A289589 a(n) is the number of permutations of length n that avoid the pattern 321 and the mesh pattern (12, 165) or the same sequence for the mesh patterns (12, 167), (12, 225), (12, 233), (12, 270), (12, 302), (12, 330), (12, 458).

1, 1, 1, 2, 5, 15, 48, 159, 538, 1850, 6446, 22712, 80794, 289804, 1047063, 3807186, 13921317, 51160389, 188858973, 699999531, 2604038517, 9719460729, 36387837723, 136609040721, 514179424239, 1939885552719, 7334783750167, 27789460372643

Offset

0,4

Links

Table of n, a(n) for n=0..27.

Shishuo Fu, Guo-Niu Han, and Zhicong Lin, k-arrangements, statistics and patterns, arXiv:2005.06354 [math.CO], 2020. Mentions this sequence.

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 B2.

Formula

From Thomas Scheuerle, Dec 10 2025: (Start)

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

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

Prog

(PARI) listA(max_n) = my(x='x+O(x^max_n)); Vec((2*x^2+(sqrt(1-4*x)-1)*x-sqrt(1-4*x)+1)/((sqrt(1-4*x)-3)*(x-1)*x)) \\ Thomas Scheuerle, Dec 10 2025

Crossrefs

Cf. A000108, A135336.

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

Sequence in context: A149926 A321467 A301994 * A071739 A390479 A268407

Adjacent sequences: A289586 A289587 A289588 * A289590 A289591 A289592

Keyword

nonn

Author

N. J. A. Sloane, Jul 08 2017

Extensions

More terms, name and offset changed by Thomas Scheuerle, Dec 10 2025

Status

approved