Kerodon

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Notation 2.5.7.2 ($(p,q)$-Shuffles). Let $p$ and $q$ be nonnegative integers. A $(p,q)$-shuffle is a strictly increasing map of partially ordered sets $\sigma : [p+q] \rightarrow [p] \times [q]$, which we will often identify with a nondegenerate $(p+q)$-simplex of the cartesian product $\Delta ^{p} \times \Delta ^{q}$.

If $\sigma$ is a $(p,q)$-shuffle, we let $\sigma _{-}: [p+q] \rightarrow [p]$ and $\sigma _{+}: [p+q] \rightarrow [q]$ denote the nondecreasing maps given by the components of $\sigma$ (so that $\sigma (i) = ( \sigma _{-}(i), \sigma _{+}(i) )$ for $0 \leq i \leq p+q$). Let $I_{-}$ denote the set of integers $1 \leq i \leq p+q$ satisfying $\sigma _{-}(i-1) < \sigma _{-}(i)$ (or equivalently $\sigma _{+}(i-1) = \sigma _{+}(i)$), and let $I_{+}$ the set of integers $1 \leq i \leq p+q$ satisfying $\sigma _{+}(i-1) < \sigma _{+}(i)$ (or equivalently $\sigma _{-}(i-1)= \sigma _{-}(i)$). We let $(-1)^{\sigma }$ denote the product

$\prod _{ (i,j) \in I_{-} \times I_{+} } \begin{cases} 1 & \text{ if } i < j \\ -1 & \text{ if } i > j. \end{cases}$

We will refer to $(-1)^{\sigma }$ as the sign of the $(p,q)$-shuffle $\sigma$.