%I #18 Jul 10 2023 13:06:48
%S 1,1,2,5,14,43,147,575,2648,14617,96696,754585,6794015,69116493,
%T 781266266,9688636317,130551322618,1897079161639,29549030800315,
%U 490880073850267,8660360684895644,161671375033644161,3183279386216962364
%N Number of permutations in S_n avoiding {bar 5}{bar 4}132 (i.e., every occurrence of 132 is contained in an occurrence of a 54132).
%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.
%K nonn
%O 0,3
%A _Lara Pudwell_, Apr 25 2008
%E a(0)=1 prepended by _Alois P. Heinz_, Jul 10 2023
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