A370653 Number of permutations of [n] having exactly three adjacent 4-cycles.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 20, 120, 836, 6700, 60360, 603960, 6646090, 79773180, 1037232420, 14523065760, 217865924620, 3486094113460, 59266711626080, 1066844378466720, 20270696788641635, 405424394055173080, 8514090075293512920

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=3 of A177252.

Cf. A370529, A370530.

Programs

PARI

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=3, N, k!*x^(k+9)/(1+x^4)^(k+1))/6))

PARI

a(n, k=3, q=4) = 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+9) / (1+x^4)^(k+1).

a(n) = (1/6) * Sum_{k=0..floor(n/4)-3} (-1)^k * (n-3*k-9)! / k!.