A370426 Number of permutations of [n] having exactly two adjacent 2-cycles.

0, 0, 0, 0, 1, 3, 9, 48, 306, 2190, 17810, 162480, 1642635, 18231465, 220420179, 2883693792, 40592133316, 611765693532, 9828843229764, 167702100599520, 3028466654021205, 57708568527002415, 1157199837194069405, 24358905149602459920, 537053113128448187766

Offset: 0

Author

Seiichi Manyama, Feb 21 2024

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..452
• R. A. Brualdi and Emeric Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.

Crossrefs

Column k=2 of A177248.

Cf. A177249, A370524, A370529.

Programs

PARI

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

PARI

a(n, k=2, q=2) = 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+2) / (1+x^2)^(k+1).

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

a(n) ~ n! / (2*n^2). - Vaclav Kotesovec, May 23 2025