Definition of multiple variables from a list

Question

I have a list of the form

{{0, 0, 0, 0, 0, 0, a, b, c, d, e, f}, {0, 0, 0, 0, 0, 1, a, c, b, d, f, e}, {2, 2, 2, 2, 1, 2, b, f, c, a, d, e}, ... } 

and I would like to translate it into variables of the form w[c1, c2, x, y] following these rules:

• the values c1 and c2 are taken from the numeric values of each sublist in order
• the values of x and y are taken from the string values of each sublist in order but each pair is reordered in alphanumeric order.

Following my example, I expect something of the form:

{w[0, 0, a, b] w[0, 0, c, d] w[0, 0, e, f], w[0, 0, a, c] w[0, 0, b, d] w[0, 1, e, f], w[2, 2, b, f] w[2, 2, a, c] w[1, 2, d, e], ...}

(notice the terms w[2, 2, c, a] must be w[2, 2, a, c] and similarly with w[0, 1, e, f])

I can use a replace function of the form /. {c1_,c2_,c3_,c4_, ..., a1_,a2_,a3_,a4,...} -> w[c1,c2,a1,a2] ... but it does not seem a very optimal solution. Besides, I will need to generalize it in case I change the number of elements in my initial list.

This question is a follow-up to a previous question: Fast enumeration of all perfect matchings in complete graph, where the solution was Ok for that purpose, but now I need to recover the variables in this particular form.

Answer 1

ClearAll[w, reShape]

SetAttributes[w, Orderless]

reShape = w @@@ Join[##, 2] & @@ ArrayReshape[#, {2, 3, 2}] &;

Example:

lst = {{0, 0, 0, 0, 0, 0, a, b, c, d, e, f}, {0, 0, 0, 0, 0, 1, a, c, 
    b, d, f, e}, {2, 2, 2, 2, 1, 2, b, f, c, a, d, e}};

Map[reShape] @ lst