A Class of Labeled Posets and the Shi Arrangement of Hyperplanes

We consider the class Pnof labeled posets onnelements which avoid certain three-element induced subposets. We show that the number of posets in Pnis (n+1)n−1by exploiting a bijection between Pnand the set of regions of the arrangement of hyperplanes in Rnof the formxi−xj=0 or 1 for 1⩽i<j⩽n. It also follows that the number of posets in Rnwithipairs (a, b) such thata<bis equal to the number of trees on {0, 1, …, n} with {n \choose 2}-i inversions.