Metraipolis › archive › dc_3fc2e93709eb

Cycle index — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Cycle_index
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-06T16:44:32.665Z
SHA-256 of the archived text: 067552b07af5fe78c72ec722b6bcb4dc55aed047e6a1501fc1fe301e64bba505
Stable URL of this page: https://metraipolis.com/documento/dc_3fc2e93709eb

How to cite

Cycle index — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Cycle_index (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_3fc2e93709eb (Metraipolis archive dc_3fc2e93709eb, sha256 067552b07af5fe78c72ec722b6bcb4dc55aed047e6a1501fc1fe301e64bba505)

BibTeX

@misc{dc3fc2e93709eb,
  title = {Cycle index — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Cycle_index},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_3fc2e93709eb; sha256 067552b07af5fe78c72ec722b6bcb4dc55aed047e6a1501fc1fe301e64bba505},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Cycle_index. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents. This compact way of storing information in an algebraic form is frequently used in combinatorial enumeration. Each permutation π of a finite set of objects partitions that set into cycles; the cycle index monomial of π is a monomial in variables a1, a2, … that describes the cycle type of this partition: the exponent of ai is the number of cycles of π of size i. The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also sometimes used in place of cycle index. Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group's action. This is the main ingredient in the Pólya enumeration theorem. Performing formal algebraic and differential operations on these polynomials and then interpreting the results combinatorially lies at the core of species theory.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Combinatorics»).
Fonte: Wikipedia in inglese, voce «Cycle index» — https://en.wikipedia.org/wiki/Cycle_index
Wikidata Q245446 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q245446
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Cycle_index?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1316905733, consultata il 2026-10-06T15:18:47.053Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 141963fe86e2a0bc2dc06ed64196568b31e19d4413b4be671329063a3cbf673e