OFFSET
0,4
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..450
R. A. Brualdi and Emeric Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.
FORMULA
G.f.: Sum_{k>=1} k! * x^(k+1) / (1+x^2)^(k+1).
a(n) = Sum_{k=0..floor(n/2)-1} (-1)^k * (n-k-1)! / k!.
EXAMPLE
The permutations of {1,2,3} having exactly one adjacent 2-cycle are (12)(3) and (1)(23). So a(3) = 2.
PROG
(PARI) my(N=30, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, k!*x^(k+1)/(1+x^2)^(k+1))))
(PARI) a(n, k=1, q=2) = sum(j=0, n\q-k, (-1)^j*(n-(q-1)*(j+k))!/j!)/k!;
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 21 2024
STATUS
approved