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 A113226 Number of permutations avoiding the pattern 12-34. 1
 1, 2, 6, 23, 107, 585, 3669, 25932, 203768, 1761109, 16595757, 169287873, 1857903529, 21823488238, 273130320026, 3627845694283, 50962676849199, 754814462534449, 11754778469338581, 191998054346198680 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) is the number of permutations on [n] that avoid the mixed consecutive/scattered pattern 12-34 (also number that avoid 43-21). LINKS A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011. Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv preprint arXiv:1108.2642 [math.CO], 2011. Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, arXiv:math/0505254 [math.CO], 2005. Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. in Appl. Math. 36 (2006), no. 2, 138-155. Steven Finch, Pattern-Avoiding Permutations [Broken link?] Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission] FORMULA 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 = 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[Binomial[b - k - If[k <= a, 1, 0], j1] Binomial[ 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[Binomial[n - b, i] Binomial[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}] CROSSREFS Sequence in context: A200405 A336071 A200403 * A071075 A007555 A101053 Adjacent sequences: A113223 A113224 A113225 * A113227 A113228 A113229 KEYWORD nonn AUTHOR David Callan, Oct 19 2005 STATUS approved

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)