0,2

Comment from Lara Pudwell (Lara.Pudwell(AT)valpo.edu), 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)

Number of ordered factorizations over the Gaussian polynomials.

Apparently, also the number of permutations in S_n avoiding {bar 3}{bar 1}542 (i.e. every occurrence of 542 is contained in an occurrence of a 31542). - Lara Pudwell (Lara.Pudwell(AT)valpo.edu), Apr 25 2008

See Andrews, Partitions, (1976) page 242 for table of Gaussian polynomials.

Lara Pudwell, Enumeration Schemes for Pattern-Avoiding Words and Permutations, Ph. D. Dissertation, Math. Dept., Rutgers University, May 2008.

a(3)=1+5+7+1=14.

Partial sums are in A026898, A003101.

Sequence in context: A137549 A014327 A137550 this_sequence A137551 A160701 A148333

Adjacent sequences: A047967 A047968 A047969 this_sequence A047971 A047972 A047973

nonn

Alford Arnold (Alford1940(AT)aol.com)