A347007 - OEIS

%I #16 Dec 23 2023 14:30:35

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

%N Number of cycle types of permutation groups with degree n.

%C 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.

%H Peter Dolland, <a href="/A347007/a347007.txt">Cycle types shared by more than one permutation group of degree n = 6..10</a>

%e 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):

%e    1. [1, 0, 0]

%e    2. [1, 1, 0]

%e    3. [1, 0, 2]

%e    4. [1, 3, 2]

%e The cycle types belong to the permutation groups {id}, C2, C3, and S3 (all subgroups of S3).

%e 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.

%o (GAP)

%o # GAP 4.11.1

%o n := 9;;

%o G := SymmetricGroup(n);

%o cc := ConjugacyClasses(G);;

%o sub := ConjugacyClassesSubgroups(G);;

%o rep := List(sub, Representative);;

%o ctlst := List( rep, x-> List( cc, c-> Size( Intersection( x, c))));;

%o Size( AsDuplicateFreeList( ctlst));

%Y Cf. A000638.

%K nonn,more

%O 0,3

%A _Peter Dolland_, Aug 10 2021