A303648 Number T(n,k) of involutions of [n] having exactly k peaks; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.

1, 1, 2, 3, 1, 4, 6, 5, 18, 3, 6, 44, 26, 7, 91, 123, 11, 8, 172, 448, 136, 9, 300, 1348, 912, 51, 10, 496, 3600, 4552, 838, 11, 781, 8704, 18476, 7441, 283, 12, 1186, 19584, 65324, 48116, 5930, 13, 1742, 41379, 206556, 250375, 66606, 1833, 14, 2492, 83210, 600456, 1120042, 536908, 47358

Offset

0,3

Links

Table of n, a(n) for n=0..56.

Wikipedia, Involution

Formula

T(2n+1,n) = A072187(n+1).

Example

T(5,0) = 5: 12345, 21345, 32145, 43215, 54321.
T(5,1) = 18: 12354, 12435, 12543, 13245, 14325, 14523, 15432, 21354, 21435, 21543, 32154, 34125, 42315, 42513, 45312, 52341, 52431, 53241.
T(5,2) = 3: 13254, 15342, 35142.
Triangle T(n,k) begins:
   1;
   1;
   2;
   3,    1;
   4,    6;
   5,   18,     3;
   6,   44,    26;
   7,   91,   123,     11;
   8,  172,   448,    136;
   9,  300,  1348,    912,     51;
  10,  496,  3600,   4552,    838;
  11,  781,  8704,  18476,   7441,   283;
  12, 1186, 19584,  65324,  48116,  5930;
  13, 1742, 41379, 206556, 250375, 66606, 1833;

Crossrefs

Row sums give A000085.

Columns k=0-1 give: A028310, A303649.

Cf. A008303 (the same for permutations), A072187, A303564 (the same for derangements).

Sequence in context: A235715 A129566 A073809 * A298364 A356769 A111776

Adjacent sequences: A303645 A303646 A303647 * A303649 A303650 A303651

Keyword

nonn,tabf

Author

Alois P. Heinz, Apr 27 2018

Status

approved