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…

Define $Q_0=1$, $Q_1=0$, and for $n\ge 2$ let … where $l(\mu)$ is the length of $\mu$ and $\mathcal{R}_\mu=\prod_{i\ge2}((i-1)R_i)^{m_i(\mu)}/m_i(\mu)!$. Kerov's Q-positivity conje…

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 $m_i( u)$ be the multiplicity o…