15 problems

Let $H_{\nu}^q=\widetilde{S_{\nu}}^q$ be the operator associated with the composition $S_{\nu}=S_{\nu_1}S_{\nu_2}\cdots S_{\nu_{\ell(\nu)}}$, and let $A_{\lambda}^{(k)}[X;q]$ denot…

Fix a prime power $q$ and $\varepsilon>0$. Let $V$ be an $n$-dimensional vector space over $\mathbb F_q$, let $\mathcal F\subseteq\operatorname{Gr}_k(V)$ be intersecting, and order…

Fix a prime power $q$. Let $V$ be an $n$-dimensional vector space over $\mathbb F_q$, let $k$ be sufficiently large, and let $\mathcal F\subseteq\operatorname{Gr}_k(V)$ be an inter…

Let $\tilde{\alpha}$ be the highest root of the Lie algebra of type $A_r$, and write $\alpha_{i,j}=\alpha_i+\alpha_{i+1}+\cdots+\alpha_j$ for $1\leq i\leq j\leq r$. Let…

Let $q$ be a prime power, let $k\in\mathbb{N}$, and let $\Gamma=\operatorname{GL}(n,\mathbb{F}_q)$. Define … The graph's independent sets are the $k$-intersecting families of inver…

A Steiner structure $\mathbb{S}_2(2,3,7)$ is a collection of $3$-dimensional subspaces of a $7$-dimensional vector space over $\mathbb{F}_2$ such that every $2$-dimensional subspac…

Let $\binom{a}{b}^q$ denote the rank-generating polynomial for partitions with distinct parts inside the partition $\langle a-1,a-3,\dots,a-(2b-1)\rangle$. A polynomial is unimodal…

Let $n$ be a prime satisfying … A coset complete 3-dimensional subspace is a 3-dimensional subspace of $\mathbb{F}_2^n$ used as a generator for the construction described in the pa…

Let a Steiner structure $\mathbb{S}_q[t,k,n]$ be a set of $k$-dimensional subspaces of $\mathbb{F}_q^n$ such that every $t$-dimensional subspace of $\mathbb{F}_q^n$ is contained in…

Let a Steiner structure $\mathbb{S}_q[t,k,n]$ be a set of $k$-dimensional subspaces of $\mathbb{F}_q^n$ such that every $t$-dimensional subspace of $\mathbb{F}_q^n$ is contained in…

A Steiner structure $\mathbb{S}_q[t,k,n]$ is a set $\mathbb{S}$ of $k$-dimensional subspaces of $\mathbb{F}_q^n$ such that each $t$-dimensional subspace of $\mathbb{F}_q^n$ is cont…

Chow–Mansour's conjecture. For every $q>0$, the polynomial $G^{\operatorname{fmaj}}_{n,k}(x,q)$ has only real roots. The source attributes this conjecture to Chow and Mansour and d…

Let $A_1,\ldots,A_k$ be standard Young tableaux with $\operatorname{sh}(A_i)=\alpha_i\vdash a_i$ for $i\in[k]$. Let $\mathcal{T}_n(A_1,\ldots,A_k)$ be the set of $k$-tuples…

Let $P^+(n,m)$ be the set of dominant weights, and let $K_{\lambda,\mu}^{\mathrm{stab}}(q)$ denote the stable $q$-analog of the weight multiplicity for the orthosymplectic Lie supe…

Let $P^+(n,m)$ be the set of dominant weights, let $\rho$ be the relevant Weyl vector, and let $K_{\lambda,\mu}(q)$ denote the $q$-analog of the weight multiplicity for…