A370528 Number of permutations of [n] having exactly two adjacent 3-cycles.

0, 0, 0, 0, 0, 0, 1, 3, 12, 57, 348, 2460, 19806, 178950, 1794420, 19778210, 237696420, 3093642300, 43350548655, 650733622665, 10417925247240, 177191430300339, 3190747212651432, 60645032890871688, 1213255040678034508, 25484737348664027532, 560785511736390349080

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=2 of A177250.

Cf. A177251, A370525, A370530.

Programs

Magma

[n le 5 select 0 else (&+[(-1)^k*Factorial(n-2*k-4)/Factorial(k): k in [0..Floor((n-6)/3)]])/2: n in [0..30]]; // G. C. Greubel, May 01 2024

Mathematica

Table[Sum[(-1)^k*(n-2*k-4)!/k!, {k,0,Floor[(n-6)/3]}]/2, {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], Vec(sum(k=2, N, k!*x^(k+4)/(1+x^3)^(k+1))/2))

PARI

a(n, k=2, 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-4)/factorial(k) for k in range(1+(n-6)//3))/2 for n in range(31)] # G. C. Greubel, May 01 2024

Formula

G.f.: (1/2) * Sum_{k>=2} k! * x^(k+4) / (1+x^3)^(k+1).

a(n) = (1/2) * Sum_{k=0..floor(n/3)-2} (-1)^k * (n-2*k-4)! / k!.