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# Summation with Multiple Indices

If your question was specifically about simplifying this particular sum, see Elaqqad's answer for a combinatorial argument on what it is equal to. If your question is more about how to manually compute the sum and how to interpret the notation, then continue reading mine.Capital Sigma Notation can have several different ways of writing what the set of values you sum over are.

In this case, for the summation: \$sumlimits_r,s,tgeq 0texts.t.rstn\$, I interpret it as summing the expression over all possible \$r,s,tinmathbbZ\$ such that the following conditions hold:\$\$begincases rgeq 0 sgeq 0 tgeq 0 rstnendcases\$\$In particular, there will be \$binomn2n\$ summands (seen from stars&bars). It doesn't matter in what way you choose to add the terms together (since there are a finite number of terms and addition is commutative), though if you wanted to do it by hand I would recommend Lexicographic order (a.

k.

a. Dictionary Order).

You would do it as: \$(0,0,n), (0,1,n-1), (0,2,n-2), dots, (0,n,0), (1,0,n-1), (1,1,n-2),dots, (n-1,0,1), (n-1,1,0), (n,0,0)\$where you order the entries according to the the size of the earliest difference.For an explicit example, with \$n2\$ you would have\$\$binomm_10binomm_20binomm_32 binomm_10binomm_21binomm_31 binomm_10binomm_22binomm_30 binomm_11binomm_20binomm_31 binomm_11binomm_21binomm_30 binomm_12binomm_20binomm_30\$\$

How would the following summation work?

\$sum_r,s,t ge 0_rstn binomm_1r binomm_2s binomm_3t\$

How would you choose the value for the next integer in the series?

For example, for n2. Either r,s, or t 2. Or r 1 and s 1 (or any combination thereof)

I think the inclusion-exclusion principle would be used but I'm don't know how it would be applied.

Sorry if this is a duplicate question. I wasn't sure how to word it.

How would the following summation work?

\$sum_r,s,t ge 0_rstn binomm_1r binomm_2s binomm_3t\$

How would you choose the value for the next integer in the series?

For example, for n2. Either r,s, or t 2. Or r 1 and s 1 (or any combination thereof)

I think the inclusion-exclusion principle would be used but I'm don't know how it would be applied.

Sorry if this is a duplicate question. I wasn't sure how to word it.

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