A370529 Number of permutations of [n] having exactly three adjacent 2-cycles.

0, 0, 0, 0, 0, 0, 1, 4, 16, 100, 730, 5940, 54160, 547540, 6077155, 73473400, 961231264, 13530711096, 203921897844, 3276281076600, 55900700199840, 1009488884673720, 19236189509000805, 385733279064689820, 8119635049867486640, 179017704376149395900

Offset: 0

Author

Seiichi Manyama, Feb 21 2024

Keywords

• nonn

Links

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

Crossrefs

Column k=3 of A177248.

Cf. A177249, A370426, A370524.

Programs

Maple

a:= proc(n) option remember; `if`(n<7, [0$6, 1][n+1], ((n-5)*(n-6)*(n-3)*a(n-1)
       -6*(n-4)*a(n-2)+(n-2)*(n-3)*((n-5)*a(n-3)+a(n-4)))/((n-5)*(n-6)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 21 2024

PARI

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0], Vec(sum(k=3, N, k!*x^(k+3)/(1+x^2)^(k+1))/6))

PARI

a(n, k=3, q=2) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;

Formula

G.f.: (1/6) * Sum_{k>=3} k! * x^(k+3) / (1+x^2)^(k+1).

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

a(n) ~ n! / (6*n^3). - Vaclav Kotesovec, Feb 21 2024