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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.
2
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
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
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Apr 27 2018
STATUS
approved