Suppose I have a set $A=\{0,1,2,\cdots,N-1\}$ and consider a partitioning of the set $A$, call it $A_p$, into $x$ pieces. Here
$$A_p=\left\{0,x,2x,\cdots,N-1\right\}=\{0,x\}\cup\{x,2x\}\cup\cdots\cup\{N-1-x,N-1\}.$$
Now suppose I want to further refine this partitioning by dividing the sets $\{0,x\},\cdots,\{N-1-x,N-1\}$ into $y$ pieces. This would be a refinement of the set $A_p$, call it $A_{p^\prime}$ given by
$$A_{p^\prime}=\{0,y,2y,\cdots ,x-y,x,x+y,\cdots,N-1-y,N-1\}.$$
Is there an elegant way in which one could sum over indexes in $A_{p^\prime}$ ensuring there is no overlapping of terms? I thought one could write
$$\sum_{a\in A_{p^\prime}}=\sum_{m=1}^{N-1}\sum_{j=(m-1)y}^{my}$$
but when $m=N-1$ we don't obtain a sum from $N-1-y\to N-1$ as I would expect and further there is an overlapping of terms that occurs. Perhaps it is not even possible to do so elegantly?
Another thought I had was that we could simply sum over indices in the set $A_p$ and emphasise that the next term in the series is computed at $a+y$ value for $a\in A_p$?