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A370524 Number of permutations of [n] having exactly one adjacent 2-cycle. 5
0, 0, 1, 2, 4, 18, 99, 612, 4376, 35620, 324965, 3285270, 36462924, 440840358, 5767387591, 81184266632, 1223531387056, 19657686459528, 335404201199049, 6056933308042410, 115417137054004820, 2314399674388138810, 48717810299204919851, 1074106226256896375532 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
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
Column k=2 of A370527.
Column k=1 of A177248
Sequence in context: A120664 A095816 A020101 * A099938 A135069 A067647
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 21 2024
STATUS
approved

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Last modified July 23 04:10 EDT 2024. Contains 374544 sequences. (Running on oeis4.)