A combinatorial interpretation for certain plethysm and Kronecker coefficients

TL;DR

This paper provides explicit combinatorial interpretations for certain plethysm and Kronecker coefficients using marked trees, offering positive counting formulas that clarify their structure and complexity.

Contribution

It introduces new combinatorial models for plethysm and Kronecker coefficients, specifically for cases with at most two-row partitions, enhancing understanding and computation.

Findings

Provides positive combinatorial interpretations for plethysm coefficients.

Yields a combinatorial interpretation for rectangular Kronecker coefficients.

Offers explicit counting formulas over marked trees.

Abstract

We give explicit positive combinatorial interpretations for the plethysm coefficients $\langle s_\mu[s_\nu], s_\lambda\rangle$, when $\lambda$ has at most two rows, as counting certain marked trees. In the special case $\mu=(n)$, this also yields a combinatorial interpretation for the corresponding rectangular Kronecker coefficient $g(\lambda, (n^k), (n^k))$. While it is easy to express these quantities as differences of counting problems in the complexity class $\mathrm{FP}$, putting the problem in $\#\mathrm{P}$, our interpretations give a positive counting formula over explicit marked trees.