5 problems

Let $\lambda=(p_1,p_2)\times(q_1,q_2)$ be a multirectangular Young diagram consisting of at most two rectangles, and let $\operatorname{Ch}_\pi^{(\alpha)}(\lambda)$ and the orienta…

Let $a_\rho^{(3)}$ be the coefficients in the monomial expansion of $g_3$. Kerov's numerical conjecture. For $k=3$, the values of $500\cdot5!\cdot7!\,a_\rho^{(3)}$ are given by the…

Let $C_i$ be the indeterminates in the $C$-expansion of Kerov polynomials. For a partition-like vector $\nu$, let $|\nu|$ be the sum of its entries, and let $F_k(\nu)$ denote the v…

Let $c_\rho^{(3)}$ be the coefficients in the monomial expansion of $f_3$. Goulden–Rattan's numerical conjecture. For $k=3$, the values of $2\cdot 6!\cdot 8!\,c_\rho^{(3)}$ are giv…

Let $K_r$ be the Kerov polynomial and write its component of weight $r-2k+1$ as $K_{r,r-2k+1}$. For a partition $u$, let $l( u)$ be its length, let $mi( u)$ be the multiplicity of…