4 problems

Cumulant-coefficient positivity conjecture. For partitions $cpi_1,\dots,cpi_\ell$, the polynomial $(-1)^{\ell-1}d^\sigma_{\pi_1,\dots,\pi_\ell}(cdelta)$ has non-negative integer co…

Kerov–Lassalle cumulant-positivity conjecture. For all integers $k_1,\dots,k_\ell\geq 1$, the polynomial which expresses this signed cumulant in terms of…

Let $cgamma=-\delta$, and let $cR_2,cR_3,\cdots$ denote the free cumulants, viewed as functions on Young diagrams. For each partition $cpi$, the Jack character…

Let $cS_n$ be the symmetric group and let $cdelta$ be the Jack deformation parameter. The structure coefficients $g^\mu_{\pi,\sigma}$ are defined by … At $cdelta=0$, these coeffici…