A289607 a(n) is the number of permutations of length n that avoid the pattern 321 and the mesh pattern (12, 279) or the same sequence for the mesh patterns (12, 309), (12, 345), (12, 465).

1, 1, 1, 2, 7, 28, 106, 382, 1345, 4706, 16504, 58229, 206933, 740786, 2670266, 9686562, 35341181, 129611902, 477573028, 1767131965, 6563858105, 24465742542, 91481514854, 343057516267, 1289899952767, 4861938012568, 18367336294636, 69533517361247, 263747884641145

Offset

0,4

Links

Table of n, a(n) for n=0..28.

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 18 2025: (Start)

G.f.: (4*x^6 - 4*x^5 + 12*x^4 - 18*x^3 + 18*x^2 - 10*x - 2*sqrt(1 - 4*x)*(x - 1)^3*(2*x - 1) + 2)/(4*(x - 1)^3*x*(2*x - 1)).

a(n) = C(n) - 2^(n-2) - 2*n - n^2 + 7*n - 17, for n > 3, where C(n) is the Catalan number A000108.

a(n) = A289606(n) + A289610(n) - A289608(n).

a(n) = A289604(n) + A289614(n) - A289610(n). (End)

Prog

(PARI) listA(max_n) = my(x='x+O(x^max_n)); Vec((4*x^6-4*x^5+12*x^4-18*x^3+18*x^2-10*x-2*sqrt(1-4*x)*(x-1)^3*(2*x-1)+2)/(4*(x-1)^3*x*(2*x-1))) \\ Thomas Scheuerle, Dec 18 2025

Crossrefs

Cf. A000108.

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

Sequence in context: A048504 A092465 A099488 * A068944 A215143 A390825

Adjacent sequences: A289604 A289605 A289606 * A289608 A289609 A289610

Keyword

nonn

Author

N. J. A. Sloane, Jul 09 2017

Extensions

More terms, name and offset changed by Thomas Scheuerle, Dec 18 2025

Status

approved