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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.

%I A303648 #17 May 09 2018 09:15:59
%S A303648 1,1,2,3,1,4,6,5,18,3,6,44,26,7,91,123,11,8,172,448,136,9,300,1348,
%T A303648 912,51,10,496,3600,4552,838,11,781,8704,18476,7441,283,12,1186,19584,
%U A303648 65324,48116,5930,13,1742,41379,206556,250375,66606,1833,14,2492,83210,600456,1120042,536908,47358
%N 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.
%H A303648 Wikipedia, <a href="https://en.wikipedia.org/wiki/Involution_(mathematics)">Involution</a>
%F A303648 T(2n+1,n) = A072187(n+1).
%e A303648 T(5,0) = 5: 12345, 21345, 32145, 43215, 54321.
%e A303648 T(5,1) = 18: 12354, 12435, 12543, 13245, 14325, 14523, 15432, 21354, 21435, 21543, 32154, 34125, 42315, 42513, 45312, 52341, 52431, 53241.
%e A303648 T(5,2) = 3: 13254, 15342, 35142.
%e A303648 Triangle T(n,k) begins:
%e A303648    1;
%e A303648    1;
%e A303648    2;
%e A303648    3,    1;
%e A303648    4,    6;
%e A303648    5,   18,     3;
%e A303648    6,   44,    26;
%e A303648    7,   91,   123,     11;
%e A303648    8,  172,   448,    136;
%e A303648    9,  300,  1348,    912,     51;
%e A303648   10,  496,  3600,   4552,    838;
%e A303648   11,  781,  8704,  18476,   7441,   283;
%e A303648   12, 1186, 19584,  65324,  48116,  5930;
%e A303648   13, 1742, 41379, 206556, 250375, 66606, 1833;
%Y A303648 Row sums give A000085.
%Y A303648 Columns k=0-1 give: A028310, A303649.
%Y A303648 Cf. A008303 (the same for permutations), A072187, A303564 (the same for derangements).
%K A303648 nonn,tabf
%O A303648 0,3
%A A303648 _Alois P. Heinz_, Apr 27 2018