A137546 Number of permutations in S_n avoiding 5234{bar 1} (i.e., every occurrence of 5234 is contained in an occurrence of a 52341).

1, 1, 2, 6, 23, 104, 535, 3082, 19763, 140885, 1117101, 9853890, 96543043, 1046549545, 12480046880, 162595262990

Offset: 0

Author

Lara Pudwell, Apr 25 2008

Keywords

• nonn
• more

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)

Examples

			a(5) = 104: There are 16 permutations that have a 5234 *pattern* that is not followed by a 1. This is different from looking for the string 5234 followed by 1.
A 5234 pattern is a string of 4 numbers abcd where b<c<d<a (i.e. the string has the same relative order as the numbers 5234.)
The 16 permutations that have a 5234 pattern not followed by an even smaller number are:
{[1, 5, 2, 3, 4], [2, 5, 1, 3, 4], [3, 5, 1, 2, 4], [4, 1, 2, 3, 5], [4, 1, 2, 5, 3], [4, 1, 5, 2, 3], [4, 5, 1, 2, 3], [5, 1, 2, 3, 4], [5, 1, 2, 4, 3], [5, 1, 3, 2, 4], [5, 1, 3, 4, 2], [5, 1, 4, 2, 3], [5, 2, 1, 3, 4], [5, 2, 3, 1, 4], [5, 3, 1, 2, 4], [5, 4, 1, 2, 3]}
For example, in 25134, 5134 forms a 5234 pattern that is not followed by something even smaller.
		

Links

• 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. A137536-A137548, A117107.

Extensions

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

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