A093374 Number of 1-2-3-avoiding permutations with exactly thrice the 1-3-2 pattern.

1, 5, 18, 57, 168, 472, 1280, 3376, 8704, 22016, 54784, 134400, 325632, 780288, 1851392, 4354048, 10158080, 23527424, 54132736, 123797504, 281542656, 637009920, 1434451968, 3215982592, 7180648448, 15971909632, 35399925760, 78198603776, 172201345024

Offset

4,2

Links

Colin Barker, Table of n, a(n) for n = 4..1000

D. Callan, A recursive bijective approach to counting permutations...

Index entries for linear recurrences with constant coefficients, signature (8, -24, 32, -16).

Formula

a(n) = C(n-3, 1)2^(n-4) + C(n-3, 1)2^(n-5) + C(n-3, 2)2^(n-7) for n<4, a(n) = 0.

G.f.: x^4*(1 - 3*x + 2*x^2 + x^3) / (1 - 2*x)^4. Corrected by Colin Barker, Feb 13 2017

From Colin Barker, Feb 13 2017: (Start)

a(n) = 2^(n-8)*(-120 + 38*n - 3*n^2 + n^3) / 3 for n>3.

a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n>7.

(End)

Mathematica

LinearRecurrence[{8, -24, 32, -16}, {1, 5, 18, 57}, 30] (* Harvey P. Dale, Apr 22 2024 *)

Prog

(PARI) a(n)=if(n<4, 0, 2^(n-4)*binomial(n-3, 1)+2^(n-5)*binomial(n-3, 2)+2^(n-7)*binomial(n-4, 3))
(PARI) Vec(x^4*(1 - 3*x + 2*x^2 + x^3) / (1 - 2*x)^4 + O(x^30)) \\ Colin Barker, Feb 13 2017

Crossrefs

Sequence in context: A307572 A325923 A335720 * A258109 A000745 A343802

Adjacent sequences: A093371 A093372 A093373 * A093375 A093376 A093377

Keyword

nonn,easy

Author

Ralf Stephan, Apr 28 2004

Status

approved