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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

How often are two permutations comparable?


Authors: Adam Hammett and Boris Pittel
Journal: Trans. Amer. Math. Soc. 360 (2008), 4541-4568
MSC (2000): Primary 05A05, 05A16, 06A07, 60C05; Secondary 20B99
Published electronically: April 4, 2008
MathSciNet review: 2403696
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Abstract: Two permutations of $ \left[ n \right]$ are comparable in the Bruhat order if one is closer, in a natural way, to the identity permutation, $ 1 \, 2 \, \cdots \, n$, than the other. We show that the number of comparable pairs is of order $ \left(n!\right)^2/n^2$ at most, and $ \left(n!\right)^2\left(0.708\right)^n$ at least. For the related weak order, the corresponding bounds are $ \left(n!\right)^2\left(0.362\right)^n$ and $ \left(n!\right)^2\prod_{i=1}^n \left(H\left(i\right)/i\right)$, where $ H\left(i\right):=\sum_{j=1}^i 1/j$. In light of numerical experiments, we conjecture that for each order the upper bound is qualitatively close to the actual number of comparable pairs.


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Additional Information

Adam Hammett
Affiliation: Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210
Email: hammett@math.ohio-state.edu

Boris Pittel
Affiliation: Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210
Email: bgp@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04478-4
PII: S 0002-9947(08)04478-4
Keywords: Bruhat ordering, weak ordering, permutations, partially ordered sets
Received by editor(s): January 31, 2006
Published electronically: April 4, 2008
Additional Notes: The first author was supported in part by NSF grant DMS-0104104.
The second author was supported in part by NSF grants DMS-0104104 and DMS-0406024.
Article copyright: © Copyright 2008 American Mathematical Society