The bidirectional ballot polytope

TL;DR

This paper introduces a geometric perspective on bidirectional ballot sequences by defining a continuous analogue called bidirectional gerrymanders, forming a convex polytope inside the unit cube, and relates its structure to counting BBS's.

Contribution

The paper defines the bidirectional ballot polytope, proves its geometric properties, and connects its vertices to BBS's, providing a new geometric understanding of their enumeration.

Findings

The bidirectional ballot polytope is a convex polytope inside the unit cube.

Vertices of the polytope correspond bijectively to bidirectional ballot sequences.

The number of BBS's of length n is asymptotically Θ(2^n/n).

Abstract

A bidirectional ballot sequence (BBS) is a finite binary sequence with the property that every prefix and suffix contains strictly more ones than zeros. BBS's were introduced by Zhao, and independently by Bosquet-M{\'e}lou and Ponty as $(1,1)$-culminating paths. Both sets of authors noted the difficulty in counting these objects, and to date research on bidirectional ballot sequences has been concerned with asymptotics. We introduce a continuous analogue of bidirectional ballot sequences which we call bidirectional gerrymanders, and show that the set of bidirectional gerrymanders form a convex polytope sitting inside the unit cube, which we refer to as the bidirectional ballot polytope. We prove that every $(2n-1)$-dimensional unit cube can be partitioned into $2n-1$ isometric copies of the $(2n-1)$-dimensional bidirectional ballot polytope. Furthermore, we show that the vertices of this polytope are all also vertices of the cube, and that the vertices are in bijection with BBS's. An immediate corollary is a geometric explanation of the result of Zhao and of Bosquet-M{\'e}lou and Ponty that the number of BBS's of length $n$ is $\Theta(2^n/n)$.