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A384631 Number of self-inverse double cosets in D_n\S_n/D_n.

%I A384631 #15 Jun 22 2025 19:35:43
%S A384631 1,2,4,8,17,52,153,482,1623,5879,21926,85436,344998,1444437,6230232,
%T A384631 27704051,126571091,593974930,2856031804,14065575098,70822693101,
%U A384631 364420818168,1913609207886,10249715874962,55938458263035,310915671908063,1758452185453926,10115287840489764
%N A384631 Number of self-inverse double cosets in D_n\S_n/D_n.
%C A384631 D_n is the dihedral group of order 2*n, seen as a subgroup of the symmetric group S_n.
%C A384631 Cosets in S_n/D_n are in bijection with polygons obtained by connecting cyclically n equally spaced points on a circle. Double cosets in D_n\S_n/D_n are in bijection with polygons up to rotation and reflection.
%H A384631 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 4.2 p.9.
%o A384631 (Python) # From Proposition 4.2 in the reference:
%o A384631 from sympy import divisors, factorial, totient
%o A384631 def A384631(n):
%o A384631     s = 0
%o A384631     if n%2==0:
%o A384631         for d in divisors(n//2):
%o A384631             if d%2==0:
%o A384631                 s += totient(d)*factorial(n//d)*(d//2)**(n//2//d)//factorial(n//2//d)
%o A384631             else:
%o A384631                 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 A384631         s += n//2*((n%4)//2+1)*factorial(2*(n//4))//factorial(n//4)
%o A384631     else:
%o A384631         for d in divisors(n):
%o A384631             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 A384631         if n%4==1:
%o A384631             s += n*factorial(n//2)//factorial(n//4)
%o A384631     return s//n//2
%Y A384631 Cf. A001710 (polygons), A000940 (polygons up to rotation and reflection), A384630 (self-inverse cycles).
%K A384631 nonn
%O A384631 3,2
%A A384631 _Ludovic Schwob_, Jun 05 2025