A370524 Number of permutations of [n] having exactly one adjacent 2-cycle.
0, 0, 1, 2, 4, 18, 99, 612, 4376, 35620, 324965, 3285270, 36462924, 440840358, 5767387591, 81184266632, 1223531387056, 19657686459528, 335404201199049, 6056933308042410, 115417137054004820, 2314399674388138810, 48717810299204919851, 1074106226256896375532
Offset: 0
Keywords
Examples
The permutations of {1,2,3} having exactly one adjacent 2-cycle are (12)(3) and (1)(23). So a(3) = 2.
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.
Programs
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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!;
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!.