Partition division maps, symmetric functions and positivity

TL;DR

This paper introduces a new linear map on symmetric functions related to partition division, explicitly determines its Schur expansion, and reveals combinatorial and positivity properties, connecting to classical and new combinatorial objects.

Contribution

It defines and analyzes a novel partition division map on symmetric functions, providing explicit expansions and combinatorial interpretations, including the introduction of $k$-Yamanouchi tableaux.

Findings

Explicit Schur expansion of the map for Schur and skew Schur functions

Introduction of $k$-Yamanouchi tableaux as combinatorial objects

Establishment of power-sum positivity and connections to Euler numbers

Abstract

We study a linear map on symmetric functions that ``divides'' a partition by a positive integer $k$, sending a Schur function indexed by a partition of $kn$ to a symmetric function indexed by partitions of $n$. We determine its Schur expansion explicitly for Schur and skew Schur functions, showing that the coefficients are enumerated by a new family of combinatorial objects, called $k$-Yamanouchi tableaux, which generalize the classical ballot (Yamanouchi) tableaux appearing in the Littlewood--Richardson rule. We also study the images of elementary symmetric functions under this map, derive the power-sum expansion of their $\omega$-images, and establish power-sum positivity. A further application establishes a connection to work of Tewodros Amdeberhan, John Shareshian, and Richard Stanley on alternating permutations and Euler numbers.