A137546 -id:A137546 - cp's OEIS frontend

A137536 Number of permutations in S_n avoiding {bar 4}2153 (i.e., every occurrence of 2153 is contained in an occurrence of a 42153).

1, 1, 2, 6, 23, 104, 532, 3002, 18375, 120559, 840480, 6184729, 47788384, 386126534, 3251434927, 28454039404

Offset: 0

Lara Pudwell, Apr 25 2008

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A137536-A137548, A117107.

a(8)-(15) from Lars Blomberg, Jun 04 2018

a(0)=1 prepended by Alois P. Heinz, Jul 10 2023

A137548 Number of permutations in S_n avoiding {bar 5}1234 (i.e., every occurrence of 1234 is contained in an occurrence of a 51234).

Original entry on oeis.org

1, 1, 2, 6, 23, 104, 535, 3088, 19937, 144065, 1165109, 10518741, 105460057, 1166159683, 14117684632, 185784598853

Offset: 0

Lara Pudwell, Apr 25 2008

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A137546.

a(8)-(15) from Lars Blomberg, Jun 05 2018

a(0)=1 prepended by Alois P. Heinz, Jul 10 2023

A117107 Number of permutations in S_n avoiding 21{bar 3}54 (i.e., every occurrence of 2154 is contained in an occurrence of a 21354) and such that the graph corresponding to the permutation is connected (see "Forest-like permutations" below).

Original entry on oeis.org

1, 1, 3, 12, 57, 304, 1765, 10943, 71519, 488186, 3456526, 25251479, 189545179

Offset: 1

Steve Butler, Apr 18 2006

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

M. Bousquet-Mélou and S. Butler, Forest-like permutations, arXiv:math/0603617 [math.CO], 2006.
Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.

Cf. A137536, A137537, A137538, A137539, A137540, A137541, A137542, A137543, A137544, A137545, A137546, A137547, A137548.

A137542 Number of permutations in S_n avoiding {bar 1}3425 (i.e., every occurrence of 3425 is contained in an occurrence of a 13425).

Original entry on oeis.org

1, 1, 2, 6, 23, 104, 533, 3038, 19128, 133112, 1028287, 8859284, 85298231, 915617310, 10887682031, 142169785210

Offset: 0

Lara Pudwell, Apr 25 2008

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A137546.

a(8)-(15) from Lars Blomberg, Jun 05 2018

a(0)=1 prepended by Alois P. Heinz, Jul 10 2023

A137537 Number of permutations in S_n avoiding 5134{bar 2} (i.e., every occurrence of 5134 is contained in an occurrence of a 51342).

Original entry on oeis.org

1, 1, 2, 6, 23, 104, 532, 3003, 18400, 120960, 845823, 6249642, 48539947, 394620211, 3346466726, 29516055880

Offset: 0

Lara Pudwell, Apr 25 2008

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A137546.

a(8)-(15) from Lars Blomberg, Jun 04 2018

a(0)=1 prepended by Alois P. Heinz, Jul 10 2023

A137538 Number of permutations in S_n avoiding 25{bar 1}34 (i.e., every occurrence of 2534 is contained in an occurrence of a 25134).

Original entry on oeis.org

1, 1, 2, 6, 23, 104, 532, 3004, 18426, 121393, 851810, 6325151, 49448313, 405298482, 3470885747, 30965656442, 287083987270, 2759838731485, 27458514900626, 282264050120512, 2993392570828096, 32704759586810036, 367673428857985261

Offset: 0

Lara Pudwell, Apr 25 2008

nonn

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)
The number of permutations of length n avoiding the dashed pattern 1-42-3. - Andrew Baxter, May 17 2011
Apparently, also the number of permutations of length n avoiding the barred pattern 23{bar 1}54, which are the same as the permutations avoiding dashed pattern 1-24-3. - Andrew Baxter, May 17 2011

			See example in A137546.

Andrew M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2011.
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv preprint arXiv:1108.2642 [math.CO], 2011.
Andrea Frosini, Veronica Guerrini, and Simone Rinaldi, Constrained Underdiagonal Paths and pattern Avoiding Permutations, Preprints:202411.1611 (2024). See pp. 13-14, 16-17.
Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A110447, A113226, A137546.

