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%I #21 Jul 10 2023 13:06:58
%S 1,1,2,5,14,43,150,617,3050,18059,125930,1013553,9246862,94226123,
%T 1059882190,13035332841
%N Number of permutations in S_n avoiding 345{bar 2}{bar 1} (i.e., every occurrence of 345 is contained in an occurrence of a 34521).
%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,more
%O 0,3
%A _Lara Pudwell_, Apr 25 2008
%E a(8)-(15) from _Lars Blomberg_, Jun 06 2018
%E a(0)=1 prepended by _Alois P. Heinz_, Jul 10 2023
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