7 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 $[m]=\{1,2,\dots,m\}$, let $\mu\vdash k$, and let $\omega_\mu$ be a fixed element of the conjugacy class $C_\mu$ in $\mathfrak{S}_k$. Let $\mathfrak{S}^{(m)}_k$ be the set of p…

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 $\sigma$ 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 $Y_k(u)$ 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…

Stanley's conjecture. For any $\pi\in S_k$,