A370525 Number of permutations of [n] having exactly one adjacent 3-cycle.

0, 0, 0, 1, 2, 6, 22, 114, 696, 4923, 39612, 357900, 3588836, 39556420, 475392840, 6187284605, 86701097310, 1301467245330, 20835850494474, 354382860600678, 6381494425302864, 121290065781743383, 2426510081356069016, 50969474697328055064, 1121571023472780698152

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 A370527.

Column k=1 of A177250.

Cf. A177251, A370528, A370530.

Programs

PARI

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

PARI

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

Formula

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

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