A370652 Number of permutations of [n] having exactly two adjacent 4-cycles.

0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 12, 60, 357, 2508, 20100, 181080, 1811886, 19938270, 239319540, 3111697260, 43569197270, 653597773860, 10458282340380, 177800134878240, 3200533135400175, 60812090365924905, 1216273182165519240, 25542270225880538760

Offset: 0

Author

Seiichi Manyama, Feb 24 2024

Keywords

• nonn

Links

• R. A. Brualdi and Emeric Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.

Crossrefs

Column k=2 of A177252.

Cf. A370426, A370528.

Programs

PARI

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=2, N, k!*x^(k+6)/(1+x^4)^(k+1))/2))

PARI

a(n, k=2, q=4) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;

Formula

G.f.: (1/2) * Sum_{k>=2} k! * x^(k+6) / (1+x^4)^(k+1).

a(n) = (1/2) * Sum_{k=0..floor(n/4)-2} (-1)^k * (n-3*k-6)! / k!.