Pattern matching product with arbitrary number of terms

Question

I am trying to write a function to convert an expression of the form

$$\alpha x_1^{n_1} x_2^{n_2} ... x_N^{n_N}$$

(with some coefficient $\alpha\in\mathbb{R}$) into a list of indices and powers

$$\{\{1,n_1\},\{2,n_2\},...,\{N,n_N\}\}.$$

for expressions of varying length (number of products $x_i^{n_i}$).

My natural choice for this was the Cases function with pattern matching. Something along the lines of, for a three-term product,

pattern = α_ x{i_}^ni_. x{j_}^nj_. x{k_}^nk_.
Cases[expr, pattern -> {{i,ni},{j,nj},{k,nk}}],

where I write x{i_} to denote Subscript[x,i_].

My question is how to match a product of arbitrary length. Is there a way to write a pattern to match

$$\alpha \prod_j x_{i_j}^{n_{i_j}}$$

for an arbitrary $j$ and $i_j\in\mathbb{Z}$ not predefined.

Alternatively, if we assume the maximum $j$ is known, is there a way I can write a single pattern (along the lines of pattern above) such that each term is optional? I.e. modify pattern to also match 2-term and 1-term products.

Answer 1

Update: "to apply the Times/Power -> List replacement only to powers of $x_i$"

expr2 = -2 a^2 expr

expr2 /. m -> 5 /. Times[a___, p : (Power[_x, _] ..)] :> 
  {Times @ a, Sequence @@ ({p} /. Power -> List /. x -> Identity)}

{-2 a^2 α, {1, n[1]}, {2, n[2]}, {3, n[3]}, {4, n[4]}, {5, n[5]}}

Original answer:

expr = α Product[Power[x[i], n[i]], {i, 1, m}]

With numeric m you can simply replace Times and Power with List:

Rest[expr /. m -> 5 /. Times | Power -> List] /. x -> Identity

 {{1, n[1]}, {2, n[2]}, {3, n[3]}, {4, n[4]}, {5, n[5]}}

Also

Block[{Power = List, Times = List}, 
   Transpose @ First @ Rest[expr /. m -> 5]] /. x -> Identity

 {{1, n[1]}, {2, n[2]}, {3, n[3]}, {4, n[4]}, {5, n[5]}}

and

First @ Cases[expr /. m -> 5, 
   Times[α, pat : __Power] :> ReplaceAll[x -> Identity] @* List @@@ {pat}, All]

{{1, n[1]}, {2, n[2]}, {3, n[3]}, {4, n[4]}, {5, n[5]}}

Answer 2

If we can assume the variables are always in the form $x[i]$ then we have

f = With[{v = #}, {First[#], Exponent[v, #]} & /@ Cases[v, x[_], ∞]] &;

EDIT: for variables in the form $x_i$ just use f = With[{v = #}, {# /. Subscript[x, a__] -> a, Exponent[v, #]} & /@ Cases[v, Subscript[x,_],∞]] &;

This handles the initial coefficient more robustly than the other answer. The idea is the Cases identifies the variables and the Map gives back the format you like. So something like

test=8a^2 x[1]^q1 x[2]^q2 x[3]^q3 x[4]
f@test

gives

{{1,q1},{2,q2},{3,q3},{4,1}}

Note that the coefficient is filtered automatically and it works for an exponent of 1.

If we cannot assume the variables are always in the form $x_i$ and, like in a now deleted comment, the coefficient may be something like $8a^2$, then it's obviously impossible since we can't know if $a$ belongs to the coefficient or the variables. That aside, we can do something like

f[exp_, cof_] := Block[{exp2, vars, ind},
  exp2 = (exp/cof) /. Times -> List;
  vars = exp2 /. Times -> List /. Power[a_, b_] -> a;
  MapIndexed[(ind = First[#2]; {ind, Exponent[#1, vars[[ind]]]}) &, exp2]]

which takes in the coefficient as an argument. Again

test=8a^2 x[1]^q1 x[2]^q2 x[3]^q3 x[4]
f@test

gives

{{1,q1},{2,q2},{3,q3},{4,1}}

as desired.

If you don't know the coefficient apriori, you can put in 1 or simply use

f[exp_] := Block[{Times=List, vars, ind},
  vars = exp /. Power[a_, b_] -> a;
  MapIndexed[(ind = First[#2]; {ind, Exponent[#1, vars[[ind]]]}) &, exp]]

then filter the coefficient later like you said you can do in the comments.