A289448 a(n) is the number of permutations of length n that avoid the pattern 231 and the mesh pattern (21, 337) or the same sequence for the mesh patterns (21, 339), (21, 369), (21, 371).

1, 1, 1, 3, 9, 29, 95, 317, 1075, 3699, 12891, 45423, 161587, 579607, 2094125, 7614371, 27842633, 102320777, 377717945, 1399999061, 5208077033, 19438921457, 72775675445, 273218081441, 1028358848477, 3879771105437, 14669567500333, 55578920745285, 210971291885013, 802235820300153

Offset

0,4

Comments

From Thomas Scheuerle, Dec 08 2025: (Start)

a(n) is the number of permutations of length n that avoid the mesh pattern:

#| |# or same sequence for: #| |# or #| |# or #| |#

-+-O- -+-O- -+-O- -+-O-

|#| #|#| |#| #|#|

-O-+- -O-+- -O-+- -O-+-

| |# | |# |#|# |#|#

The binary encoding of the mesh patterns, as it is used here in the name, is explained by Christian Sievers in the SeqFan thread. (End)

Links

Table of n, a(n) for n=0..29.

Christian Sievers, RFE Dec 2025: Mesh patterns avoiding 321, SeqFan thread.

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B1.

Formula

From Thomas Scheuerle, Dec 08 2025: (Start)

G.f.: -(-4*x^2 - 3*sqrt(1 - 4*x)*x + 5*x + 2*sqrt(1 - 4*x) - 2)/((sqrt(1 - 4*x) - 3)*(x - 1)*x).

a(n) = 2*(A000108(n) - A135336(n))+1. (End)

Prog

(PARI) listA(max_n) = my(x='x+O(x^max_n)); Vec(-(-4*x^2-3*sqrt(1 - 4*x)*x+5*x+2*sqrt(1-4*x)-2)/((sqrt(1-4*x)-3)*(x-1)*x)) \\ Thomas Scheuerle, Dec 08 2025

Crossrefs

Cf. A000108, A135336.

Related to mesh patterns: A280891, A289446-A289453, A289587-A289616, A289652-A289654.

Sequence in context: A071728 A036550 A290897 * A071732 A389933 A289804

Adjacent sequences: A289445 A289446 A289447 * A289449 A289450 A289451

Keyword

nonn

Author

N. J. A. Sloane, Jul 08 2017

Extensions

More terms, name and offset changed by Thomas Scheuerle, Dec 08 2025

Status

approved