Partition division maps, symmetric functions and positivity

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 $\ddot{I}‰$-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.

32 pages, comments welcome!