Shellability of the higher pinched Veronese posets

The pinched Veronese poset \({\mathcal {V}}^{\bullet }_n\) is the poset with ground set consisting of all nonnegative integer vectors of length \(n\) such that the sum of their coordinates is divisible by \(n\) with exception of the vector \((1,\ldots ,1)\) . For two vectors \(\mathbf {a}\) and \(\mathbf {b}\) in \({\mathcal {V}}^{\bullet }_n\) , we have \(\mathbf {a}\preceq \mathbf {b}\) if and only if \(\mathbf {b}- \mathbf {a}\) belongs to the ground set of \({\mathcal {V}}^{\bullet }_n\) . We show that every interval in \({\mathcal {V}}^{\bullet }_n\) is shellable for \(n \ge 4\) . In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in \({\mathcal {V}}^{\bullet }_n\) has consequences in commutative algebra. As a corollary, we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for \(n \ge 4\) . (This also follows from a result by Conca, Herzog, Trung, and Valla.)