37 problems

Let $C_0=1$, $C_1=0$, and for $k\geq 2$ define $C_k$ on Young diagrams by … Let $L_k$ be the polynomial with rational coefficients determined by $K_k-R_{k+1}=L_k(C_2,C_3,\ldots)$;…

Let $Lie^{(2)}$ be the symmetric-function species used in the plethysm below, and let $V\!h_k(n)$ be the $S_n$-module whose Frobenius characteristic is the degree-$n$ term of…

Let $\boldsymbol{p}=(p_1,\ldots,p_m)$ and $\boldsymbol{q}=(q_1,\ldots,q_m)$ encode a partition with $p_i$ parts of size $q_i$, where $p_i$ are positive integers and…

Let ${\mathsf {BNM}}_k(r,s)$ be the bipartite matching complex on two parts of sizes $r$ and $s$, with the natural action of $S_r\times S_s$. Let $H(n,l)$ denote the irreducible…

Let $n=2k$ be even. The complexes ${\mathsf {NPM}}_n$ and the poset ${\mathcal L}_n^{(2)}$ have reduced homology groups carrying actions of $S_n$, where $\varepsilon_n$ denotes the…

Let $\sigma$ be the rectangular partition parametrized by positive integers $p_1,\dots,p_m,q_1,\dots,q_m$, let $\mu\vdash k$ be fixed, and let $\widehat{\chi}^\sigma(\mu,1^{n-k})$…

Let $h\colon [k]\rightarrow [k]$ be a Hessenberg function, let $S$ be a regular semisimple operator on $\mathbb C^k$, and let $\mathfrak S_k$ act on $H^*(\operatorname{Hess}(S,h))$…

Let $k$ be a positive integer, let $h\colon [k]\rightarrow [k]$ be a Hessenberg function, let $S$ be a regular semisimple operator on $\mathbb C^k$, and let $\mathfrak S_k$ act on…

Let $r\in\mathbb{N}$, and let $\Phi^1[r+2,r]$ be the $\mathfrak{S}_{r+2}\times\mathfrak{S}_r$-module appearing in the relative polynomial-degree-$2$ Lie algebra homology calculatio…

Let $\lambda \vdash n$. Write $Ev(\lambda)$ for the set of even refinements of $\lambda$, let $\mathcal{R}_{2N+1}(2|\lambda|)$ and $\mathcal{R}^{c}_{2N}(2|\lambda|)$ be the partiti…

Irreducible-support stability conjecture. For $t\geqslant3$, every irreducible appearing in $\tilde{H}_{n-2}(B_n^{(t)})$ also appears in $\tilde{H}_{n-2}(B_n^{(2)})$.

Equivariant inclusion conjecture. For $t\geqslant1$, there are $\mathfrak{S}_n$-equivariant inclusions

Top elementary coefficient conjecture. The coefficient of $e_n$ is always $1$, that is, $a_{(n)}=1$.

Let $M$ be an $\fS_n$-module, and let $X$ be a set of combinatorial objects equipped with functions $\operatorname{stat}:X\to\mathbb{N}$ and…

Let $\falR_{\fmu}$ denote the Garsia–Procesi module associated with a partition $\fmu$, and let $\fY={x^{\alpha}\mid \alpha\text{ is sub-Yamanouchi}}$ be its specified monomial bas…

Let $k\geq 0$ and let $\lambda$ be a partition. The polynomial $a_{\mathrm{id}_k}^\lambda(n)$ is the character coefficient polynomial associated with the identity pattern of length…

Single-row root-at-minus-one conjecture. When $\lambda$ has a single row, the polynomials $a_\sigma^\lambda(n)$ have $n=-1$ as a root.

Positivity and real-rootedness conjecture. For all $k,n\in\mathbb{N}$, $M_{\textnormal{id}_k,n}$ is a nonnegative linear combination of irreducible symmetric group characters. Furt…

Let $\lambda$ be an integer partition of $n+1$, and let $\mu^{(1)},\ldots,\mu^{(k)}$ be partitions of $n$, each obtained from $\lambda$ by removing a removable cell from its Young…

Let $Y_k(u)$ be the set of expanders with $k$ edges, $u-1$ white vertices and one black vertex. Let $X_k(i,j)$ be the set of expanders with $k$ edges, $i+j-2$ white vertices and tw…

For a Young diagram $\lambda$, let $R_2(\lambda),R_3(\lambda),\ldots$ be its free cumulants, and let $K_k(R_2,R_3,\ldots)$ be the universal Kerov character polynomial satisfying…

Let $\Omega_n$ be the superspace algebra with commuting variables $x_1,\ldots,x_n$ and anticommuting variables $\theta_1,\ldots,\theta_n$, acted on by the symmetric group…

Erasing marks conjecture. The collection

Let $A$ be an alphabet of size $n\ge 2$, and let $S_n$ act on $A^*$ by permuting the alphabet. A rank-selected subposet is obtained by retaining words whose ranks belong to a finit…

Rank-inequality conjecture. One has