Abstract
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.
Highlights
We introduce ballot matrices, a signed combinatorial structure whose definition naturally follows from the generating function for labeled interval orders
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
Recent work has employed the use of sign reversing involutions in the study of unlabeled interval orders
Summary
Recent work has employed the use of sign reversing involutions in the study of unlabeled interval orders. A combinatorial structure consisting of signed, upper triangular, non-row empty matrices whose entries are ballots. The definition of such matrices follows naturally from the generating function of labeled interval orders. A sign reversing involution is used to identify fixed points for which there is exactly one per equivalence class. The decomposition of any single fixed point into a pair consisting of a permutation and an inversion table is provided. This allows for the main result of the paper, that the set of labeled interval orders on [n] is in bijection with two separate sets.
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