36 problems

Type-C identities.

Let $t=\varphi(s)$ be a meromorphic function on $\mathbb{C}$, and let $\zeta_{q,r}(s,t,z;\boldsymbol{\omega})$ and $\zeta_r(s,z;\boldsymbol{\omega})$ denote the corresponding $q$-a…

For integers $k$ and $n$ in the range where the polynomial $\hat{E}_{k,n}(q)$ is defined, let $[q]\hat{E}_{k,n}(q)$ denote the coefficient of $q$ in $\hat{E}_{k,n}(q)$. Linear-coef…

Generalized Q-Kostka positivity conjecture. If $\bar{\mu^*}$ is a partition, then $L_{\lambda;\mu^*}(q)$ is a polynomial in $q$ with non-negative integer coefficients. These coeffi…

Let ${\cal OP}_n^k$ be the set of ordered $k$-partitions of $[n]$, and let $\mathop{\rm bmaj}$ and $\mathop{\rm binv}$ be the block-major index and block-inversion statistics. Let…

Let ${\cal P}_n^k$ denote the set of partitions of $[n]$ into $k$ blocks, and let $\mathop{\rm mak}$, $\mathop{\rm lmak}$, $\mathop{\rm mak}{}'$ and ${\mathop{\rm lmak}}'$ be the f…

Let $\tau$ be a preorder of size $n$, let $P_\tau$ be the poset of lattice points associated with its preorder polytope, and let $\mathcal Z_q(P_\tau,t)$ be its $q$-zeta polynomial…

Let $n$ be an even positive integer and let $F_q^n$ be an $n$-dimensional vector space over the finite field with $q$ elements. Write $\mathrm{sub}(F_q^n)$ for its subspaces, and l…

For positive integers $a,b$ and $k$, define … where $c_d(k)_q=\sum_{s\mid\gcd(d,k)}\mu(d/s)[s]_q$. Here $[n]_q$ is the $q$-integer, $\mu$ is the Möbius function, and…

Let $V$ be an $n$-dimensional vector space over $\mathbb{F}_q$, and let $k \ge d \ge 3$ be integers satisfying … For a family $\mathcal{F} \subset \genfrac{[}{]}{0pt}{}{V}{k}$, cal…

Refined type A Hilbert-series conjecture. For $1\leq j\leq n-1$,

Zabrocki's conjecture. For $0\leq k\leq n-1$,

Let $\mathcal{B}=\\{s\in\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{Z}}:s\text{ is balanced}\\}$ be the set of all balanced bi-infinite binary sequences, and let … be their language of fi…

Let $\mu_q$ be the $q$-analog of the monoid homomorphism used to define $q$-Markoff numbers, and let $\mu_q(w)_{12}\in\mathbb{Z}[q]$ denote the corresponding entry above the diagon…

Let $\tilde{\mathsf{F}}(q,y,z)$ be the generalized forest matrix, and define $\bar{\mathsf{F}}(q,y,z)=\tilde{\mathsf{F}}(q,y,z)\operatorname{diag}((q^{k(k-1)/2})_{k\geq0})$. Let…

Bachmann's bi-bracket reduction conjecture. Every bi-bracket can be written as a $\mathbb{Q}$-linear combination of mono-brackets; equivalently,

Let $B_n$ be the hyperoctahedral group of signed permutations, and for $\sigma\in B_n$ let $\mathsf{fwex}(\sigma)$, $\mathsf{\neg}(\sigma)$, $\mathsf{nest}_B(\sigma)$, and…

Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$, and let $\lambda$ and $\mu$ be weights such that the multiplicity $m(\lambda,\mu)$ of $\mu$ in the irreducible representatio…

The signed log-concavity conjecture. The polynomial $\tilde{A}_n(t,-1)$ is log-concave for $n\geq1$.

Inversion-major-index equidistribution conjecture. For $n\geq 1$,

Fix a partition $\lambda\vdash n$, and let $c_\lambda(q,r)$ be the coefficient of $e_\lambda$ in the Frobenius characteristic $\Phi_{n,r}(q)$. Write $\lambda_i=0$ for $i$ larger th…

Let $w\in\mathfrak{S}_n$ be a $123$-avoiding permutation, meaning that it contains no subsequence order-isomorphic to $123$, and let $I_w$ be its diagram. Write…

For a positive integer $n$, let $v=(2n-1)(2n)(2n-3)(2n-2)\cdots3412$ be the indicated permutation, and let $I_v$ be its diagram. Write $M_{2n}(\overline{I_v},q)$ for the matrix cou…

Let $B\subseteq[m]\times[n]$ be a board, let $r$ be a rank, and let $q$ be a prime power. The $q$-hit number $H_r(B,q)$ is defined for this board and rank. Weak positivity conjectu…

Let $m$ and $n$ be coprime positive integers. An $(m,n)$-core is a partition that is simultaneously an $m$-core and an $n$-core; denote the set of such cores by $\mathcal P_{m,n}$.…