A179974 Irregular triangle T(n,k) = (n-A049085(n,k))!, read by rows, 1 <= k <= A000041(n).

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 6, 1, 1, 2, 2, 6, 6, 24, 1, 1, 2, 6, 2, 6, 24, 6, 24, 24, 120, 1, 1, 2, 6, 2, 6, 24, 24, 6, 24, 120, 24, 120, 120, 720, 1, 1, 2, 6, 24, 2, 6, 24, 24, 120, 6, 24, 120, 120, 720, 24, 120, 720, 120, 720, 720, 5040, 1, 1, 2, 6, 24, 2, 6, 24, 120, 24, 120, 720, 6, 24, 120, 120, 720, 720, 24, 120, 720, 720, 5040, 120, 720, 5040

Offset

0,7

Comments

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

Since A049085 is a resortment of A036043 both A179972 and this sequence have row sums equal to A179973.

Links

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

Example

Triangle begins:
  1;
  1;
  1,1;
  1,1,2;
  1,1,2,2,6;
  1,1,2,2,6,6,24;
  1,1,2,6,2,6,24,6,24,24,120;
  1,1,2,6,2,6,24,24,6,24,120,24,120,120,720;
  1,1,2,6,24,2,6,24,24,120,6,24,120,120,720,24,120,720,120,720,720,5040;
  ...

Prog

(PARI)
C(sig)={if(!#sig, 0, vecsum(sig)-vecmax(sig))!}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 02 2025

Crossrefs

Cf. A000041 (row lengths), A179973 (row sums), A036036, A036042, A049085 (max part).

Sequence in context: A046772 A348291 A359652 * A387203 A246402 A114551

Adjacent sequences: A179971 A179972 A179973 * A179975 A179976 A179977

Keyword

nonn,tabf

Author

Alford Arnold, Aug 05 2010

Extensions

a(0)=1 prepended by Andrew Howroyd, Oct 02 2025

Status

approved