Decomposing labeled interval orders as pairs of permutations

Claesson, Anders and Hannah, Stuart A. (2014) Decomposing labeled interval orders as pairs of permutations. The Electronic Journal of Combinatorics, 21 (4). P4.16. ISSN 1077-8926 (

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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.