A005225 Number of permutations of length n with equal cycles.
1, 2, 3, 10, 25, 176, 721, 6406, 42561, 436402, 3628801, 48073796, 479001601, 7116730336, 88966701825, 1474541093026, 20922789888001, 400160588853026, 6402373705728001, 133991603578884052, 2457732174030848001, 55735573291977790576, 1124000727777607680001
Offset: 1
Examples
For example, a(4)=10 since, of the 24 permutations of length 4, there are 6 permutations with consist of a single 4-cycle, 3 permutations that consist of two 2-cycles and 1 permutation with four 1-cycles. Also, a(7)=721 since there are 720 permutations with a single cycle of length 7 and 1 permutation with seven 1-cycles.
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- D. P. Walsh, A differentiation-based characterization of primes, Abstracts Amer. Math. Soc., 25 (No. 2, 2002), p. 339, #975-11-237.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..450
- R. K. Guy, Letter to N. J. A. Sloane, Jul 1988
- D. P. Walsh, Primality test based on the generating function
- D. P. Walsh, A differentiation-based characterization of primes
- H. S. Wilf, Three problems in combinatorial asymptotics, J. Combin. Theory, A 35 (1983), 199-207.
Crossrefs
Programs
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Maple
a:= n-> n!*add((d/n)^d/d!, d=numtheory[divisors](n)): seq(a(n), n=1..30); # Alois P. Heinz, Nov 07 2012
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Mathematica
Table[n! Sum[((n/d)!*d^(n/d))^(-1), {d, Divisors[n]}], {n, 21}] (* Jean-François Alcover, Apr 04 2011 *) -
Maxima
a(n):= n!*lsum((d!*(n/d)^d)^(-1),d,listify(divisors(n))); makelist(a(n),n,1,40); /* Emanuele Munarini, Feb 03 2014 */
Formula
a(n) = n!*sum(((n/k)!*k^(n/k))^(-1)) where sum is over all divisors k of n. Exponential generating function [for a(1) through a(n)]= sum(exp(t^k/k)-1, k=1..n).
a(n) = (n-1)! + 1 iff n is a prime.
Extensions
Additional comments from Dennis P. Walsh, Dec 08 2000
More terms from Vladeta Jovovic, Dec 01 2001