Catalan Numbers and Interval Structures

Abstract

There are many interesting families of intervals counted by the Catalan numbers. First, however, let us make a remark about antichains in the poset Int([n]) of intervals on [n]. If $\mathcal{A}$ is such an antichain, then no two intervals in $\mathcal{A}$ can share a common left endpoint or a common right endpoint. Moreover, if we order the intervals so that the left endpoints are strictly increasing, then the right endpoints must also be strictly increasing. In fact, a family

$$ \mathrm{\mathcal{F}}=\left\{\left[{a}_i,{b}_i\right]\Big|1\le i\le n\right\} $$

of intervals in Int([n]) is an antichain if and only if both the left endpoint sequence ${a}_1\cdots {a}_n$ and the right endpoint sequence ${b}_1\cdots {b}_n$ can be ordered at the same time in strictly increasing order, in symbols, ℱ is an antichain if and only if (after reindexing if necessary)

$$ {a}_1<\cdots <{a}_n, {b}_1<\cdots <{b}_n, {a}_i\le {b}_i $$

for all i.