Séminaire Lotharingien de Combinatoire, 95B.33 (2026), 12 pp.
Siddarth Kannan and Terry Dekun Song
Pólya Enumeration, Wreath Product Symmetric Functions, and Moduli Spaces of Curves
Abstract.
We develop a calculus for Sn-equivariant Euler characteristics of moduli spaces of stable curves and stable maps. Our approach involves an enrichment of Pólya's cycle index polynomial of a graph to a certain algebra Λ[2] of wreath product symmetric functions. Building on foundational work of Macdonald, we prove that Λ[2] may be viewed as the Grothendieck ring of the category of polynomial functors which map symmetric sequences of vector spaces to vector spaces. This interpretation gives rise to an action of Λ[2] on the ordinary ring of symmetric functions Λ, which is described concretely in terms of Adams operations and skewing by power sums. This action lets us deduce appealing formulas, involving only ordinary symmetric functions, for generating functions of Sn-equivariant Euler characteristics.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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