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

1, 1, 2, 3, 6, 14, 34, 98, 294, 952, 3246, 11698, 43732, 170752, 689996, 2888034, 12458784, 55406422, 253142182, 1187934740, 5712033368, 28131119956, 141645386202, 728841303696, 3827217750492, 20499431084644, 111876916526070, 621831335167486, 3516904353610572

Offset: 1

Author

Ludovic Schwob, Jun 05 2025

Keywords

• nonn

Comments

Z_n is the cyclic group of order n, seen as a subgroup of the symmetric group S_n.

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.

Links

• Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See Proposition 4.1 p. 9.

Crossrefs

Cf. A000142 (cycles), A002619 (cycles up to rotation), A384631 (self-inverse polygons).

Programs

Python

# From Proposition 4.1 in the reference:
from sympy import factorial,divisors,totient
def A384630(n):
    s = 0
    if n%2==0:
        for d in divisors(n//2):
            if d%2==0:
                s += totient(d)*(d//2)**(n//2//d)*factorial(n//d)//factorial(n//2//d)
            else:
                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))
    else:
        for d in divisors(n):
            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))
    return s//n