A324133 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 12 and t = 12.
1, 1, 2, 6, 24, 114, 608, 3554, 22480, 152546, 1103200, 8456994, 68411632, 581745250, 5183126016, 48245682338, 467988498064, 4720072211938, 49400302118560, 535546012710434, 6004045485933104, 69507152958422370, 829789019700511040, 10202854323325253538, 129061753086335478736
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..558
- Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.
Programs
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Maple
b:= proc(n, m) option remember; `if`(n=0, ceil(2^(m-2)), m*b(n-1, m)+b(n-1, m+1)) end: a:= n-> b(n, 0): seq(a(n), n=0..24); # Alois P. Heinz, Aug 25 2021 -
Mathematica
b[n_, m_] := b[n, m] = If[n == 0, Ceiling[2^(m-2)], m*b[n-1, m] + b[n-1, m+1]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Apr 15 2022, after Alois P. Heinz *)
Formula
a(n) = -2^(n-1) + 2*Sum_{i = 0..n-1} binomial(n-1,i) * a(i) with a(0) = 1. [It follows from Kitaev's recurrence for C_n on p. 220 of his paper.] - Petros Hadjicostas, Oct 30 2019
From Alois P. Heinz, Aug 25 2021: (Start)
G.f.: Sum_{k>=0} ceiling(2^(k-2))*x^k / Product_{j=1..k} (1-j*x).
a(n) = Sum_{j=0..n} Stirling2(n,j)*ceiling(2^(j-2)). (End)
Extensions
More terms from Petros Hadjicostas, Oct 30 2019
Comments