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%I #21 Jul 10 2023 11:40:42
%S 1,1,2,6,23,104,535,3082,19763,140885,1117101,9853890,96543043,
%T 1046549545,12480046880,162595262990
%N Number of permutations in S_n avoiding 5234{bar 1} (i.e., every occurrence of 5234 is contained in an occurrence of a 52341).
%C From _Lara Pudwell_, Oct 23 2008: (Start)
%C 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.
%C 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.
%C 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.
%C 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)
%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/pudwell_thesis.pdf">Enumeration Schemes for Pattern-Avoiding Words and Permutations</a>, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.
%H Lara Pudwell, <a href="https://doi.org/10.37236/301">Enumeration schemes for permutations avoiding barred patterns</a>, El. J. Combinat. 17 (1) (2010) R29.
%e 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.
%e 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.)
%e The 16 permutations that have a 5234 pattern not followed by an even smaller number are:
%e {[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]}
%e For example, in 25134, 5134 forms a 5234 pattern that is not followed by something even smaller.
%Y Cf. A137536-A137548, A117107.
%K nonn,more
%O 0,3
%A _Lara Pudwell_, Apr 25 2008
%E a(8)-(15) from _Lars Blomberg_, Jun 05 2018
%E a(0)=1 prepended by _Alois P. Heinz_, Jul 10 2023
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