Explicit formulas for permutation pattern character polynomials

Given permutations $\pi \in S_n$ and $\sigma \in S_k$, let $N_\sigma(\pi)$ denote the number of occurrences of $\sigma$ in $\pi$. While pattern avoidance and the distribution of pattern occurrences in permutations have been extensively studied, their interactions with the group structure on $S_n$ are still poorly understood. Gaetz and Ryba showed that the expected value of $\chi^{\lambda[n]}(\pi)N_\sigma(\pi)$ for $\pi \in S_n$ is given by a polynomial $a_\sigma^\lambda(n)$. More recently, Gaetz and Pierson derived explicit formulas for $a_{\mathrm{id}_k}^\lambda(n)$ when $\lvert\lambda\rvert \le 2$, which led them to conjecture that the polynomials $a_{\mathrm{id}_k}^\lambda(n)$ are real-rooted and nonnegative for $n \ge k$. We show that for all partitions $\lambda$, the polynomials $a_{\mathrm{id}_k}^\lambda(n)$ admit explicit closed forms in $n$ and $k$. These formulas allow us to exhibit counterexamples to Gaetz and Pierson's real-rootedness conjecture as well as to prove special cases of their nonnegativity conjecture. Lastly, we note that our results imply that the expected value of $f \cdot N_{\mathrm{id}_k}$ on $S_n$ admits a closed form whenever $f$ is a permutation statistic expressible as a polynomial in the functions $m_j \colon \bigsqcup_{n \ge 0} S_n \to \mathbb{Z}$ which count $j$-cycles in their inputs.