A324130 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 1 and t = 132.

1, 1, 2, 6, 24, 116, 652, 4178, 30070, 240164, 2107606, 20156458, 208639514, 2323794632, 27709659880, 352203163790, 4753474785808, 67889631514128, 1022936113573148, 16216615869916570, 269816176058513398, 4701111255106851632, 85599799432794431978, 1625828159969984754538

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Formula

From Petros Hadjicostas, Oct 29 2019: (Start)

Let b(n) = A111004(n) = number of permutations avoiding a consecutive 132 pattern. Then a(n) = 2*a(n-1) - b(n-1) + Sum_{i = 1..n-1} binomial(n-1,i) * b(i) * a(n-1-i) for n >= 1 with a(0) = b(0) = 1. [See the recurrence for C_n on p. 220 of Kitaev (2005).]

E.g.f.: If A(x) is the e.g.f. of (a(n): n >= 0) and B(x) is the e.g.f. of (b(n): n >= 0) (i.e., B(x) = 1/(1 - Int(exp(-t^2/2), t = 0..x))), then A'(x) = (1 + B(x)) * A(x) - B(x) with A(0) = B(0) = 1. [Theorem 16, p. 219, in Kitaev (2005)] (End)

Crossrefs

Cf. A000142, A111004.

Sequence in context: A342141 A266332 A007405 * A324131 A221988 A329788

Adjacent sequences: A324127 A324128 A324129 * A324131 A324132 A324133

Keyword

nonn

Author

N. J. A. Sloane, Feb 16 2019

Extensions

More terms from Petros Hadjicostas, Oct 29 2019 using a recurrence by Kitaev (2005)

Status

approved