A276837 Number A(n,k) of permutations of [n] such that for each cycle c the smallest integer interval containing all elements of c has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 6, 5, 1, 0, 1, 1, 2, 6, 12, 8, 1, 0, 1, 1, 2, 6, 24, 25, 13, 1, 0, 1, 1, 2, 6, 24, 60, 57, 21, 1, 0, 1, 1, 2, 6, 24, 120, 150, 124, 34, 1, 0, 1, 1, 2, 6, 24, 120, 360, 399, 268, 55, 1, 0

Offset: 0

Author

Alois P. Heinz, Sep 20 2016

Keywords

• nonn
• tabl

Comments

The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.

Examples

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,    1,     1,     1, ...
  0, 1,  1,   1,    1,    1,    1,     1,     1, ...
  0, 1,  2,   2,    2,    2,    2,     2,     2, ...
  0, 1,  3,   6,    6,    6,    6,     6,     6, ...
  0, 1,  5,  12,   24,   24,   24,    24,    24, ...
  0, 1,  8,  25,   60,  120,  120,   120,   120, ...
  0, 1, 13,  57,  150,  360,  720,   720,   720, ...
  0, 1, 21, 124,  399, 1050, 2520,  5040,  5040, ...
  0, 1, 34, 268, 1145, 3192, 8400, 20160, 40320, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..30, flattened
• Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019

Crossrefs

Columns k=0-10 give: A000007, A000012, A000045(n+1), A214663, A276838, A276839, A276840, A276841, A276842, A276843, A276844.

Main diagonal gives A000142.

Cf. A263757, A276719.

Formula

A(n,k+1) - A(n,k) = A263757(n,k) for n>0.