How would an expression that looks like this be written?
$$\underbrace{\sum_{m_1=0}^{n-1}\sum_{m_2=0}^{m_1-1}\dots\sum_{m_{n-2}=0}^{m_{n-3}-1}\sum_{m_{n-1}=0}^{m_{n-2}-1}m_{n-1}}_{n-1\sum-symbols}$$
What is the correct notation for nested sigma (summation) symbols of this nature? Is this the most efficiently this equation can be written out?
Best Answer
One can write more compactly $$\sum_{\substack{0\le m_1<n\\0\le m_2<m_1\\\vdots\\0\le m_{n-2}<m_{n-3}\\0\le m_{n-1}<m_{n-2}}}m_{n-1}.$$ Personally, I would first define a set $$S=\{(m_1,\dots,m_{n-1})\in\mathbb N^{n-1}:n>m_1>m_2>\dots>m_{n-2}>m_{n-1}\ge0\}$$ and then write $$\sum_{(m_1,\dots,m_{n-1})\in S}m_{n-1}.$$