A370530 Number of permutations of [n] having exactly three adjacent 3-cycles.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 20, 116, 820, 6600, 59650, 598140, 6592740, 79232140, 1031214100, 14450182880, 216911207555, 3472641749080, 59063810100120, 1063582404217144, 20215010963623896, 404418346892678160, 8494912449554675844, 186928503912130116360

Offset: 0

Author

Seiichi Manyama, Feb 21 2024

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..450
• R. A. Brualdi and Emeric Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.

Crossrefs

Column k=3 of A177250.

Cf. A177251, A370525, A370528.

Programs

Magma

[n le 8 select 0 else (&+[(-1)^k*Factorial(n-2*k-6)/Factorial(k): k in [0..Floor((n-9)/3)]])/6: n in [0..30]]; // G. C. Greubel, May 01 2024

Mathematica

Table[Sum[(-1)^k*(n-2*k-6)!/k!, {k,0,Floor[(n-9)/3]}]/6, {n,0,30}] (* G. C. Greubel, May 01 2024 *)

PARI

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=3, N, k!*x^(k+6)/(1+x^3)^(k+1))/6))

PARI

a(n, k=3, q=3) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;

SageMath

[sum((-1)^k*factorial(n-2*k-6)/factorial(k) for k in range(1+(n-9)//3))/6 for n in range(31)] # G. C. Greubel, May 01 2024

Formula

G.f.: (1/6) * Sum_{k>=3} k! * x^(k+6) / (1+x^3)^(k+1).

a(n) = (1/6) * Sum_{k=0..floor(n/3)-3} (-1)^k * (n-2*k-6)! / k!.