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

Author

Alois P. Heinz, Apr 27 2018

Keywords

• nonn
• tabf

Examples

			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;
		

Links

• Wikipedia, Involution

Crossrefs

Row sums give A000085.

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

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

Formula

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