A369098 Number of permutations of [n] having exactly one adjacent 4-cycle.

0, 0, 0, 0, 1, 2, 6, 24, 118, 714, 5016, 40200, 362163, 3623772, 39876540, 478639080, 6223394516, 87138394540, 1307195547720, 20916564680760, 355600269756485, 6401066270800350, 121624180731849810, 2432546364331038480, 51084540451761077514, 1123879093137556106358

Offset: 0

Author

Seiichi Manyama, Feb 24 2024

Keywords

• nonn

Examples

			The permutations of {1,2,3,4,5} having exactly one adjacent 4-cycle are (1234)(5) and (1)(2345). So a(5) = 2.
		

Links

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

Crossrefs

Column k=4 of A370527.

Column k=1 of A177252.

Programs

PARI

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, k!*x^(k+3)/(1+x^4)^(k+1))))

PARI

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

Formula

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

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