A Simple Proof of a Conjecture of Simion

TL;DR

This paper provides a straightforward proof of Hildebrand's result that the number of certain lattice paths, related to Simion's unimodality conjecture, are log concave, simplifying the understanding of this combinatorial property.

Contribution

The paper offers a simple and accessible proof of Hildebrand's log concavity result, advancing the understanding of lattice path enumeration related to Simion's conjecture.

Findings

Confirmed the log concavity of lattice path counts

Simplified the proof of Hildebrand's result

Enhanced understanding of combinatorial path properties

Abstract

Simion had a unimodality conjecture concerning the number of lattice paths in a rectangular grid with the Ferrers diagram of a partition removed. Hildebrand recently showed the stronger result that these numbers are log concave. Here we present a simple proof of Hildebrand's result.