The positivity and real-rootedness conjecture for increasing-subsequence character polynomials

Let $S_n$ be the symmetric group, let $\textnormal{id}_k=12\ldots k\in S_k$ be the identity permutation, and let $M_{\textnormal{id}_k,n}$ be the $n$-th moment class function associated with occurrences of the increasing pattern $\textnormal{id}_k$. For a partition $\lambda$, let $a_{\textnormal{id}_k}^\lambda(n)$ be the polynomial agreeing with the irreducible-character coefficient $\langle\chi^{\lambda[n]},M_{\textnormal{id}_k,n}\rangle$ whenever defined.

Positivity and real-rootedness conjecture. For all $k,n\in\mathbb{N}$, $M_{\textnormal{id}_k,n}$ is a nonnegative linear combination of irreducible symmetric group characters. Furthermore, the polynomials $a_{\textnormal{id}_k}^\lambda(n)$ are real-rooted, with all roots less than $k$.

This conjecture combines a positivity assertion for the character expansion of the increasing-subsequence moment with a root-location assertion for its stable character coefficients. It is the main motivation of the paper; the surrounding results establish polynomiality of the coefficients, while the stated positivity and real-rootedness remain conjectural.