Positivity of permutation pattern character polynomials

Let $N_\sigma(\pi)$ denote the number of occurrences of a permutation pattern $\sigma\in S_k$ in a permutation $\pi\in S_n$. Gaetz and Ryba (2021) showed using partition algebras that the $d$-th moment $M_{\sigma,d,n}(\pi)$ of $N_\sigma$ on the conjugacy class of $\pi$ is given by a polynomial in $n,m_1,\dots,m_{dk}$, where $m_i$ denotes the number of $i$-cycles of $\pi$. They also showed that the coefficient $\langle \chi^{\lambda[n]}, M_{\sigma,d,n}\rangle$ agrees with a polynomial $a_{\sigma,d}^\lambda(n)$ in $n$. This work is motivated by the conjecture that when $\sigma=\text{id}_k$ is the identity permutation, all of these coefficients are nonnegative. We directly compute closed forms for the polynomials $a_{\text{id}_k}^{\lambda}(n)$ in the cases $\lambda=(1),(1,1),$ and $(2)$, and use this to verify the positivity conjecture for those cases by showing that the polynomials are real-rooted with all roots less than $k$. We also study the case $a_{\sigma}^{(1)}(n)$, for which we give a formula for the polynomials and their leading coefficients.