The spider poset is Macaulay
Let \(Q(k, \ell)\) be the partially ordered set whose Hasse diagram can be obtained from \(k+1\) copies of a chain with \(l+1\) vertices by identifying their top vertices. Let \(Q^n(k,\ell)\) be the \(n\)th cartesian power of \(Q(k,\ell)\). It is proved that \(Q^n(k,\ell)\) is a Macaulay poset for any positive integers \(k, \ell\) and \(n\). That is, there exists a (new) linear order on the ground set of \(Q^n(k,\ell)\) such that the shadow of the first \(m\) elements (\(m\) arbitrary) of any level consists of the first elements of the neighbouring level and has minimum size with respect to the shadow of any \(m\)-element subset of the given level.
scientific article; zbMATH DE number 1443247