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.

Keywords

• shuffles
• permutations
• enumeration
• catalan numbers
• shuffle algebra
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]