Enumeration of the distinct shuffles of the permutations

A shuffle of two words is a word obtained by concatenating the two original words in either order and then sliding any letters from the second word back past letters of the first word, in such a way that the letters of each original word remain spelled out in their original relative order. Examples of shuffles of the words 1234 and 5678 are, for instance, 15236784 and 51236748. In this dissertation, we enumerate the distinct shuffles of two permutations of any two lengths, where the permutations are written as words in the letters 1, 2, 3, . . . , m and 1, 2, 3, . . . , n, respectively.