The pinched Veronese poset Vn∙\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}^{\bullet }_n$$\end{document} is the poset with ground set consisting of all nonnegative integer vectors of length n\documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document} such that the sum of their coordinates is divisible by n\documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document} with exception of the vector (1,…,1)\documentclass[12pt]{minimal}
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\begin{document}$$(1,\ldots ,1)$$\end{document}. For two vectors a\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {a}$$\end{document} and b\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {b}$$\end{document} in Vn∙\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}^{\bullet }_n$$\end{document}, we have a⪯b\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {a}\preceq \mathbf {b}$$\end{document} if and only if b-a\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {b}- \mathbf {a}$$\end{document} belongs to the ground set of Vn∙\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}^{\bullet }_n$$\end{document}. We show that every interval in Vn∙\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}^{\bullet }_n$$\end{document} is shellable for n≥4\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 4$$\end{document}. 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 Vn∙\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {V}}^{\bullet }_n$$\end{document} 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≥4\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 4$$\end{document}. (This also follows from a result by Conca, Herzog, Trung, and Valla.)
