I am trying to evaluate the following term. I already have asked a question to a similar problem. $$\sum_{c_1}^N\sum_{c_2\ge c_1}^N\sum_{c_3\ge c_1}^N \delta_{c_2,c_k}P_{c_1,c_2}\delta_{c_3,c_l}P_{c_1,c_3}$$ I tried it myself: So one can group some terms: $$\sum_{c_1}^N\sum_{c_2\ge c_1}^N\delta_{c_2,c_k}P_{c_1,c_2}\sum_{c_3\ge c_1}^N \delta_{c_3,c_l}P_{c_1,c_3}$$ The terms inside the last two sums are only non-zero if $c_{2,3}=c_{k,l}$ because of the kronecker deltas. But how can i simplify this further? Edit: So I analyzed myself, but I am not sure if it is correct. Basically as I said the terms inside the last two sums contribute only if the kronecker delta doesn't vanish. But it doesn't vanish trivially if $c_{2,3}=c_{k,l}$, because the sums are not over all elements. So one gets in addition the conditions $c_1\le c_l, c_1\le c_k$. Which can be summarized as: $$c_1\le min(c_l,c_k)$$ So the final results must be: $$\sum_{c_1\le min(g_l,g_k)}P_{c_1,c_l}P_{c_1,c_k}$$
Could somebody please confirm my result?