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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJun 26th 2016
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   Author: Mike Shulman
   Format: MarkdownItexA $(p,q)$-[[shuffle]] is a permutation of $p+q$ things that preserves the internal order of the first $p$ things and of the last $q$ things. As remarked on [wikipedia](https://en.wikipedia.org/wiki/Riffle_shuffle_permutation), since a $(p,q)$-shuffle is uniquely determined by the choice of the $p$ images of the first $p$ things, there are $\begin{pmatrix} p+q\\ p\end{pmatrix} = \frac{(p+q)!}{p! q!}$ of them. They are not a group; the product of two shuffles is not a shuffle. The *direct sum* of a permutation $\pi$ of $p$ things and a permutation $\sigma$ of $q$ things is a permutation of $p+q$ things that rearranges the first $p$ things among themselves according to $\pi$ and the last $q$ things among themselves according to $\sigma$. This defines an injective (non-normal) group homomorphism $S_p \times S_q \hookrightarrow S_{p+q}$. Note that $S_p \times S_q$ has cardinality $p!q!$, while $S_{p+q}$ has cardinality $(p+q)!$. What exactly is the relationship between these two things? It seems to me that the $(p,q)$-shuffles should be a set of canonical representatives for the cosets of $S_p\times S_q$ in $S_{p+q}$, and the cardinalities cited above bear that out. In particular, any permutation of $p+q$ should decompose uniquely into the product of a $(p,q)$-shuffle and something in the image of $S_p\times S_q$. But if so, I'm surprised that I haven't seen this stated anywhere; can anyone give a reference? I'm also interested in the obvious generalization to $(p_1,\dots,p_n)$-shuffles. And is there any group-theoretic (or category-theoretic) way to characterize the "canonicalness" of these representatives?

   A $(p,q)$-shuffle is a permutation of $p+q$ things that preserves the internal order of the first $p$ things and of the last $q$ things. As remarked on wikipedia, since a $(p,q)$-shuffle is uniquely determined by the choice of the $p$ images of the first $p$ things, there are $\begin{pmatrix} p+q\\ p\end{pmatrix} = \frac{(p+q)!}{p! q!}$ of them. They are not a group; the product of two shuffles is not a shuffle.

   The direct sum of a permutation $\pi$ of $p$ things and a permutation $\sigma$ of $q$ things is a permutation of $p+q$ things that rearranges the first $p$ things among themselves according to $\pi$ and the last $q$ things among themselves according to $\sigma$. This defines an injective (non-normal) group homomorphism $S_p \times S_q \hookrightarrow S_{p+q}$. Note that $S_p \times S_q$ has cardinality $p!q!$, while $S_{p+q}$ has cardinality $(p+q)!$.

   What exactly is the relationship between these two things? It seems to me that the $(p,q)$-shuffles should be a set of canonical representatives for the cosets of $S_p\times S_q$ in $S_{p+q}$, and the cardinalities cited above bear that out. In particular, any permutation of $p+q$ should decompose uniquely into the product of a $(p,q)$-shuffle and something in the image of $S_p\times S_q$. But if so, I’m surprised that I haven’t seen this stated anywhere; can anyone give a reference? I’m also interested in the obvious generalization to $(p_1,\dots,p_n)$-shuffles. And is there any group-theoretic (or category-theoretic) way to characterize the “canonicalness” of these representatives?

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeJun 26th 2016
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   Author: Tim_Porter
   Format: MarkdownItexThere are results on the poset of (p,q)-shuffles that may help. It seems that Dan Kan (nearly 50 years ago) worked with a lexiordering on the set of shuffles to get a description of the Samelson and Whitehead products within a simplicial group context. A brief summary was included in the survey article of Curtis on simplicial homotopy theory. I have a draft proof of the resulting formula that involves an analysis of this order and the resulting decomposition of the shuffle poset. I will send it to you by e-mail as it is too long to put here.

   There are results on the poset of (p,q)-shuffles that may help. It seems that Dan Kan (nearly 50 years ago) worked with a lexiordering on the set of shuffles to get a description of the Samelson and Whitehead products within a simplicial group context. A brief summary was included in the survey article of Curtis on simplicial homotopy theory. I have a draft proof of the resulting formula that involves an analysis of this order and the resulting decomposition of the shuffle poset. I will send it to you by e-mail as it is too long to put here.