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%I #29 May 09 2018 09:55:26
%S 1,1,2,5,1,15,7,1,1,52,39,13,12,2,1,1,203,211,112,103,41,24,17,5,2,1,
%T 1,877,1168,843,811,492,337,238,122,68,39,28,8,5,2,1,1,4140,6728,6089,
%U 6273,4851,3798,2956,1960,1303,859,594,314,204,110,64,43,17,8,5,2,1,1
%N Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 12-3; triangle T(n,k), n>=0, 0<=k<=(n-1)*(n-2)/2-[n=0], read by rows.
%C Patterns 1-23, 3-21, 32-1 give the same triangle.
%H Alois P. Heinz, <a href="/A260665/b260665.txt">Rows n = 0..50, flattened</a>
%H A. Claesson and T. Mansour, <a href="https://arxiv.org/abs/math/0110036">Counting occurrences of a pattern of type (1,2) or (2,1) in permutations</a>, arXiv:math/0110036 [math.CO], 2001
%F Sum_{k>0} k * T(n,k) = A001754(n).
%e T(4,1) = 7: 1324, 1342, 2134, 2314, 2341, 3124, 4123.
%e T(4,2) = 1: 1243.
%e T(4,3) = 1: 1234.
%e T(5,3) = 12: 12534, 12543, 13245, 13425, 13452, 21345, 23145, 23415, 23451, 31245, 41235, 51234.
%e T(5,4) = 2: 12435, 12453.
%e T(5,5) = 1: 12354.
%e T(5,6) = 1: 12345.
%e Triangle T(n,k) begins:
%e 0 : 1;
%e 1 : 1;
%e 2 : 2;
%e 3 : 5, 1;
%e 4 : 15, 7, 1, 1;
%e 5 : 52, 39, 13, 12, 2, 1, 1;
%e 6 : 203, 211, 112, 103, 41, 24, 17, 5, 2, 1, 1;
%e 7 : 877, 1168, 843, 811, 492, 337, 238, 122, 68, 39, 28, 8, 5, 2, 1, 1;
%p b:= proc(u, o) option remember;
%p `if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
%p add(expand(b(u+j-1, o-j)*x^(o-j)), j=1..o))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p seq(T(n), n=0..10);
%t b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1], {j, 1, u}] + Sum[Expand[b[u + j - 1, o - j]*x^(o - j)], {j, 1, o}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0] ]; Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jul 10 2017, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000110, A092923, A264451, A264452, A264453, A264454, A264455, A264456, A264457, A264458, A264459.
%Y Row sums give A000142.
%Y Cf. A000217, A001754, A260670, A263776.
%K nonn,tabf
%O 0,3
%A _Alois P. Heinz_, Nov 14 2015
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