Applying Burnside's Lemma to a One-Dimensional Escher Problem

2006 ◽  
Vol 79 (3) ◽  
pp. 167 ◽  
Author(s):  
Tomaz̆ Pisanski ◽  
Doris Schattschneider ◽  
Brigitte Servatius
Keyword(s):  
One Dimensional ◽  
Burnside's Lemma ◽  
Burnside’S Lemma

Applying Burnside's Lemma to a One-Dimensional Escher Problem

2006 ◽  
Vol 79 (3) ◽  
pp. 167-180
Author(s):  
Tomaž Pisanski ◽  
Doris Schattschneider ◽  
Brigitte Servatius
Keyword(s):  
One Dimensional ◽  
Burnside's Lemma ◽  
Burnside’S Lemma

scholarly journals The ‘Burnside Process’ Converges Slowly

2002 ◽  
Vol 11 (1) ◽  
pp. 21-34 ◽  
Author(s):  
LESLIE ANN GOLDBERG ◽  
MARK JERRUM
Keyword(s):  
Markov Chain ◽  
Mixing Time ◽  
Significant Proportion ◽  
Main Tool ◽  
Infinite Family ◽  
Combinatorial Structures ◽  
Special Groups ◽  
Sampling Problem ◽  
Burnside's Lemma ◽  
Burnside’S Lemma

We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.

4. A combinatorial zoo

2016 ◽  
pp. 50-71
Author(s):  
Robin Wilson
Keyword(s):  
Fibonacci Numbers ◽  
Exclusion Principle ◽  
Pigeonhole Principle ◽  
Tower Of Hanoi ◽  
Wide Range ◽  
The World ◽  
Burnside's Lemma ◽  
Burnside’S Lemma

‘A combinatorial zoo’ presents a menagerie of combinatorial topics, ranging from the box (or pigeonhole) principle, the inclusion–exclusion principle, the derangement problem, and the Tower of Hanoi problem that uses combinatorics to determine how soon the world will end to Fibonacci numbers, the marriage theorem, generators and enumerators, and counting chessboards, which involves symmetry. The method used to average the numbers of colourings that remain unchanged by each symmetry in this latter problem is often called ‘Burnside’s lemma’. This concept has since been developed into a much more powerful result, which has been used to count a wide range of objects with a degree of symmetry, such as graphs and chemical molecules.

Enumeration of structural isomers in alicyclic hydrocarbons and in porphyrins

1977 ◽  
Vol 55 (15) ◽  
pp. 2773-2777 ◽  
Author(s):  
Brian Alspach ◽  
S. Aronoff
Keyword(s):  
Structural Isomers ◽  
Polya's Theorem ◽  
Pólya’S Theorem ◽  
Burnside's Lemma ◽  
Burnside’S Lemma

The number of structural isomers obtainable from all possible substitutions in an alicyclic hydrocarbon is found by the use of Burnside's Lemma. Four classes of compounds exist according to the number of vertices of the ring, and the number and type of substituents on the carbons. The theory is also applicable to other types of compounds in which paired substituents occur, e.g. in the biologically-synthesisable porphyrins. Where the pairing is not mandatory, Polya's Theorem is utilized for enumeration. Examples are provided for each of the classes of compounds.

Characterization of polymer structures based on Burnside’s lemma

2011 ◽  
Vol 84 (1) ◽  
Author(s):  
Kentaro Doi ◽  
Keigo Kato ◽  
Satoyuki Kawano
Keyword(s):  
Burnside's Lemma ◽  
Burnside’S Lemma
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