We introduce ballot matrices, a signed combinatorial structure whose
definition naturally follows from the generating function for labeled interval
orders. A sign reversing involution on ballot matrices is defined. We show that
matrices fixed under this involution are in bijection with labeled interval
orders and that they decompose to a pair consisting of a permutation and an
inversion table. To fully classify such pairs, results pertaining to the
enumeration of permutations having a given set of ascent bottoms are given.
This allows for a new formula for the number of labeled interval orders
