OFFSET
0,4
COMMENTS
Conjecture: a(n) = number of degree-n permutations of prime order.
The conjecture is false. Cf. A214003. This sequence gives the number of n-permutations whose cycle lengths are restricted to the prime numbers. - Geoffrey Critzer, Nov 08 2015
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..450
Ljuben Mutafchiev, A Note on the Number of Permutations whose Cycle Lengths Are Prime Numbers, arXiv:2108.05291 [math.CO], 2021.
EXAMPLE
E.g.f.: A(x) = 1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 44*x^5/5! + 55*x^6/6! + 1434*x^7/7! + ...
where
log(A(x)) = x^2/2 + x^3/3 + x^5/5 + x^7/7 + x^11/11 + x^13/13 + x^17/17 + x^19/19 + x^23/23 + x^29/29 + ... + x^prime(n)/prime(n) + ...
a(5) = 44 because there are 5!/5 = 24 permutations that are 5-cycles and there are 5!/(2*3) = 20 permutations that are the disjoint product of a 2-cycle and a 3-cycle. - Geoffrey Critzer, Nov 08 2015
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(isprime(j),
a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
end:
seq(a(n), n=0..25); # Alois P. Heinz, May 12 2016
MATHEMATICA
f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n], Apply[And, PrimeQ[#]] &]]], {n, 0, 23}] (* Geoffrey Critzer, Nov 08 2015 *)
PROG
(PARI) {a(n)=n!*polcoeff(exp(sum(k=1, n, x^prime(k)/prime(k))+x*O(x^n)), n)}
for(n=0, 31, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 17 2012
STATUS
approved