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).

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

Author

Lara Pudwell, Apr 25 2008

Keywords

• nonn

Comments

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

Examples

			See example in A137546.
		

Links

• 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.

Crossrefs

Cf. A110447, A113226, A137546.

Extensions

Edited by Andrew Baxter, May 17 2011

Offset corrected by Alois P. Heinz, Jul 10 2023