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 en and the homogeneous symmetric functions hn. Analogous to the Pieri rule for sμhn and the dual Pieri rule for sμen, Grinberg showed that the Schur coefficients for the “Pieri rule” of sμG(k,n) can be determined by the determinant petk(λ,μ) of Petrie matrices. Cheng, Chou, Eu, Fu, and Yao provided a ribbon tiling interpretation for the coefficient petk(λ,∅). We offer a more transparent combinatorial perspective on the structure and behavior of Petrie symmetric functions. First, we provide a weighted formula for the determinant of a Petrie matrix in terms of certain orientations of the associated Petrie graph derived from the matrix. We then give a formula for petk(λ,μ) for all skew shapes using purely combinatorial methods. 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. As a side discovery, we find a connection between 2-colored noncrossing set partitions and 1234-avoiding permutations.