On the average rank of an element in a filter of the partition lattice (Q1317456)

Let \(F\) be the filter generated by the antichain \(A\) in the partition lattice on \(n\) elements \(P_ n\), and let \(r\) be the rank function of \(P_ n\). The author studies the colouring monotonicity and proves that \[ \bigl( 1/ | F | \bigr) \sum_{\pi \in F} r(\pi) \geq \bigl( 1/ | P_ n | \bigr) \sum_{\pi \in P_ n} r(\pi). \] Some interesting open problems are posed, too.