A257562 -id:A257562 - cp's OEIS frontend

A165542 Number of permutations of length n which avoid the patterns 4231 and 4123.

1, 1, 2, 6, 22, 89, 380, 1677, 7566, 34676, 160808, 752608, 3548325, 16830544, 80234659, 384132724, 1845829988, 8897740300, 43010084460, 208409687323, 1012046126532, 4923952560917, 23997719075657, 117136530812812, 572552052378494, 2802078324448067

Offset: 0

Vincent Vatter, Sep 21 2009

nonn

G.f. conjectured to be non-D-finite (see Albert et al link). - Jay Pantone, Oct 01 2015

			There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.

David Bevan, Jay Pantone, and Nathaniel Shar, Table of n, a(n) for n = 0..1000 (terms 1 through 40 by David Bevan, terms 41 through 70 by Nathaniel Shar)
Michael H. Albert, Cheyne Homberger, Jay Pantone, Nathaniel Shar, Vincent Vatter, Generating Permutations with Restricted Containers, arXiv:1510.00269 [math.CO], 2015.
C. Bean, M. Tannock and H. Ulfarsson, Pattern avoiding permutations and independent sets in graphs, arXiv:1512.08155 [math.CO], 2015.
Darla Kremer and Wai Chee Shiu, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
Wikipedia, Permutation classes avoiding two patterns of length 4.

Cf. A053614, A106228, A165542, A165545, A257561, A257562.

More terms from David Bevan, Feb 04 2014

a(0)=1 prepended by Jay Pantone, Oct 01 2015

A165545 Number of permutations of length n which avoid the patterns 2341 and 3421.

Original entry on oeis.org

1, 1, 2, 6, 22, 89, 382, 1711, 7922, 37663, 182936, 904302, 4535994, 23034564, 118209806, 612165222, 3195359360, 16795435994, 88825567814, 472356139660, 2524292893556, 13549955878141, 73026827854516, 395017112175542, 2143881709415478, 11671226062503926

Offset: 0

Vincent Vatter, Sep 21 2009

nonn

These permutations have an enumeration scheme of depth 4.

G.f. is conjectured to be non-D-finite (see Albert et al link). - Jay Pantone, Oct 01 2015

			There are 22 permutations of length 4 which avoid these two patterns, so a(4)=22.

Jay Pantone, Table of n, a(n) for n = 0..1000
Michael H. Albert, Cheyne Homberger, Jay Pantone, Nathaniel Shar, Vincent Vatter, Generating Permutations with Restricted Containers, arXiv:1510.00269 [math.CO], 2015.
Darla Kremer and Wai Chee Shiu, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
V. Vatter, Enumeration schemes for restricted permutations, Combin., Prob. and Comput. 17 (2008), 137-159.
Wikipedia, Permutation classes avoiding two patterns of length 4.

Cf. A053614, A106228, A165542, A165545, A257561, A257562.

a(0)=1 prepended by Jay Pantone, Oct 01 2015

A053617 Number of permutations of length n which avoid the patterns 1234 and 1324.

Original entry on oeis.org

1, 1, 2, 6, 22, 90, 396, 1837, 8864, 44074, 224352, 1163724, 6129840, 32703074, 176351644, 959658200, 5262988330, 29057961666, 161374413196, 900792925199, 5050924332096, 28434661250454, 160644331001476, 910455895039056, 5174722258676440, 29486753617569684

Offset: 0

Moa Apagodu, Mar 20 2000

nonn

These permutations have an "enumeration scheme" of depth 4, see D. Zeilberger's article in the links.
G.f. conjectured to be non-D-finite (see Albert et al. link). - Jay Pantone, Oct 01 2015
a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 2>4, 3>4} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the first element is the largest and the fourth element is the smallest. - Sergey Kitaev, Dec 10 2020

Andrew Baxter and Jay Pantone, Table of n, a(n) for n = 0..600 (terms n=1..100 from Andrew Baxter)
Michael H. Albert, Cheyne Homberger, Jay Pantone, Nathaniel Shar, Vincent Vatter, Generating Permutations with Restricted Containers, arXiv:1510.00269 [math.CO], 2015.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Kremer, Darla and Shiu, Wai Chee, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
Wikipedia, Permutation classes avoiding two patterns of length 4.
D. Zeilberger, Enumeration schemes and more importantly their automatic generation, Annals of Combinatorics 2 (1998) 185-195. The link is to an overview on Doron Zeilberger's home page; there is a local copy here [Pdf file only, no active links]

Cf. A032351, A053614, A106228, A165542, A165545, A257561, A257562.

More terms from Andrew Baxter, May 20 2011

cp's OEIS Frontend

A165542 Number of permutations of length n which avoid the patterns 4231 and 4123.

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A165545 Number of permutations of length n which avoid the patterns 2341 and 3421.

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A053617 Number of permutations of length n which avoid the patterns 1234 and 1324.

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