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arXiv · 2511.02312

A combinatorial interpretation for certain plethysm and Kronecker coefficients

Igor Pak·Greta Panova·Joshua P. Swanson
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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.

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Igor Pak, Greta Panova, Joshua P. Swanson. 2025-11-04. A combinatorial interpretation for certain plethysm and Kronecker coefficients. https://arxiv.org/abs/2511.02312

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