The term ‘shuffle’ conjures up the idea of shuffling a pack of cards. In fact the mathematical idea is nearer to shuffling two packs of cards one through the other. Suppose we have a pack of pp cards and a pack of qq cards and we build a pack of p+qp+q cards, whilst retaining the order on the two ‘sub-packs’. The result is a (p,q)(p,q)-shuffle.




For p,qp,q \in \mathbb{N} two natural numbers, a (p,q)(p,q)-shuffle is a permutation

(μ 1,,μ p,ν 1,,ν q) (\mu_1, \cdots, \mu_p, \nu_1, \cdots, \nu_q)

of (1,2,,p+q)(1,2, \cdots, p+q) subject to the condition that

μ 1<μ 2<<μ p \mu_1 \lt \mu_2 \lt \cdots \lt \mu_p


ν 1<ν 2<<ν q. \nu_1 \lt \nu_2 \lt \cdots \lt \nu_q \,.

The signature of a (p,q)(p,q)-shuffle is the signature of the corresponding permutation.

Equivalent characterizations

Two other equivalent (and dual) ways of defining the notion of (p,q)(p,q)-shuffle are as follows (e.g. Hoffbeck-Moerdijk 17, section 1.1):

  • Consider pp and qq as the linear orders [1,p]={1<<p}[1,p] = \{ 1 \lt \dots \lt p \} and [1,q]={1<<q}[1,q] = \{ 1 \lt \dots \lt q \}. Then a (p,q)(p,q)-shuffle is a way of extending the partial order on the coproduct [1,p]+[1,q][1,p] + [1,q] to a linear order, or equivalently, a surjective monotone function
    [1,p]+[1,q][1,p+q].[1,p] + [1,q] \to [1,p+q].
  • Consider pp and qq as non-empty linear orders [0,p]={0<<p}[0,p] = \{ 0 \lt \dots \lt p \} and [0,q]={0<<q}[0,q] = \{ 0 \lt \dots \lt q \}. Then a (p,q)(p,q)-shuffle is a maximal chain? within the product partial order [0,p]×[0,q][0,p] \times [0,q], or equivalently, an injective monotone function
    [0,p+q][0,p]×[0,q].[0,p+q] \to [0,p] \times [0,q].


The same concept viewed from the other end leads to unshuffles . These are just shuffles but are used in dual situations in the applications. The definition that follows is ‘from the literature’. It is equivalent to that of shuffle that we gave above. (Although not needed, it is important to note the different terminology used in certain applications of the idea for when original source material is consulted.)


We say that a permutation σS n\sigma\in S_n is a (j,nj)(j,n-j)-unshuffle, ojno\leq j\leq n if σ(1)<σ(j)\sigma(1)\lt \ldots \sigma(j) and σ(j+1)<<σ(n)\sigma(j+1)\lt \ldots \lt \sigma(n).

You can also say that σ\sigma is a (j,nj)(j,n-j)-unshuffle if σ(i)<σ(i+1)\sigma(i) \lt \sigma(i+1) when iji\neq j.


Products of simplices

Shuffles control the combinatorics of products of simplices. See products of simplices for details.

Eilenberg-Zilber map

Related to the product of simplices: shuffles control the Eilenberg-Zilber map. See there for details.

Differential graded coalgebras

Shuffles are used in defining the pre-cgc structure on V\bigwedge V in the theory of differential graded coalgebras

Differential graded Hopf algebras

Shuffles are also used for defining the shuffle product on T(V)T(V), see differential graded Hopf algebra.

In L L_\infty-algebras

In the definition of L-∞ algebras the unshuffle side of shuffles is used.


(p,q)(p,q)-shuffles (called (s,t)(s,t)-permutations) are discussed in section 2.2.3 of

The two dual equivalent characterizations of (p,q)(p,q)-shuffles (called shuffles of linear trees or shuffles of linear orders) are discussed in section 1.1 of

Revised on October 23, 2017 08:04:14 by Noam Zeilberger (