A264173 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 1324; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.

1, 1, 2, 6, 23, 1, 110, 10, 632, 86, 2, 4229, 782, 29, 32337, 7571, 407, 5, 278204, 78726, 5856, 94, 2659223, 882997, 84351, 2215, 14, 27959880, 10657118, 1251246, 48234, 322, 320706444, 137977980, 19318314, 984498, 14322, 42, 3985116699, 1910131680, 311306106

Offset: 0

Author

Alois P. Heinz, Nov 06 2015

Keywords

• nonn
• tabf

Comments

Pattern 4231 gives the same triangle.

Examples

			T(4,1) = 1: 1324.
T(6,2) = 2: 132546, 142536.
T(8,3) = 5: 13254768, 13264758, 14253768, 14263758, 15263748.
T(10,4) = 14: 132547698(10), 132548697(10), 132647598(10), 132648597(10), 132748596(10), 142537698(10), 142538697(10), 142637598(10), 142638597(10), 142738596(10), 152637498(10), 152638497(10), 152738496(10), 162738495(10).
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        2;
03 :        6;
04 :       23,      1;
05 :      110,     10;
06 :      632,     86,     2;
07 :     4229,    782,    29;
08 :    32337,   7571,   407,    5;
09 :   278204,  78726,  5856,   94;
10 :  2659223, 882997, 84351, 2215, 14;
		

Links

• Alois P. Heinz, Rows n = 0..120, flattened

Crossrefs

Columns k=0-10 give: A113228, A264174, A264175, A264176, A264177, A264178, A264179, A264180, A264181, A264182, A264183.

Row sums give A000142.

T(2n+2,n) gives A000108(n) for n>0.

Cf. A004526, A061206, A264319 (pattern 3412).

Programs

Maple

b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
      add(b(u-j, o+j-1, `if`(t>0 and j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);

Mathematica

b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1, Sum[b[u - j, o + j - 1, If[t > 0 && j < t, -j, 0]], {j, 1, u}] + Sum[b[u + j - 1, o - j, j] * If[t < 0 && -j <= t, x, 1], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0, 0]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Apr 30 2017, translated from Maple *)

Formula

Sum_{k>0} k * T(n,k) = ceiling((n-3)*n!/4!) = A061206(n-3) (for n>3).