A347007 Number of cycle types of permutation groups with degree n.

1, 1, 2, 4, 11, 19, 55, 93, 285, 535, 1514, 2934

Offset: 0

Author

Peter Dolland, Aug 10 2021

Keywords

• nonn
• more

Comments

A000638 gives the number of permutation groups of degree n. Each permutation group is assigned a cumulative cycle type resulting from the cycle types of its member permutations.

Examples

			The 4 cycle types of the 4 permutation groups with degree 3 may be represented by arrays of length 3 (the number of partitions of 3, A000041(3)), indicating the quantity of member permutations, whose cycle type yields a specific partition of n. The partitions are listed in graded lexicographical ordering (see A193073), here (1^3), (2,1), (3):
   1. [1, 0, 0]
   2. [1, 1, 0]
   3. [1, 0, 2]
   4. [1, 3, 2]
The cycle types belong to the permutation groups {id}, C2, C3, and S3 (all subgroups of S3).
Note: For degree n < 6 all permutation groups have different cycle types, so a(n) = A000638(n). For n = 6 there are exactly two permutation groups with the same cycle type (namely [1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], both groups isomorphic with C2^2), so a(6) = 55 = A000638(6) - 1.
		

Links

• Peter Dolland, Cycle types shared by more than one permutation group of degree n = 6..10

Crossrefs

Cf. A000638.

Programs

GAP

# GAP 4.11.1
n := 9;;
G := SymmetricGroup(n);
cc := ConjugacyClasses(G);;
sub := ConjugacyClassesSubgroups(G);;
rep := List(sub, Representative);;
ctlst := List( rep, x-> List( cc, c-> Size( Intersection( x, c))));;
Size( AsDuplicateFreeList( ctlst));