On a Pieri-like rule for the Petrie symmetric functions

TL;DR

This paper connects k-ribbon tilings with Petrie symmetric functions, providing a combinatorial interpretation for coefficients in a Pieri-like rule, extending previous results and enabling effective specializations.

Contribution

It establishes a new combinatorial interpretation for coefficients in a Pieri-like rule for Petrie symmetric functions using k-ribbon tilings.

Findings

Provides a combinatorial interpretation for the coefficients.

Extends previous results by Cheng, Chou, and Eu.

Enables derivation of certain specializations.

Abstract

A $k$-ribbon tiling is a decomposition of a connected skew diagram into disjoint ribbons of size $k$. In this paper, we establish a connection between a subset of $k$-ribbon tilings and Petrie symmetric functions, thus providing a combinatorial interpretation for the coefficients in a Pieri-like rule for the Petrie symmetric functions due to Grinberg (Algebr. Comb. 5 (2022), no. 5, 947-1013). This also extends a result by Cheng, Chou and Eu et al. (Proc. Amer. Math. Soc. 151 (2023), no. 5, 1839-1854). As a bonus, our findings can be effectively utilized to derive certain specializations.