Identifying transitions in an array

Question

I have two arrays of elements. The second array is generated by the transfer of one or more element to another position. I want to know which element is transferred to which position. Following is the simple example

A1 = {a, a, a, a, a, a};
A2 = {2a, 0, 2a, 0, a, a};

In this example the o/p I am looking for is {2->1 and 4->3}. How to achieve this ? Also, I want to have preference order in transitions. For example above example could also give {2->3, 4-> 1}. But I want to put conditions suggesting some transitions are less likely than others. So in above case, 2-> 3 is less probable than 2-> 1. so the o/p should be {2->1 and 4->3}.

Will be thankful for any suggestion.

Answer 1

This works for the simple example in OP:

Rule @@@ Transpose[PositionIndex[A2] /@ {0, 2 a}]

{2 -> 1, 4 -> 3}

A generalization for cases where A1 is a constant array:

ClearAll[posMap]
posMap = Module[{pos1 = Flatten[Position[#, 0]], 
  pos2 = Flatten @ MapIndexed[ConstantArray[#2[[1]], #] &, 
     Ramp[Coefficient[#, Variables[#][[1]]] - 1]]},
 Thread[pos1 -> pos2]] &;

Examples:

posMap[A2]

{2 -> 1, 4 -> 3}

posMap[{3 a, 0, a, 0, a, a}]

{2 -> 1, 4 -> 1}

posMap[{3 a, 0, 0, a, 0, 2 a, a}]

{2 -> 1, 3 -> 1, 5 -> 6}