A378177 Triangle read by rows: T(n,k) is the number of subgroups of S_n isomorphic to S_k up to conjugacy, where S_n is the n-th symmetric group.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 4, 4, 2, 1, 1, 3, 5, 5, 3, 1, 1, 1, 4, 7, 10, 4, 2, 1, 1, 1, 4, 10, 13, 5, 2, 2, 1, 1, 1, 5, 12, 22, 9, 4, 2, 2, 1, 1, 1, 5, 15, 27, 11, 4, 3, 2, 2, 1, 1, 1, 6, 20, 47, 17, 9, 3, 3, 2, 2, 1, 1, 1, 6, 23, 56, 19, 9, 4, 3, 3, 2, 2, 1, 1

Offset

1,8

Links

Jianing Song, Rows n = 1..17

Formula

T(n,2) = A004526(n) = floor(n/2): we need to classify the order-2 elements (i.e., product of disjoint transpositions) in S_n up to conjugacy. As the conjugation of (a_1,...,a_k) in S_n by sigma is (sigma(a_1),...,sigma(a_k)), two order-2 elements are conjugate if and only if they are the product of the same number of transpositions. In S_n the number of disjoint transpositions can range from 1 to floor(n/2), so T(n,2) = floor(n/2).

Example

Table begins
  1
  1, 1
  1, 1, 1
  1, 2, 1, 1
  1, 2, 2, 1, 1
  1, 3, 4, 4, 2, 1
  1, 3, 5, 5, 3, 1, 1
  1, 4, 7, 10, 4, 2, 1, 1
  1, 4, 10, 13, 5, 2, 2, 1, 1
  1, 5, 12, 22, 9, 4, 2, 2, 1, 1

Prog

(GAP) A378177 := function(n, k)
return Length(IsomorphicSubgroups(SymmetricGroup(n), SymmetricGroup(k)));
end;

Crossrefs

Cf. A004526, A378266, A378273, A378274.

Sequence in context: A274190 A322596 A037306 * A194799 A291119 A366309

Adjacent sequences: A378174 A378175 A378176 * A378178 A378179 A378180

Keyword

nonn,tabl,hard,more

Author

Jianing Song, Nov 18 2024

Status

approved