inversion-generating function for linear extensions of the rectangular poset with parameters $s,t$ that avoid the permutation pattern $\sigma$,…

Let $EN_{s,t}(\sigma)(q)$ denote the inversion-generating function for pattern-avoiding linear extensions of the rectangular poset with parameters $s,t$. The 2143 two-column invers…

Let $F_s(q)$ be defined by … and, for $s\ge 2$, … Let $EN_{s,t}(\sigma)(q)$ denote the inversion-generating function for pattern-avoiding linear extensions of the rectangular poset…

Gaussian-product unimodality conjecture. For every $0\leq k<n$, this polynomial is symmetric and unimodal. The source states that, using known results together with its preceding t…

Let $G$ and $G'$ be the symmetry groups specified in the claim, and write $M_G(a,b,c)_{-1}$ and $A_G(a,b,c)_{-1}$ for the corresponding matrices specialized at $q=-1$. Let…

For the explicitly defined $q$-weighted Kasteleyn and Kasteleyn-Percus matrices $M(a,b,c;q)$, $M_\rho(a,a,a;q)$, $A_\tau(a,b,b;q)$, $\widetilde{A}_\tau(a,b,b;q)$, and…

Stanton's conjecture. The polynomials $f(n,k;q)$ and $a_\mu(q)$ satisfy

Let $\delta_n$ denote the staircase shape, let $\operatorname{RPP}(\delta_{n+2k}/\delta_n)$ be the reverse plane partitions of the skew shape $\delta_{n+2k}/\delta_n$, and let…

Let $EN_{s,t}(\sigma)[q]$ and $NE_{s,t}(\sigma)[q]$ denote the major-index generating functions for the indicated pattern-avoiding linear extensions of rectangular posets. Let…

The coefficient-of-$q^{2s-2}$ conjecture. For all $s\ge2$, the coefficient of $q^{2s-2}$ in $F_s(q)$ is the number of compositions of $s+9$ into $s$ parts, none of which is $2$ or…

Given an ordered set partition $\pi\in\mathcal{OP}_n$, define its shape $\operatorname{shape}(\pi)$ to be the composition whose $i$th part is the size of the $i$th block of $\pi$ f…

Let $w$ be a permutation in the class avoiding $4231$, $35142$, $42513$, and $351624$. Let $PF^{{\rm L}}_w(q)$ and $PF^{\rotatebox[origin=c]{180}{\scriptsize {\rm L}}}_w(q)$ be the…

Fix $a,b\in\mathbb{N}$. Let $\mathrm{R}(a,b)$ be the set of partitions fitting in the $b\times a$ rectangle, let $\mathsf{ml}_{b,a}(\mu)$ be the minimum level on the frontier of…

For a permutation $\sigma\in S_n$, let $\operatorname{inv}(\sigma)$ and $\operatorname{exc}(\sigma)$ denote its inversion number and excedance number, respectively, and define … Th…

For each integer $k\geq 0$, let $M_{2^k-1}(321;q)$ denote the major-index generating polynomial associated with 321-avoiding permutations of length $2^k-1$, and write…