3 problems

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 $Y_k(u)$ 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…

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 $K_k-R_{k+1}=L_k(C_2,C_3,\ldots)$;…