61 problems

For a positive integer $n$, a tuple ${\bf r}=(r_1,\ldots,r_m)$ of nonnegative integers, and the polynomial ${\sf{M}}^n_{\bf r}(q)$ defined in the paper, the claim concerns its coef…

Ismail's conjecture. The sequence $(P_n)_{n\geq 0}$ satisfies

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$. Let $\genfrac{[}{]}{0pt}{}{V}{1}_q$ denote the family of $1$-dimensional subspaces of $V$, and let…

Ismail's conjecture. If

Let $G=SO_{2n+1}$, $Sp_{2n}$, or $SO_{2n}$, let $frak{V}( u)$ denote the restriction to $G$ of the irreducible finite-dimensional $GLN$-module of highest weight $u$, and let…

Let $G=GL_n$, let $I$ specify a Levi subgroup, and let $K_{\lambda,\mu}^{GL_n,I}(q)$ be the corresponding Kostant-type $q$-analogue of the generalized Littlewood–Richardson coeffic…

Let $u_{\lambda,\mu}(q)$ be the $q$-multiplicity associated with the type-$C$ crystal, and let a one-dimensional sum mean the corresponding polynomial obtained from the crystal ene…

Type-C identities.

Let $X_{\frac{1}{}}{\lambda,\mu}(q)$, $U_{\lambda,\mu}(q)$, and $K^{(2)}_{\lambda,\mu}(q)$ denote the polynomial $q$-multiplicities and one-dimensional sums introduced for the rele…

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…

Conjecture on nontrivial-zero trajectories. The limit

Conjecture on trivial-zero trajectories. The limit

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…

Parabolic positivity conjecture. If $\bar{\mu^*}$ is a partition, then $K_{\lambda;\mu^*}(q)$ has non-negative integer coefficients. These polynomials are introduced as $q$-analogs…

Let $G_{\lambda}[X;q]$ be the $q$-analog of the Schur $Q$-function indexed by a strict partition $\lambda$, and write the transition from the $G$-basis to the $Q$-basis as … The co…

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…

For each $n$, let $G_n(t)=\sum_{\rho}t^{\operatorname{exc}(\rho)}$, where the sum is over $123$-avoiding derangements of length $n$ and $\operatorname{exc}(\rho)$ is the number of…

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…

Let $\mathcal{Z}_q$ be the algebra of $q$-multiple zeta values, and let $\zeta_q(k_1,\ldots,k_d)$ denote multiple $q$-zeta values of the particular kind specified in the source.…

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 $m_q(\lambda,\mu)$ denote the $q$-multiplicity associated with a pair of weights $(\lambda,\mu)$ from the list provided by Berenshteu{}in and Zelevinskiu{}. Harris et al.'s con…

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