Note on \(1\)-Crossing Partitions

Abstract

It is shown that there are \(\binom{2n-r-1}{n-r}\) noncrossing partitions of an \(n\)-set together with a distinguished block of size \(r\), and \(\binom{n}{k-1}\binom{n-r-1}{k-2}\) of these have \(k\) blocks, generalizing a result of Béna on partitions with one crossing. Furthermore, specializing natural \(q\)-analogues of these formulae with \(q\) equal to certain \(d^{th}\) roots of unity gives the number of such objects having \(d\)-fold rotational symmetry.