22 problems

Proctor's conjecture. The minuscule posets are the only posets for which $F_P(m;q)$ has a product formula of this form.

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…

The q-analogue conjecture. For every such $n$ and $k$,

Let $\mathrm{PP}^{m}(n\times n)$ denote plane partitions in an $n\times n$ box of height at most $m$, let $\mathrm{Tr}$ denote transposition, and for a transpose-invariant plane pa…

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…

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…

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 $FS_{s,t}$ be the set of $t$-Fuss–Catalan paths of semilength $s$, and let $FC_{s,t}$ denote the same path set. Let $\operatorname{istat}$ be a statistic on these paths, and le…

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 $\sigma$,…

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…

Let $w$ be a permutation, and let the counting function for matrices of rank $r$ whose support avoids the permutation diagram $O_w$ be the associated $q$-analogue of placements of…

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…

Let $n$ be a positive integer, let $\mu=(\mu_1,\mu_2,\ldots,\mu_k)\vdash n$, and let $E_1,E_2,\ldots,E_k$ be successive segments of the word $123\cdots n$ of respective lengths…

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…

Let $\operatorname{Av}_n(321)$ denote the permutations of length $n$ avoiding the pattern $321$, and let the major-index polynomial sequences be the corresponding generating polyno…

Gaussian poset characterization conjecture. Every Gaussian poset is minuscule. The converse—that every minuscule poset is Gaussian—is also asserted in the source. This conjecture p…