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

0,3

Comments

Pattern 4231 gives the same triangle.

Links

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

Formula

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

Example

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;

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<t, -j, 0)), j=1..u)+
      add(b(u+j-1, o-j, j)*`if`(t<0 and -j<=t, x, 1), j=1..o)))
    end:
T:= n-> (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 *)

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).

Sequence in context: A350274 A350273 A264319 * A220183 A177252 A168270

Adjacent sequences: A264170 A264171 A264172 * A264174 A264175 A264176

Keyword

nonn,tabf

Author

Alois P. Heinz, Nov 06 2015

Status

approved