0,2

a(n) is the number of permutations on [n] that avoid the mixed consecutive/scattered pattern 12-34 (also number that avoid 43-21).

Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. in Appl. Math., to appear.

Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns..

In the recurrence coded in Mathematica below, w[n] = # (12-34)-avoiding permutations on [n]; v[n, a] is the number that start with a descent and have first entry a; u[n, a, k, b] is the number that start with an ascent and that have (i) first entry a, (ii) other than a, all ascent initiators <k, (iii) second entry b. The summation index c denotes the next ascent initiator after a. The indices j1, j2, i, j all count entries lying strictly between a and c in position and with value in the intervals: j1 in [k, b), j2 in (c, k), i in (b, n], j in (c, b).

523146 contains 2346 as a 12-34 pattern because the 23 and 46 are

adjacent in the permutation and the reduced form of 2346 is 1234.

Clear[u, v, w]; w[0]=w[1]=1; w[n_]/; n>=2 := w[n] = u[n]+v[n]; v[n_]/; n>=2 := v[n] = Sum[v[n, a], {a, 2, n}]; v[1, 1] = 1; v[n_, a_]/; 2<=a<=n := v[n, a] = Sum[u[n-1, b], {b, a-1}] + Sum[v[n-1, b], {b, 2, a-1}]; u[1] = 1; u[n_]/; n>=2 := u[n] = Sum[u[n, a], {a, n-1}]; u[1, 1] = 1; u[n_, a_]/; a==n := 0; u[n_, a_]/; 1<=a<n := u[n, a, n]; u[1, 1, k_] := 1; u[2, 1, k_] := 1; u[n_, a_, k_]/; a>=n := 0; u[n_, a_, k_]/; 1<=a<n && n>=3 := u[n, a, k] = Sum[u[n, a, k, b], {b, a+1, n}]; u[n_, a_, k_, b_]/; 1<=a<b<=n && k>=b+2 := u[n, a, b+1, b]; u[n_, a_, k_, b_]/; 1<=a<n && b==n && k==n+1 := u[n, a, n, n]; u[n_, a_, k_, b_]/; 1==a<b==n && k==2 := 1; u[n_, a_, k_, b_]/; 1<=a<b<=n && k<=b := u[n, a, k, b] = Sum[bi[b-k-If[k<=a, 1, 0], j1]bi[k-1-If[a<k, 1, 0]-c, j2]* u[n-2-j1-j2, c, k-If[a<k, 1, 0]-j2], {c, k-1-If[a<k, 1, 0]}, {j1, 0, b-k-If[k<=a, 1, 0]}, {j2, 0, k-1-If[a<k, 1, 0]-c}]; u[n_, a_, k_, b_]/; 1<=a<b<n && k==b+1 && {a, b}=={1, 2} := 1; u[n_, a_, k_, b_]/; 1<=a<b<n && k==b+1 && {a, b}!={1, 2} := u[n, a, k, b] = Sum[bi[n-b, i]bi[b-2-c, j]u[n-2-i-j, c, b-1-j], {c, b-2}, {i, 0, n-b}, {j, 0, b-2-c}]; Table[w[n], {n, 0, 15}]

Adjacent sequences: A113223 A113224 A113225 this_sequence A113227 A113228 A113229

Sequence in context: A125273 A130908 A000772 this_sequence A071075 A007555 A101053

nonn

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