Shellability of the higher pinched Veronese posets

Abstract

The pinched Veronese poset $${\mathcal {V}}^{\bullet }_n$$ V n ? is the poset with ground set consisting of all nonnegative integer vectors of length $$n$$ n such that the sum of their coordinates is divisible by $$n$$ n with exception of the vector $$(1,\ldots ,1)$$ ( 1 , ? , 1 ) . For two vectors $$\mathbf {a}$$ a and $$\mathbf {b}$$ b in $${\mathcal {V}}^{\bullet }_n$$ V n ? , we have $$\mathbf {a}\preceq \mathbf {b}$$ a ? b if and only if $$\mathbf {b}- \mathbf {a}$$ b - a belongs to the ground set of $${\mathcal {V}}^{\bullet }_n$$ V n ? . We show that every interval in $${\mathcal {V}}^{\bullet }_n$$ V n ? is shellable for $$n \ge 4$$ n ? 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 $${\mathcal {V}}^{\bullet }_n$$ V 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 $$n \ge 4$$ n ? 4 . (This also follows from a result by Conca, Herzog, Trung, and Valla.)