The h, k-equal Partition Lattice is EL-shellable when h $\geq$ k

Abstract

It has been shown that the h, k-equal partition lattice \(\tilde \Pi_n^{h, k}\) is EL-shellable when h < k. We produce an EL-shelling for \(\tilde \Pi_n^{h, k}\) when n ≥ h ≥ k ≥ 2 and observe that, in this shelling, there are no weakly decreasing chains. This shows that \(\tilde \Pi_n^{h, k}\) is contractible for such values of h and k, which can also be seen by the fact that \(\tilde \Pi_n^{h, k}\) is noncomplemented.