A Ratio Inequality for Binary Trees and the Best Secretary

  • Authors:

  • Grzegorz Kubicki;Jenö Lehel;Michał Morayne

  • Affiliations:

  • Department of Mathematics, University of Louisville, Louisville, KY 40292, USA (e-mail: gkubicki@louisville.edu);Department of Mathematics, University of Louisville, Louisville, KY 40292, USA (e-mail: gkubicki@louisville.edu);Institute of Mathematics, Wrocław Univ. of Tech., Wybrzeże Wyspiańskiego 27, 50–370 Wrocław and Inst. of Math., Polish Acad. Sci., Kopernika 18, 51–617 Wrocław ...

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2002

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Abstract

Let Tn 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 ***** insert equation here ***** 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.