Symmetric brace algebras - unshuffle sequences I'm studying brace algebras in this article: Symmetric Brace Algebras.
In the following definition, what do the authors mean by "unshuffle sequences"?

Definition 2. A symmetric brace algebra is a graded vector space $B$ together with a collection of degree $0$ multilinear braces $x\langle x_1,\ldots,x_n\rangle$ that are graded symmetric in $x_1,\ldots,x_n$ and satisfy the identities
  $$x\langle\rangle=x$$
  and
  $$x\langle x_1,\ldots,x_m\rangle\langle y_1,\ldots,y_n\rangle=\sum\epsilon\cdot x\langle x_1\langle y_{i_1^1},\ldots,y_{i_{t_1}^1}\rangle,x_2\langle y_{i_1^2},\ldots,y_{i_{t_2}^2}\rangle,\ldots,x_m\langle y_{i_1^m},\ldots,y_{i_{t_m}^m}\rangle,y_{i_1^{m+1}},\ldots,y_{i_{t_{m+1}}^{m+1}}\rangle$$
  where the sum is taken over all unshuffle sequences
  $$i_1^1<\cdots<i_{t_1}^1,\ldots,i_1^{m+1}<\cdots<i_{t_{m+1}}^{m+1}$$
  of $\{1,\ldots,n\}$ and where $\epsilon$ is the Koszul sign of the permutation
  $$(x_1,\ldots,x_m,y_1\ldots,y_n)\longmapsto(x_1,y_{i_1^1},\ldots,y_{i_{t_1}^1},\ldots,x_2,y_{i_1^2},\ldots,y_{i_{t_2}^2},\ldots,x_m,y_{i_1^m},\ldots,y_{i_{t_m}^m},y_{i_1^{m+1}},\ldots,y_{i_{t_{m+1}}^{m+1}})$$
  of elements of $B$.

Thank you!
 A: The definition of shuffles or unshuffles sequences changes in the books. I prefer the definition that appears  in the book Algebraic Operads by Loday and Vallette. 
A  $ (p, q) $ - shuffle is a sequence of integers
  $$(i_1,\ \ldots,\ i_p \ | \ j_1,\ \ldots,\ j_q)$$ 
which is a permutation of $ \{ 1, \ldots, p + q \} $ such that
  $ i_1 <\ldots <i_p $ and $ j_1 <\ldots <j_q $.
 
  The permutation associated with $\sigma \in S_{p + q}$ given by
  $$ \sigma (1) = i_1, \ldots, \sigma (p) = i_p, \sigma(p + 1) = j_1, \ldots, \sigma (p + q) = j_q $$
  is, by abuse of notation, also called $ (p, q) $ - shuffle. 
We will denote by $Sh(p,q)$ the subset of $ S_{p + q}$ formed by $(p, q) $ - shuffles.
 
  Let's look at a example of a $(p,q)$ - shuffle. The only 
  $ (1,2) $ - shuffles are $ (1 | 23) $, $ (2 | 13) $, $ (3 | 12) $. Note that identity permutation is both a
$ (1,2) $ -shuffle as a $ (2,1) $ -shuffle. 
The notion of  $ (i_1 ,\ \ldots, \ i_k) $ -shuffles  extends naturally.
So what the authors call of "unshuffle sequences" is the set of  $(t_1, \ldots, t_{m+1})$-shuffles of  $\{1, \ldots, n \} $, with $t_j \geq 0$ for all $1\leq j \leq m+1$. 
