A317639 Number of equivalence classes of Dyck paths of semilength n for the consecutive pattern UDUDD, where U=(1,1) and D=(1,-1).

1, 1, 1, 2, 4, 6, 10, 19, 32, 54, 98, 170, 292, 520, 909, 1577, 2787, 4883, 8515, 14998, 26299, 45984, 80863, 141844, 248381, 436406, 765649, 1341844, 2356500, 4134749, 7249981, 12728630, 22335110, 39174776, 68766785, 120670190, 211689586, 371558266, 652014636

Offset

0,4

Comments

Two Dyck paths of the same length are equivalent with respect to a given pattern if they have equal sets of occurrences of this pattern.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..4096

J.-L. Baril, A. Petrossian, Equivalence classes of Dyck paths modulo some statistics, 2014.

J.-L. Baril, A. Petrossian, Equivalence Classes of Motzkin Paths Modulo a Pattern of Length at Most Two, J. Int. Seq. 18 (2015) 15.7.1

K. Manes, A. Sapounakis, I. Tasoulas, P. Tsikouras, Equivalence classes of ballot paths modulo strings of length 2 and 3, arXiv:1510.01952 [math.CO], 2015.

Wikipedia, Counting lattice paths

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
     `if`(y=0, b(x-2, y)+b(x-6, y+2), b(x-1, y-1))+b(x-5, y+1)))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..42);

Mathematica

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, If[y == 0, b[x - 2, y] + b[x - 6, y + 2], b[x - 1, y - 1]] + b[x - 5, y + 1]]];
a[n_] := b[2n, 0];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Aug 20 2018, from Maple *)

Crossrefs

Cf. A000108, A001519, A177528, A244885, A244886, A274114, A274115, A274289.

Sequence in context: A133140 A349592 A026680 * A164141 A034872 A032362

Adjacent sequences: A317636 A317637 A317638 * A317640 A317641 A317642

Keyword

nonn

Author

Alois P. Heinz, Aug 02 2018

Status

approved