A289450 Related to number of mesh patterns of length 2 that avoid the pattern 231 or 321. For details see the comment section.

1, 1, 1, 3, 10, 33, 109, 364, 1233, 4236, 14740, 51868, 184298, 660400, 2383928, 8661433, 31649818, 116242093, 428878333, 1588858033, 5908076563, 22042959973, 82495136173, 309605919163, 1164967889197, 4393950529675, 16609453053051, 62913704495451, 238760752257655, 907721466242659, 3456717799933235

Offset

0,4

Comments

From Thomas Scheuerle, Dec 23 2025: (Start)

a(n) is the number of permutations of length n that avoid the pattern 231 and the mesh pattern (12, 283) or the same sequence for the mesh patterns (12, 285), (12, 315), (12, 317), (12, 347), (12, 349), (12, 379), (12, 381), (12, 405), (12, 407), (12, 414), (12, 427), (12, 433), (12, 435), (12, 469), (12, 471), (12, 478), (12, 491), (12, 497), (12, 499).

a(n) is the number of permutations of length n that avoid the pattern 231 and the mesh pattern (21, 205) or the same sequence for the mesh patterns (21, 207), (21, 237), (21, 239), (21, 345), (21, 347), (21, 377), (21, 379), (21, 460), (21, 462), (21, 465), (21, 467), (21, 492), (21, 494), (21, 497), (21, 499).

a(n) is the number of permutations of length n that avoid the pattern 321 and the mesh pattern (12, 247) or the same sequence for the mesh patterns (12, 253), (12, 382), (12, 478).

a(n) is the number of permutations of length n that avoid the pattern 321 and the mesh pattern (21, 163) or the same sequence for the mesh patterns (21, 167), (21, 169), (21, 173), (21, 179), (21, 183), (21, 185), (21, 189), (21, 227), (21, 231), (21, 233), (21, 237), (21, 243), (21, 247), (21, 249), (21, 253), (21, 298), (21, 302), (21, 314), (21, 318), (21, 362), (21, 366), (21, 378), (21, 382), (21, 394), (21, 398), (21, 410), (21, 414), (21, 458), (21, 462), (21, 474), (21, 478). (End)

Links

Table of n, a(n) for n=0..30.

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 05 2025: (Start)

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

a(n) = A000108(n) - A014137(n-2), for n >= 2. (End)

Prog

(PARI) listA(max_n) = my(x='x+O(x^max_n)); Vec(-((sqrt(1-4*x)-1)*(x^2+x-1))/(2*(x-1)*x)) \\ Thomas Scheuerle, Dec 05 2025

Crossrefs

Cf. A000108, A014137.

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

Sequence in context: A255116 A020704 A006190 * A113299 A126931 A257178

Adjacent sequences: A289447 A289448 A289449 * A289451 A289452 A289453

Keyword

nonn

Author

N. J. A. Sloane, Jul 08 2017

Extensions

More terms, offset and name changed by Thomas Scheuerle, Dec 23 2025

Status

approved