A173869 Irregular triangle T(n,k) = A164341(n,k) * A036039(n,k), read by rows, 1 <= k <= A000041(n).

1, 1, 2, 2, 6, 6, 4, 24, 24, 18, 24, 10, 120, 120, 120, 120, 90, 80, 26, 720, 720, 720, 480, 720, 720, 300, 480, 540, 300, 76, 5040, 5040, 5040, 5040, 5040, 5040, 3360, 3780, 3360, 5040, 2100, 2100, 2520, 1092, 232, 40320, 40320, 40320, 40320, 25200, 40320

Offset

0,3

Comments

Rows have A000041(n) entries, with partitions in Abramowitz and Stegun order (A036036).

T(n,k) is the number of pairs of involutions that when composed yield a permutation whose cycle type is the associated partition. - Andrew Howroyd, Oct 05 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)

T. Kyle Petersen and Bridget Eileen Tenner, How to write a permutation as a product of involutions (and why you might care), arXiv:1202.5319 [math.CO], 2012.

Example

Triangle begins:
  0 | 1;
  1 | 1;
  2 | 2, 2;
  3 | 6, 6, 4;
  4 | 24, 24, 18, 24, 10;
  5 | 120, 120, 120, 120, 90, 80, 26;
  6 | 720, 720, 720, 480, 720, 720, 300, 480, 540, 300, 76;
   ...

Prog

(PARI)
B(n, e)=sum(k=0, n\2, binomial(n, 2*k)*e^(n-k)*(2*k)!/(k!*2^k))
C(sig)={my(S=Set(sig)); prod(k=1, #S, my(c=#select(t->t==S[k], sig)); B(c, S[k]))*vecsum(sig)!/(vecprod(sig)*prod(k=1, #S, (#select(t->t==S[k], sig))!))}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 05 2025

Crossrefs

Cf. A000041 (row lengths), A000085, A036036, A036039, A104778, A111883 (row sums), A164341.

Sequence in context: A033754 A171661 A276418 * A247377 A167909 A368745

Adjacent sequences: A173866 A173867 A173868 * A173870 A173871 A173872

Keyword

easy,nonn,tabf

Author

Alford Arnold, Mar 12 2010

Extensions

Definition rephrased by R. J. Mathar, Mar 26 2010

a(0)=1 prepended by Andrew Howroyd, Oct 05 2025

Status

approved