A333726 Triangle read by rows: T(n,k) is the number of permutations in S_n for which all cycles are length k or greater.

1, 2, 1, 6, 2, 2, 24, 9, 6, 6, 120, 44, 24, 24, 24, 720, 265, 160, 120, 120, 120, 5040, 1854, 1140, 720, 720, 720, 720, 40320, 14833, 8988, 6300, 5040, 5040, 5040, 5040, 362880, 133496, 80864, 58464, 40320, 40320, 40320, 40320, 40320

Offset: 1

Author

Peter Kagey, Apr 23 2020

Keywords

• nonn
• tabl

Examples

			Triangle begins:
n\k|      1      2     3     4     5     6     7     8     9
---+--------------------------------------------------------
  1|      1
  2|      2      1
  3|      6      2     2
  4|     24      9     6     6
  5|    120     44    24    24    24
  6|    720    265   160   120   120   120
  7|   5040   1854  1140   720   720   720   720
  8|  40320  14833  8988  6300  5040  5040  5040  5040
  9| 362880 133496 80864 58464 40320 40320 40320 40320 40320
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened)
• Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Crossrefs

Columns 1-4: A000142, A000166, A038205, A047865.

Cf. A068424.

Formula

T(n,k) = 0 when n < k,

T(n,k) = (n-1)! when n = k, and otherwise

T(n,k) = (n-1) * T(n-1, k) + A068424(n-1, k-1) * T(n-k, k).