DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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The cycle-index series of a graded family of S_n-actions

Definition

Let $(A_{n})_{n \geq 0}$ be a sequence of finite sets such that each $A_{n}$ carries an action of the symmetric group $S_{n}$ . For $σ \in S_{n}$ , write ${Fix}_{A_{n}} (σ)$ for the set of structures in $A_{n}$ fixed by $σ$ .

The cycle-index series of the family is the formal power series

$Z_{A} (t) : = \sum_{n \geq 0} Z_{A_{n}} (s_{1}, \dots, s_{n}) t^{n},$

where

$Z_{A_{n}} (s_{1}, \dots, s_{n}) : = \frac{1}{n!} \sum_{σ \in S_{n}} ∣ {Fix}_{A_{n}} (σ) ∣ \prod_{d = 1}^{n} s_{d}^{j_{d} (σ)} .$

For each fixed $n$ , the coefficient of $t^{n}$ lies in $Q [s_{1}, \dots, s_{n}]$ .

Depends on

Used by

The symmetric-group cycle-index series is coefficientwise exponential Corollary

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ben Lynn, Polya Theory: The Cycle Index Polynomial (standard reference, not scraped)