# Sum over multiple indices that take specific values

Pretty simple question: how do I sum over multiple indices that can only take specific values all together?

To clarify: let's say I have the following sum:

$\sum p_{a,b,c} f(a,b,c)$

where $f$ is a function and $p$ is a coefficient (number) still in symbolic form, both dependent on indices $a,b,c$. Here {$a$, $b$, $c$} can take values specified by a list of three numbers, like {1,2,3}, {3,6,7} and so on. My idea is that I have a list of triplets, and I would like the sum to run on these triplets by assigning $a$ to the first number, $b$ to the second and $c$ to the third. So let's say that for a list of triplets

list={{1,2,3},{3,6,7},{2,6,9}}

The outcome should be

$p_{1,2,3}f[1,2,3]+p_{2,6,9}f[2,6,9]+p_{3,6,7}f[3,6,7]$

I was able to do it with Apply if there was only a function, but the coefficients $p$ need to carry the indices so it becomes tricky.

• I'm not understanding what you mean by p_{1,2,3} as that translates to {p_,2 p_, 3 p_}. But if p_ is really a function that takes a list as an argument, then you might want to try Sum[p[list[[i]]] f[list[[i, 1]], list[[i, 2]], list[[i, 3]]], {i, Length[list]}] which gets you f[1, 2, 3] p[{1, 2, 3}] + f[2, 6, 9] p[{2, 6, 9}] + f[3, 6, 7] p[{3, 6, 7}].
– JimB
Jun 9, 2015 at 13:47
• So $p$ is actually also a function of $a,b,c$? Jun 9, 2015 at 13:51
• no, $p$ carries indices $a,b,c$ but they are just labels. i edited the question. Jun 9, 2015 at 13:55
• So $p$ is symbolic and not numerical? Jun 9, 2015 at 14:00
• If p is subscripted (and using subscripts can have definite but sometimes unexpected consequences), then the following might be what you want: Sum[Subscript[p, list[[i, 1]], list[[i, 2]], list[[i, 3]]] f[list[[i, 1]], list[[i, 2]], list[[i, 3]]], {i, Length[list]}] with output f[1, 2, 3] Subscript[p, 1, 2, 3] + f[2, 6, 9] Subscript[p, 2, 6, 9] + f[3, 6, 7] Subscript[p, 3, 6, 7].
– JimB
Jun 9, 2015 at 14:10

Here's one way:

Total[Subscript[p, #] & /@ lst f @@@ lst]


or

Total[Subscript[p, #] & /@ lst f /@ lst]


But you're probably making a mistake defining your p's this way, especially if you later need to do something with them. If you are willing to define p as a function, then

Total[p @@@ lst f @@@ lst]


would work.

• (p @@@ lst).(f @@@ lst) will also work. Jun 9, 2015 at 22:41
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