0,3

Patterns 1-23, 3-21, 32-1 give the same triangle.

Alois P. Heinz, Rows n = 0..50, flattened

Anders Claesson and Toufik Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001.

Sum_{k>0} k * T(n,k) = A001754(n).

T(4,1) = 7: 1324, 1342, 2134, 2314, 2341, 3124, 4123.

T(4,2) = 1: 1243.

T(4,3) = 1: 1234.

T(5,3) = 12: 12534, 12543, 13245, 13425, 13452, 21345, 23145, 23415, 23451, 31245, 41235, 51234.

T(5,4) = 2: 12435, 12453.

T(5,5) = 1: 12354.

T(5,6) = 1: 12345.

Triangle T(n,k) begins:

0 : 1;

1 : 1;

2 : 2;

3 : 5, 1;

4 : 15, 7, 1, 1;

5 : 52, 39, 13, 12, 2, 1, 1;

6 : 203, 211, 112, 103, 41, 24, 17, 5, 2, 1, 1;

7 : 877, 1168, 843, 811, 492, 337, 238, 122, 68, 39, 28, 8, 5, 2, 1, 1;

b:= proc(u, o) option remember;

`if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+

add(expand(b(u+j-1, o-j)*x^(o-j)), j=1..o))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):

seq(T(n), n=0..10);

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 *)

Columns k=0-10 give: A000110, A092923, A264451, A264452, A264453, A264454, A264455, A264456, A264457, A264458, A264459.

Row sums give A000142.

Cf. A000217, A001754, A260670, A263776.

Sequence in context: A381529 A101895 A260670 * A110220 A349106 A119518

Adjacent sequences: A260662 A260663 A260664 * A260666 A260667 A260668

nonn,tabf,changed

Alois P. Heinz, Nov 14 2015

approved