Let $n\geq 1$, let $[n]=\{1,\dots,n\}$, and let $\overline{[n]}=\{\bar{1},\dots,\bar{n}\}$ be a disjoint barred copy of $[n]$. Let $NC(n)$ be the lattice of noncrossing partitions of $[n]$ ordered by refinement, so that $\pi \leq \sigma$ means every block of $\pi$ is contained in a block of $\sigma$. Let $D$ be a closed disc whose boundary contains the $2n$ labelled points in clockwise order

\begin{align*} 1,\bar{1},2,\bar{2},\dots,n,\bar{n}. \end{align*}

For each $\rho \in NC(n)$, draw every block of $\rho$ as the closed convex polygonal cell with that block as its cyclically ordered vertex set, with singleton blocks drawn as their boundary vertices. Let $K:NC(n)\to NC(n)$ denote the Kreweras complement: $K(\rho)$ is the unique maximal partition $\kappa$ of $\overline{[n]}$ such that the combined partition $\rho \cup \kappa$ is noncrossing in this interlaced disc model. Equivalently, the blocks of $K(\rho)$ are the barred boundary vertices lying in the same connected complementary component of the unbarred polygonal drawing of $\rho$. After identifying $\overline{[n]}$ with $[n]$ by $\bar{i}\mapsto i$, regard $K(\rho)$ as an element of $NC(n)$. Then, for all $\pi,\sigma \in NC(n)$,

\begin{align*} \pi \leq \sigma \implies K(\sigma) \leq K(\pi). \end{align*}

Moreover, for every $\pi \in NC(n)$,

\begin{align*} |\pi| + |K(\pi)| = n+1, \end{align*}

where $|\rho|$ denotes the number of blocks of a partition $\rho$.