I am currently trying to get my hands on double coset rings. I use the definition given in wikipedia, namely if $G$ is a finite group and $H\subset G$ is a subgroup, then for any double coset $HgH$ we can form the element $[HgH]=\sum_{x\in HgH}x\in \mathbb{Z}[G]$, and the subgroup generated by the $[HgH]$ is actually a subring.
I'd like to understand the structure constants of this ring. It is clear that the double cosets that appear in the decomposition of $[HgH]\cdot [Hg'H]$ are exactly those of the form $Hghg'H$ with $h\in H$, and I computed that the multiplicity of $[Hgg'H]$ in this product is $|H|\cdot N(g,g')$, where $$ N(g,g') = \frac{|H\cap H^{gg'}|\cdot |H\cap H^{g^{-1}}H^{g'}|}{|H\cap H^{g}|\cdot |H\cap H^{g'}|}$$ and $H^g=gHg^{-1}$ is the conjugate of $H$ given by $g$.
I am led by several motivations to believe that $N(g,g')$ is itself an integer ($|H|\cdot N(g,g')$ must certainly be an integer unless I made a mistake), but I can't really find a group-theoretical reason why. Is there a well-known formula that takes care of this?
And if it is indeed an integer, how much does it depend on $g$ and $g'$? When $H$ is a normal subgroup then clearly $N(g,g')=1$, and this seemed to be the case in all the other cases I looked at, but it seems a little too good to be true.