Enumeration of the distinct shuffles of permutations

Abstract : 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 paper, we enumerate the distinct shuffles of two permutations of any two lengths, where the permutations are written as words in the letters $1,2,3,\ldots ,m$ and $1,2,3,\ldots ,n$, respectively.
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Camillia Smith Barnes. Enumeration of the distinct shuffles of permutations. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.155-166. ⟨hal-01185410⟩

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