This analysis demonstrates the number of irreducibles in Schur functions, highlighting connections to lattices and Foulkes' conjecture.
We give a formula for the number of irreducibles (with multiplicity) in the decomposition of the plethysm s_λ[sₘ] of Schur functions in terms of the number of lattice points in certain rational polytopes. In the case where λ = n consists of a single part, we will give a combinatorial interpretation of this number as the cardinality of a set of matrices modulo permutation equivalence. This is also the setting of Foulkes' conjecture, and our results allow us to state a weaker version that only involves comparing the cardinalities of such sets, rather than the multiplicities of irreducible representations.
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May Lim (2025) studied this question.
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