Definition:Summation/Inequality/Multiple Indices/Examples

Examples of Summation by Inequality over Multiple Indices

Sum of Subscripts

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $a$ be an $n$-tuply subscripted variable.

Consider the expression:

$\ds \sum_{\substack {j_1 \mathop + j_2 \mathop + \mathop \cdots \mathop + j_n \mathop = n \\ j_i \mathop \ge j_2 \mathop \ge \mathop \cdots \mathop \ge j_n \mathop \ge 0} } a_{j_1 j_2 \ldots j_n}$

For $n = 5$, this means:

$a_{11111} + a_{21110} + a_{22100} + a_{31100} + a_{32000} + a_{41000} + a_{50000}$