cp's OEIS Frontend

A384630 Number of self-inverse double cosets in Z_n\S_n/Z_n.

%I A384630 #13 Jun 17 2025 19:28:37
%S A384630 1,1,2,3,6,14,34,98,294,952,3246,11698,43732,170752,689996,2888034,
%T A384630 12458784,55406422,253142182,1187934740,5712033368,28131119956,
%U A384630 141645386202,728841303696,3827217750492,20499431084644,111876916526070,621831335167486,3516904353610572
%N A384630 Number of self-inverse double cosets in Z_n\S_n/Z_n.
%C A384630 Z_n is the cyclic group of order n, seen as a subgroup of the symmetric group S_n.
%C A384630 Cosets in S_n/Z_n are in bijection with cycles obtained by connecting cyclically n equally spaced points on a circle. Double cosets in Z_n\S_n/Z_n are in bijection with cycles up to rotation.
%H A384630 Ludovic Schwob, <a href="https://arxiv.org/abs/2506.04007">On the enumeration of double cosets and self-inverse double cosets</a>, arXiv:2506.04007 [math.CO], 2025. See Proposition 4.1 p. 9.
%o A384630 (Python) # From Proposition 4.1 in the reference:
%o A384630 from sympy import factorial,divisors,totient
%o A384630 def A384630(n):
%o A384630     s = 0
%o A384630     if n%2==0:
%o A384630         for d in divisors(n//2):
%o A384630             if d%2==0:
%o A384630                 s += totient(d)*(d//2)**(n//2//d)*factorial(n//d)//factorial(n//2//d)
%o A384630             else:
%o A384630                 s += totient(d)*sum(factorial(n//d)*d**i//2**i//factorial(i)//factorial(n//d-2*i) for i in range(n//2//d+1))
%o A384630     else:
%o A384630         for d in divisors(n):
%o A384630             s += totient(d)*sum(factorial(n//d)*d**i//2**i//factorial(i)//factorial(n//d-2*i) for i in range(n//d//2+1))
%o A384630     return s//n
%Y A384630 Cf. A000142 (cycles), A002619 (cycles up to rotation), A384631 (self-inverse polygons).
%K A384630 nonn
%O A384630 1,3
%A A384630 _Ludovic Schwob_, Jun 05 2025