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- Counting Eulerian orientations
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Counting Eulerian orientations Guttmann, Tony
Description
Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations for the ogfs for the number of planar Eulerian orientations counted by the edges, $U(x),$ and the number of 4-valent planar Eulerian orientations counted by the number of vertices, $A(x)$. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for the coefficients of the generating function. From these algorithms we have obtained 100 terms for $U(x)$ and 90 terms for $A(x).$ Analysis of these series suggests that they both behave as $ \cdot (1 - \mu x)^2/\log(1 - \mu x),$ where we make the confident conjectures that $\mu = 4\pi$ for Eulerian orientations counted by edges and $\mu=4\sqrt{3}\pi$ for 4-valent Eulerian orientations counted by vertices. (Joint work with Andrew Elvey Price).
Item Metadata
Title |
Counting Eulerian orientations
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-09-21T15:30
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Description |
Inspired by the paper of Bonichon, Bousquet-M\'elou, Dorbec and Pennarun, we give a system of functional equations for the ogfs for the number of planar Eulerian orientations counted by the edges, $U(x),$ and the number of 4-valent planar Eulerian orientations counted by the number of vertices, $A(x)$. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for the coefficients of the generating function. From these algorithms we have obtained 100 terms for $U(x)$ and 90 terms for $A(x).$
Analysis of these series suggests that they both behave as $ \cdot (1 - \mu x)^2/\log(1 - \mu x),$ where we make the confident conjectures that $\mu = 4\pi$ for Eulerian orientations counted by edges and $\mu=4\sqrt{3}\pi$ for 4-valent Eulerian orientations counted by vertices. (Joint work with Andrew Elvey Price).
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Extent |
29 minutes
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Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Melbourne
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Series | |
Date Available |
2018-04-01
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0364607
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International