Combinatorial proofs of Petrie Pieri rule and Plethystic Pieri rule

Petrie symmetric functions $G(k,n)$, also known as truncated homogeneous symmetric functions or modular complete symmetric functions, form a class of symmetric functions interpolating between the elementary symmetric functions $e_n$ and the homogeneous symmetric functions $h_n$. Analogous to the Pieri rule for $s_\mu h_n$ and the dual Pieri rule for $s_\mu e_n$, Grinberg showed that the Schur coefficients for the ``Pieri rule'' of $s_\mu G(k,n)$ can be determined by the determinant $\mathbf{pet}_k(\lambda,\mu)$ of Petrie matrices. Cheng, Chou, Eu, Fu, and Yao provided a ribbon tiling interpretation for the coefficient $\mathbf{pet}_k(\lambda,\varnothing)$, which was later generalized by Jin, Jing, and Liu to $\mathbf{pet}_k(\lambda,\mu)$ in the case where $\lambda/\mu$ is connected. The goal of this paper is to offer a more transparent combinatorial perspective on the structure and behavior of Petrie symmetric functions. First, we provide a refined combinatorial formula for the determinant of a Petrie matrix in terms of certain orientations of the associated graph derived from the matrix. We then generalize the result of JJL to arbitrary skew shapes using purely combinatorial proofs. In addition, we investigate the generating function of these orientations with respect to certain statistics. As an application of our method, we present a combinatorial proof of the plethystic Pieri rule.