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I am trying to input $\sum_{a_1=1}^{n-1}$$\sum_{a_2=1}^{{a_1}-1}$$\sum_{a_3=1}^{{a_2}-1}$$\sum_{a_4=1}^{{a_3}-1}$$...f({{a_1},{a_2},{a_3},{a_4}...})$ into Mathematica. The number of sums should be a variable, not a constant.

I do not know how to nest a variable amount of sums without having to modify the code every time that variable changes. Could someone please give me an example?

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Something like this ought to work for you:

With[{m = 4}, 
     Sum[F @@ Array[K, m], 
         Evaluate[Sequence @@ Prepend[Table[{K[j + 1], 1, K[j] - 1}, {j, m - 1}],
                                      {K[1], 1, n - 1}]]]]

which yields

$$\sum_{\tt K[1]=1}^{\tt -1+n} \sum_{\tt K[2]=1}^{\tt -1+K[1]} \sum_{\tt K[3]=1}^{\tt -1+K[2]} \sum_{\tt K[4]=1}^{\tt -1+K[3]} {\tt F[K[1],K[2],K[3],K[4]]}$$

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