Highly Influenced

9 Excerpts

- Published 2002 in Combinatorics, Probability & Computing

Let T be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n;T ) = |{S ⊆ Tn : 1n ∈ S, S ∼= T }|, and B(n;T ) = |{S ⊆ Tn : 1n / ∈ S, S ∼= T }|. In this note we prove that A(n;T1) B(n;T1) 6 A(n;T2) B(n;T2) for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.

@article{Kubicki2002ARI,
title={A Ratio Inequality For Binary Trees And The Best Secretary},
author={Grzegorz Kubicki and Jen{\"{o} Lehel and Michal Morayne},
journal={Combinatorics, Probability & Computing},
year={2002},
volume={11},
pages={149-161}
}