Some Monotonicity Properties of Partial Orders

  title={Some Monotonicity Properties of Partial Orders},
  author={Ronald L. Graham and Andrew Chi-Chih Yao and F. Frances Yao},
  journal={SIAM J. Matrix Analysis Applications},
A fundamental quantity which arises in the sorting of n numbers al, a2.''", an is Pr (ai < ai]P), the probability that ai < ai assuming that all linear extensions of the partial order P are equally likely. In this paper, we establish various properties of Pr (ai < ajlP) and related quantities. In particular, it is shown that Pr (ai <bjlP')>=Pr (a<b.iP) if the partial order P consists of two disjoint linearly ordered sets A ={al<a2<" "<a,,,,},B={bl<b2<" '<bn} and P'=PU{any relations of the form… CONTINUE READING

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Publications referenced by this paper.

The Art of Computer Programming Sorting and Searching

  • D E Knuth
  • The Art of Computer Programming Sorting and…
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2 Excerpts

On representations o]subsets

  • P Hall
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A simple calculation shows that 4

  • A simple calculation shows that 4

Pr (.'1') 1 / 2 < - Pr (') which violates

  • R L Graham, A C Yao, F F Yao Thus
  • Pr (.'1') 1 / 2 < - Pr (') which violates

We should also note (as pointed out by D. Kleitman and J. Shearer) that the conjecture is not true if we allow P to have even one relation of the form ai < b as the following example shows

  • Let a={al

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