A class of labeled posets and the Shi arrangement of hyperplanes

摘要

We consider the class P-n of labeled posets on n elements which avoid certain three-element induced subposets. Wc show that the number of posets in P-n is (n + 1)(n - 1) by exploiting a bijection between P-n and the set of regions of the arrangement of hyperplanes in R-n of the form x(i) - x(j) = 0 or 1 for 1 less than or equal to i < j less than or equal to n. It also follows that the number of posets in P-n with i pairs (a, b) such that a < b is equal to the number of trees on {0, 1, ..., n} with ((n)(2)) - i inversions. (C) 1997 Academic Press.