Edited by Andrew Baxter, May 17 2011

Offset corrected by Alois P. Heinz, Jul 10 2023

A137539 Number of permutations in S_n avoiding 1{bar 5}324 (i.e., every occurrence of 1324 is contained in an occurrence of a 15324).

Original entry on oeis.org

1, 1, 2, 6, 23, 104, 532, 3005, 18453, 121853, 858239, 6406401, 50422167, 416666542, 3602203962, 32481902733

Offset: 0

Lara Pudwell, Apr 25 2008

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A137546.

a(8)-(15) from Lars Blomberg, Jun 04 2018

a(0)=1 prepended by Alois P. Heinz, Jul 10 2023

A137540 Number of permutations in S_n avoiding {bar 4}1253 (i.e., every occurrence of 1253 is contained in an occurrence of a 41253).

Original entry on oeis.org

1, 1, 2, 6, 23, 104, 533, 3026, 18744, 125284, 895936, 6809075, 54697741, 462353643, 4097020158, 37936453443

Offset: 0

Lara Pudwell, Apr 25 2008

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A137546.

a(8)-(15) from Lars Blomberg, Jun 05 2018

a(0)=1 prepended by Alois P. Heinz, Jul 10 2023

A137541 Number of permutations in S_n avoiding 1{bar 5}234 (i.e., every occurrence of 1234 is contained in an occurrence of a 15234).

Original entry on oeis.org

1, 1, 2, 6, 23, 104, 533, 3027, 18769, 125687, 901366, 6876307, 55498146, 471743013, 4207097145, 39237139709

Offset: 0

Lara Pudwell, Apr 25 2008

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

Cf. A137546.

a(8)-(15) from Lars Blomberg, Jun 05 2018

a(0)=1 prepended by Alois P. Heinz, Jul 10 2023

A137543 Number of permutations in S_n avoiding {bar 1}3245 (i.e., every occurrence of 3245 is contained in an occurrence of a 13245).

Original entry on oeis.org

1, 1, 2, 6, 23, 104, 534, 3060, 19445, 136976, 1072110, 9344781, 90711178, 977610989, 11627993555, 151488021053

Offset: 0

Lara Pudwell, Apr 25 2008

From Lara Pudwell, Oct 23 2008: (Start)
A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.
Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.
A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.
For example, if q = 5{bar 1}32{bar 4}, then q1 = 532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

			See example in A137546.

Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
Lara Pudwell, Enumeration schemes for permutations avoiding barred patterns, El. J. Combinat. 17 (1) (2010) R29.

a(8)-(15) from Lars Blomberg, Jun 05 2018

a(0)=1 prepended by Alois P. Heinz, Jul 10 2023

cp's OEIS Frontend

A137536 Number of permutations in S_n avoiding {bar 4}2153 (i.e., every occurrence of 2153 is contained in an occurrence of a 42153).

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A137548 Number of permutations in S_n avoiding {bar 5}1234 (i.e., every occurrence of 1234 is contained in an occurrence of a 51234).

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A117107 Number of permutations in S_n avoiding 21{bar 3}54 (i.e., every occurrence of 2154 is contained in an occurrence of a 21354) and such that the graph corresponding to the permutation is connected (see "Forest-like permutations" below).

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A137542 Number of permutations in S_n avoiding {bar 1}3425 (i.e., every occurrence of 3425 is contained in an occurrence of a 13425).

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A137537 Number of permutations in S_n avoiding 5134{bar 2} (i.e., every occurrence of 5134 is contained in an occurrence of a 51342).

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A137538 Number of permutations in S_n avoiding 25{bar 1}34 (i.e., every occurrence of 2534 is contained in an occurrence of a 25134).

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A137539 Number of permutations in S_n avoiding 1{bar 5}324 (i.e., every occurrence of 1324 is contained in an occurrence of a 15324).

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A137540 Number of permutations in S_n avoiding {bar 4}1253 (i.e., every occurrence of 1253 is contained in an occurrence of a 41253).

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A137541 Number of permutations in S_n avoiding 1{bar 5}234 (i.e., every occurrence of 1234 is contained in an occurrence of a 15234).

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