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

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

Offset

0,3

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.

Links

Table of n, a(n) for n=0..11.

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

Example

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.

Prog

(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));

Crossrefs

Cf. A000638.

Sequence in context: A139785 A283173 A283254 * A000638 A039824 A204519

Adjacent sequences: A347004 A347005 A347006 * A347008 A347009 A347010

Keyword

nonn,more

Author

Peter Dolland, Aug 10 2021

Status

approved