0,3

a(n) is the number of permutations on [n] that avoid the consecutive pattern 3412 (also number that avoid 2143).

Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. in Appl. Math., 36 (2006), no. 2, 138-155.

Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.

Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns.

In the recurrence coded in Mathematica below, w[n, a] = #3412-avoiding permutations on [n] with first entry a; u[n, a, b] is the number that start with an ascent a<b, and v[n, a] is the number that start with a descent from a (n>=2). The main sum for u[n, a, b] counts by length k of the longest initial increasing subsequence. The cases k=2, k=3, k>=4 are considered separately.

The 5!-a[5] = 10 permutations on [5] not counted by a[5] are

14523, 24513, 34125, 34512, 35124, 43512, 45123, 45132, 45231, 53412.

Adjacent sequences: A113226 A113227 A113228 this_sequence A113230 A113231 A113232

Sequence in context: A002136 A117226 A117156 this_sequence A113228 A063255 A117158

nonn

David Callan (callan(AT)stat.wisc.edu), Oct 19 2005