592 problems

Let $\Gamma$ be a connected graph on $[n]=\{1,2,\ldots,n\}$, and identify each edge of $\Gamma$ with the corresponding transposition of $S_n$. Write $T(\Gamma)$ for the resulting s…

For each $n$, let $a_{n,k}$ be the number of permutations in ${\mathfrak{S}}_n$ whose longest increasing subsequence has length $k$. A sequence $(a_1,a_2,\dots,a_N)$ of positive re…

Let $F$ be a field of characteristic $p$, and let two Specht modules for a symmetric group correspond to partitions. The $p$-core of a partition is obtained by removing rim $p$-hoo…

Let $e$ be a prime, let $n$ be a positive integer, and let $\lambda$ be an $e$-regular partition of $n$. Over a field of characteristic $e$, write $D^{\lambda}$ for the irreducible…

Let $h$ be the quantum characteristic, let $\boldsymbol{S}^{\alpha}$ and $\mathbf{D}^{\beta}$ be the spin modules indexed by an $h$-strict partition $\alpha$ and a restricted $h$-s…

Haiman's conjecture. For every $z\in S_n$ and $\mu\vdash n$, the nonzero coefficients of $\alpha_{\mu}^{z}$ are all positive and form a unimodal sequence.

Let $S_{\bullet}=\{S_N\}_{N\ge 1}$, and let $\chi^{\mu[\bullet]}=\{\chi^{\mu[N]}\}$ denote the stable irreducible character associated with a partition $\mu$. For a word $w$ in the…

For each permutation $\pi$, let $K_{\pi}$ be the universal Kerov character polynomial defined by … For a cycle of length $k$, write $K_k$ for the corresponding polynomial. Kerov's…

Let $G$ be a group and $(V_1,\dots,V_N)$ a sequence of $G$-modules. The sequence is $G$-log-concave if there are $G$-module embeddings … for every $1<i<N$. For a graded $G$-module…

Let $F_r$ be a free group of rank $r$, let $w\in F_r$ be a word, and let $\lambda\vdash k$ be a partition. For sufficiently large $n$, write $\lambda^{+}(n)$ for the stable partiti…

Let $B^{\rho,d}$ be a RoCK block of a symmetric group, and let $T^{\mathsf{Br}(p)}(d,d)$ be Turner's algebra, which contains an idempotent corner isomorphic to…

Basis conjecture. One has

Albertson–Collins conjecture. If $n\geq 4$, then

For $1\leq r<k<n$, let $C(n,k;r)$ be the set of $k$-cycles $\sigma\in S_n$ whose fixed points satisfy $\operatorname{fix}(\sigma)\subseteq\{r+1,\ldots,n\}$. Let…

Let $n$ be a positive integer, and let $\mathfrak{S}_n$ be the symmetric group. A complex irreducible character of $\mathfrak{S}_n$ has a square containing all irreducible characte…

Dihedral symmetry conjecture. The well-known lattice triangulation of $\Delta_{k,n}$ is invariant under the dihedral group $\langle a,b\rangle=D_{2n}\leq S_n$, and $D_{2n}$ is the…

Let $n$ be a positive integer, and let a partition be self-conjugate if it equals its conjugate. Its Durfee size is the side length of its largest square contained in its Young dia…

Let $S_n$ be the symmetric group, and let $\psi_w$ denote the unnormalized steady state probability associated with a permutation $w\in S_n$ in the inhomogeneous TASEP. Assume that…

Parity-unimodality conjecture. The fake-degree polynomials $f^{clambda}(q)$ are parity-unimodal for all partitions $clambda$.

Let $\EuScript{E}_m$ be the Fomin–Kirillov algebra over a field $\mathbf{k}$ associated with the symmetric group $\mathbb{S}_m$. Fomin–Kirillov's conjecture. The algebra…

Suppose that $\alpha, \beta, \gamma, \lambda, \mu, \nu$ are partitions such that $|\alpha|=|\beta|=|\gamma|$ and $|\lambda|=|\mu|=|\nu|$. For $n\in\mathbb{Z}_{\geq 0}$, consider th…

Let $n\geq 4$ be an integer and set … The star graph is $\operatorname{Cay}(S_n,S)$. Abdollahi–Vatandoost's conjecture. The star graph is integral, and its set of distinct eigenval…

Let $S_n$ be the symmetric group. A family $\mathcal{A} \subset S_n$ is intersecting if for every $\sigma,\pi \in \mathcal{A}$ there is an $i \in [n]$ such that $\sigma(i)=\pi(i)$.…

Let $S_n$ be the symmetric group, and let $T_n$ be its transposition graph. For $\mathcal{A} \subset S_n$, write $\partial \mathcal{A}$ for its edge-boundary in $T_n$. Let…

A Gelfand $S_n$-graph is a graph whose elements are partitioned into molecules, and whose cells are the corresponding combinatorial cell structures. Marberg–Zhang's conjecture. Eve…