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A372579
Number of permutations of [n] such that the number of cycles of length k is zero or equals k for every k.
5
1, 1, 0, 0, 3, 15, 0, 0, 0, 2240, 22400, 0, 0, 4804800, 67267200, 0, 3405402000, 57891834000, 0, 0, 49497518070000, 1039447879470000, 0, 0, 0, 56947245360343962624, 1480628379368943028224, 0, 0, 4057662073660588368847872, 121729862209817651065436160, 0, 0, 0
OFFSET
0,5
LINKS
Wikipedia, Permutation
FORMULA
a(n) = 0 <=> n in { A001422 }.
a(n) > 0 <=> n in { A003995 }.
EXAMPLE
a(5) = 15 = 5*3: (1)(23)(45), (1)(24)(35), (1)(25)(34), ..., (1,2)(3,4)(5),
(1,3)(2,4)(5), (1,4)(2,3)(5).
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(j=0 or j=i, b(n-i*j, i-1)*(i-1)!^j/j!*
combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jul 04 2024
STATUS
approved