The e-positivity of $\mathrm{rowDiv}_k(e_μ)$

Source abstract

We prove that the partition division operator $\mathrm{rowDiv}_k$ preserves e-positivity, resolving a conjecture of Alexandersson and Dai. For $μ\vdash kn$ and $ν\vdash n$, we express the coefficient of $e_ν$ in $\mathrm{rowDiv}_k(e_μ)$ as a normalized sum of positive rational weights over closed walks on a finite directed graph of $k$-restricted partitions. The proof combines the Pieri rule and a root of unity factorization for Macdonald polynomials. As a corollary, we recover the h-positivity of the symmetric functions $A_{n,k}$ of Amdeberhan, Shareshian, and Stanley.