A264781 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 45321; triangle T(n,k), n >= 0, 0 <= k <= max(0, floor((n-1)/4)), read by rows.

1, 1, 2, 6, 24, 119, 1, 708, 12, 4914, 126, 38976, 1344, 347765, 15110, 5, 3447712, 180736, 352, 37598286, 2308548, 9966, 447294144, 31481472, 225984, 5764747515, 457520055, 4753185, 45, 80011430240, 7068885600, 97954080, 21280, 1189835682714, 115808906178

Offset: 0

Author

Alois P. Heinz, Nov 24 2015

Keywords

• nonn
• tabf

Comments

Consecutive patterns 12354, 21345, 54312 give the same triangle.

The attached Maple program gives a recurrence for the o.g.f. of each row in terms of u. Using that recurrence we may get any row or column from this irregular triangular array T(n,k). The recurrence follows from manipulation of the bivariate o.g.f./e.g.f. 1/W(u,z) = Sum_{n, k >= 0} T(n, k)*u^k*z^n/n!, whose reciprocal W(u,z) is the solution of the o.d.e. in Theorem 3.2 in Elizalde and Noy (2003) (with m = a = 3). - Petros Hadjicostas, Nov 05 2019

Examples

			T(5,1) = 1: 45321.
T(6,1) = 12: 156432, 256431, 356421, 453216, 456321, 463215, 546321, 563214, 564213, 564312, 564321, 645321.
T(9,2) = 5: 786549321, 796548321, 896547321, 897546321, 897645321.
Triangle T(n,k) begins:
00 :           1;
01 :           1;
02 :           2;
03 :           6;
04 :          24;
05 :         119,          1;
06 :         708,         12;
07 :        4914,        126;
08 :       38976,       1344;
09 :      347765,      15110,        5;
10 :     3447712,     180736,      352;
11 :    37598286,    2308548,     9966;
12 :   447294144,   31481472,   225984;
13 :  5764747515,  457520055,  4753185,    45;
14 : 80011430240, 7068885600, 97954080, 21280;
		

Links

• Alois P. Heinz, Rows n = 0..170, flattened
• A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes, 2011.
• Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116) with m = a = 3.
• Petros Hadjicostas, Maple program for a recurrence.

Crossrefs

Columns k=0-1 give: A202213, A264896.

Row sums give A000142.

T(4n+1,n) gives A007696.

Cf. A002265, A007696, A062199.

Programs

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
       b(u+j-1, o-j, `if`(u+j-30, -1, `if`(t=-1, -2, 0)))), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..17);

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[
     b[u+j-1, o-j, If[u+j-3 < j, 0, j]], {j, 1, o}] + Expand[
     If[t == -2, x, 1]*Sum[b[u-j, o+j-1, If[j < t || t == -2, 0,
     If[t > 0, -1, If[t == -1, -2, 0]]]], {j, 1, u}]]];
T[n_] := CoefficientList[b[n, 0, 0], x];
T /@ Range[0, 17] // Flatten (* Jean-François Alcover, Feb 19 2021, after Alois P. Heinz *)

Formula

Sum_{k > 0} k * T(n,k) = A062199(n-5) for n > 4.