Combinatorics of symmetric group:S4

From Groupprops
Jump to: navigation, search
This article gives specific information, namely, combinatorics, about a particular group, namely: symmetric group:S4.
View combinatorics of particular groups | View other specific information about symmetric group:S4

This page discusses some of the combinatorics associated with symmetric group:S4, 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.

In our case n = 4:

Partition Partition in grouped form Formula calculating size of fiber for \mathcal{B}(n) \to P(n) Size of fiber for \mathcal{B}(n) \to P(n) Formula calculating size of fiber for S_n \to \mathcal{B}(n) Size of fiber for S_n \to \mathcal{B}(n) Formula calculating size of fiber for S_n \to P(n), which is precisely the conjugacy class size for that cycle type (product of previous two formulas) Size of fiber for S_n \to P(n), which is precisely the conjugacy class size for that cycle type (product of previous two sizes)
1 + 1 + 1 + 1 1 (4 times) \frac{4!}{(1!)^44!} 1 (1 - 1)!^4 1 \frac{4!}{1^44!} 1
2 + 1 + 1 2 (1 time), 1 (2 times) \frac{4!}{(2!)(1!)^2(2!)} 6 (2 - 1)!(1 - 1)!^2 1 \frac{4!}{(2)(1)^2(2!)} 6
2 + 2 2 (2 times) \frac{4!}{(2!)^2(2!)} 3 (2 - 1)!^2 1 \frac{4!}{(2)^2(2!)} 3
3 + 1 3 (1 time), 1 (1 time) \frac{4!}{(3!)^1(1!)(1!)^1(1!)} 4 (3 - 1)!(1 - 1)! 2 \frac{4!}{(3)^1(1!)(1^1)(1!)} 8
4 4 (1 time) \frac{4!}{(4!)^1(1!)} 1 (4 - 1)! 6 \frac{4!}{(4^1(1!)} 6
Total (5 rows -- number of rows equals number of unordered integer partitions of 4) -- -- 15 (equals the size of \mathcal{B}(n), termed the Bell number, for n = 4) -- 11 (see Oeis:A107107) -- 24 (equals 4!, the size of the symmetric group)