A387061 Number of Sylow permutations in S_n.

0, 0, 1, 5, 23, 99, 479, 2645, 19599, 154007, 1271519, 11688489, 126123095, 1481833859, 15162417087, 126294191309, 2497347563039, 53642575418415, 937621220224319, 17116389710781136, 281286727706878100

Offset: 0

Author

Joan Thibault, Aug 15 2025

Keywords

• nonn
• more
• new

Comments

Sylow permutations are permutations for which the induced group is a Sylow group.

Equivalently, a permutation x is Sylow if there exists a prime number p, such that for each cycle c in the cycle decomposition of x, c has length some power of p.

Not all cycles need to have the same length; e.g., ((1 2)(3 4 5 6)) is a Sylow permutation in S_6.

We exclude the identity from Sylow permutations.

All permutation groups are uniquely characterized by the Sylow permutations they contain (and one can reconstruct said permutation group by computing the group induced by this set).

Examples

			For a(3) = 5, the solutions are { ((1 2)) ((1 3)) ((2 3)) ((1 2 3)) ((1 3 2)) } (1-length cycles are omitted from the decomposition).
For a(4) = 23, the solutions are (up to permutation) { ((1 2)) ((1 2)(3 4)) ((1 2 3 4)) ((1 2 3)) }.
For a(6) = 479, the solutions include (for example) ((1 2)(3 4 5 6)) but not ((1 2)(3 4 5)).
		

Crossrefs

Cf. A000142, A186202.