Combinatorics of symmetric group:S6

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This article gives specific information, namely, combinatorics, about a particular group, namely: symmetric group:S6.
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This page discusses some of the combinatorics associated with symmetric group:S6, that relies specifically on viewing it as a symmetric group on a finite set.

Partitions, subset partitions, and cycle decompositions

Denote by \mathcal{B}(n) the set of all unordered set partitions of \{ 1,2,\dots,n\} into subsets and by P(n) the set of unordered integer partitions of n. There are natural combinatorial maps:

S_n \to \mathcal{B}(n) \to P(n)

where the first map sends a permutation to the subset partition induced by its cycle decomposition, which is equivalently the decomposition into orbits for the action of the cyclic subgroup generated by that permutation on \{ 1,2,\dots,n\}. The second map sends a subset partition to the partition of n given by the sizes of the parts. The composite of the two maps is termed the cycle type, and classifies conjugacy classes in S_n, because cycle type determines conjugacy class.

Further, if we define actions as follows:

  • S_n acts on itself by conjugation
  • S_n acts on \mathcal{B}(n) by moving around the elements and hence changing the subsets
  • S_n acts on P(n) trivially

then the maps above are S_n-equivariant, i.e., they commute with the S_n-action. Moreover, the action on \mathcal{B}(n) is transitive on each fiber above P(n) and the action on S_n is transitive on each fiber above the composite map to P(n). In particular, for two elements of \mathcal{B}(n) that map to the same element of P(n), the fibers above them in S_n have the same size.

There are formulas for calculating the sizes of the fibers at each level.