Definition:Summation/Inequality/Multiple Indices/Examples/Sum of Subscripts
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Example of Summation by Inequality over Multiple Indices
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:
- $a_{11111} + a_{21110} + a_{22100} + a_{31100} + a_{32000} + a_{41000} + a_{50000}$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: $(22)$