128 problems

Prove the six affine Jacobi–Trudi identities conjectured by Ole Warnaar for the relevant specialisations of multiparameter Hall–Littlewood polynomials of types $\mathrm{B}_n$,…

For every positive integer $k$ and every $k$-bounded partition $\lambda$, the $K$-$k$-Schur function $\mathsf{KSchur}^{(k)}_{\lambda}$ is a nonnegative integer linear combination o…

For every admissible multiset parameter $\mathbf{s}$, the associated multiset Eulerian–Narayana polynomial $E_{\mathbf{s}}(t)$ is real-rooted. More precisely, after removing its fa…

For every integer $n\ge 1$, every integer $1\le r\le n$, every real matrix $A=(a_{ij})\in\mathbb{R}^{n\times n}$ whose minors of orders at most $r$ are nonnegative, and every parti…

For every fixed arity $k$, every $k$-ary constraint-satisfaction problem having a Mal'tsev extension has linear non-redundancy; equivalently, its non-redundancy is $O(n)$ in the re…

Let $(W,S)$ be a Coxeter system with positive roots $\Phi^+$. For biclosed subsets $A,B\subseteq\Phi^+$ whose join exists in the biclosed-set order, Dyer's conjecture asserts that…

For every $n\ge 1$ and every pair of weak compositions $\alpha,\beta\in\mathbb{Z}_{\ge 0}^n$, the product of key polynomials admits a nonnegative expansion in the basis of Demazure…

For every $k,n\ge 1$, let $A_{k,n}^{(1)}(x)$ and $A_{k,n}^{(2)}(x)$ denote each specified pair of antichain-polynomial sums associated with $[k]\times Q$, where $Q=[2]\times[n]$. T…

Let $\mathcal{C}_n^k$ denote the $k$-uniform hypercycle on $n$ vertices. A hypergraph is called integral if all of its adjacency eigenvalues are integers. The conjecture states tha…

For every pair of coprime integers $1<a<b$, the matrix-counting $q$-series $Z_{a,b}(q)$ equals the explicit infinite product $P_{a,b}(q)$, namely $Z_{a,b}(q)=P_{a,b}(q)$. Equivalen…

For a nonzero real polynomial $r$ of degree $d$, define $\mathcal{W}(r)$ by $\sum_{n\ge 0} r(n)t^n=\frac{\mathcal{W}(r)(t)}{(1-t)^{d+1}}$. If the coefficients of both…

For every nonnegative integer $t$, every integer $m\ge 2$, and every family of pairs of finite sets $(A_i,B_i)_{i=1}^m$ satisfying $|A_i\cap B_i|\le t$ for all $i$ and…

For every pair of integers $1\le k\le n$, consider the refined Narayana polynomial and its formal degree-$k$ symmetric decomposition, written in the form…

Let $A$ be a finite-dimensional algebra and let $M$ be a finite-dimensional $A$-module. A brick-chain filtration of $M$ is a filtration whose factors are filtered by copies of bric…

For every finite connected weighted graph $G=(V,E,w)$, the spectral gap of the interchange process on $G$ equals the spectral gap of the continuous-time random walk on $G$:…

Does there exist a Neumaier graph whose coherent rank is exactly $5$? Equivalently, determine whether there exists a finite simple graph $G$ that is a Neumaier graph and whose cohe…

For every polynomial $f(t)\in\mathbb{Z}[t]$ that is monic, has nonnegative coefficients, is real-rooted, and is palindromic in the sense that $f(t)=t^d f(t^{-1})$ for $d=\deg f$, t…

For every toric $g$-contribution polynomial $p(t)$ in the family defined by Ehrenborg, Hetyei, and Readdy, all zeros of $p(t)$ are real; equivalently, $p(t)$ is real-rooted.

Let $k$ be a field, let $R_m=k[x_1,\ldots,x_m]$, and let $I(C_n)\subseteq R_n$ be the edge ideal of the cycle graph $C_n$. For $n\ge 4$ and $r\ge 0$, set…

For every lattice polytope $P$, the associated $q$-Ehrhart series $E_P(q,t)$ is a rational function and satisfies the corresponding $q$-reciprocity relation; moreover, the associat…

For every binary matrix $A$, let $A^\dagger$ be its transpose-complement, and let $\operatorname{ins}(A)$ and $\operatorname{rec}(A)$ denote the insertion and recording tableaux in…

The conjecture concerns a determinantal expression for a related antisymmetrizer arising in the study of symmetric functions and alternating-sign-matrix enumeration. The supplied s…

For every finite preorder $\tau$, the $h$-polynomial of $\tau$ equals that of its dual preorder: $h_{\tau}(t)=h_{\tau^{*}}(t)$, where $\tau^{*}$ denotes the dual preorder.

For every finite Coxeter group $W$ of rank $r$, the $h$-polynomial of the order complex of its noncrossing-partition lattice admits a nonnegative real-rooted symmetric decompositio…

The conjectures concern permutation statistics on $\mathfrak{S}_n$. First, the generating function formed from…