Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-08-04478-4
Published electronically: April 4, 2008
MathSciNet review: 2403696
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05A05, 05A16, 06A07, 60C05, 20B99

Retrieve articles in all journals with MSC (2000): 05A05, 05A16, 06A07, 60C05, 20B99


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

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