OFFSET
0,3
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
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.
FORMULA
Sum_{k > 0} k * T(n,k) = A062199(n-5) for n > 4.
EXAMPLE
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;
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(
b(u+j-1, o-j, `if`(u+j-3<j, 0, j)), j=1..o)+ expand(
`if`(t=-2, x, 1)*add(b(u-j, o+j-1, `if`(j<t or t=-2, 0,
`if`(t>0, -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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Nov 24 2015
STATUS
approved