DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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Substitution of combinatorial classes

Definition

Let $A$ and $B$ be combinatorial classes. Their substitution $A \circ B$ is the class of pairs

$(a, (b_{1}, \dots, b_{r}))$

such that $a \in A$ has size $r$ and each $b_{i} \in B$ . Its size is

$∣ (a, (b_{1}, \dots, b_{r})) ∣ : = ∣ b_{1} ∣ + \dots + ∣ b_{r} ∣ .$

The point is that an object of size $r$ in $A$ supplies exactly $r$ ordered slots to be filled by $B$ -objects.

If $B$ has a size-zero object, then for suitable outer classes $A$ fixed total size can arise from arbitrarily large values of $r$ , so the substituted class need not be combinatorial. The substitution theorem therefore assumes $[x^{0}] B (x) = 0$ .

Depends on

Combinatorial classes, counting sequences and ordinary generating functions

Used by

If B(0)=0 then A circB has generating function A(B(x)) Theorem

Dependency tree · two levels

4 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

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics (standard reference, not scraped)
Stephen Melczer, An Invitation to Enumeration, Chapter 5: Combinatorial Constructions (standard reference, not scraped)