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

%I A347007 #16 Dec 23 2023 14:30:35
%S A347007 1,1,2,4,11,19,55,93,285,535,1514,2934
%N A347007 Number of cycle types of permutation groups with degree n.
%C A347007 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 A347007 Peter Dolland, <a href="/A347007/a347007.txt">Cycle types shared by more than one permutation group of degree n = 6..10</a>
%e A347007 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 A347007    1. [1, 0, 0]
%e A347007    2. [1, 1, 0]
%e A347007    3. [1, 0, 2]
%e A347007    4. [1, 3, 2]
%e A347007 The cycle types belong to the permutation groups {id}, C2, C3, and S3 (all subgroups of S3).
%e A347007 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 A347007 (GAP)
%o A347007 # GAP 4.11.1
%o A347007 n := 9;;
%o A347007 G := SymmetricGroup(n);
%o A347007 cc := ConjugacyClasses(G);;
%o A347007 sub := ConjugacyClassesSubgroups(G);;
%o A347007 rep := List(sub, Representative);;
%o A347007 ctlst := List( rep, x-> List( cc, c-> Size( Intersection( x, c))));;
%o A347007 Size( AsDuplicateFreeList( ctlst));
%Y A347007 Cf. A000638.
%K A347007 nonn,more
%O A347007 0,3
%A A347007 _Peter Dolland_, Aug 10 2021