Mark Jerrum, Computational Polya theory, Surveys in combinatorics,  Home/Search   Document Details and Download   Summary   Related Articles   Check

....usually is small compared with the total number of classes, but some classes are very small, so a random search for outer classes as proposed in [5] will not necessarily work. We also see that these groups usually have very many classes in total, so even a modified random search along the lines of [12] is therefore not suitable. A notable exception among the otherwise satisfactory runtimes is 1 2 [S 2 11 ]2 for which the new algorithm yields only a minimal improvement. In this group (The wreath product S 11 # 2 has 3 normal subgroups of index 2 namely [ 1 2 S 2 11 ]2, 1 2 [S 2 11 ]2 and ....

Mark Jerrum, Computational Polya theory, Surveys in combinatorics,

....fundamental to the theory of enumeration under group action, but has other aspects too. The Orbit Counting Lemma is proved by counting in two ways the edges 2 of the bipartite graph B on X [G, where X is the set on which G acts, and fx; gg is an edge if and only if g fixes the point x. Jerrum [64] has extended this simple idea to a Markov chain to choose a random orbit of G on X. Start with a point of X chosen from any distribution. One step in the chain consists in moving to a random neighbour g of x, and then to a random neighbour x 0 of g. This Markov chain is irreducible and ....

M. R. Jerrum, Computational P'olya theory, pp. 103--118 in Surveys in Combinatorics, 1995 (Peter Rowlinson, ed.), London Math. Soc. Lecture Notes 218, Cambridge University Press, Cambridge, 1995.

....the stationary distribution of M B . Then (g) is proportional to the degree of vertex g in the bipartite graph corresponding to Burnside s Lemma, which is j Fix gj = k c(g) where c(g) denotes the number of cycles in the permutation g. We have therefore established the following Lemma from [8]: Lemma 1 Let be the stationary distribution of the Markov chain M B (G; Sigma ) Then (g) k c(g) j Upsilon ( Sigma ; G)j for all g 2 G. Although the Markov chain M B on G is the most convenient one for us to work with, it is clear that we can invert the order of steps (B1) and (B2) to ....

....is clear that we can invert the order of steps (B1) and (B2) to obtain a dual Markov chain M 0 B (G; Sigma ) with state space Sigma m . The dual Markov chain 4 has greater practical appeal, as it gives a uniform sampler for orbits (i.e. unlabelled structures) 4 In references [7] and [8], the primed and unprimed versions are reversed. Lemma 2 Let 0 be the stationary distribution of the Markov chain M 0 B (G; Sigma ) Then 0 (ff) jGj jff G j j Upsilon ( Sigma ; G)j for all ff 2 Sigma m ; in particular, 0 assigns equal probability to each orbit ff G . ....

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Mark Jerrum, Computational P'olya theory. In "Surveys in Combinatorics 1995," London Mathematical Society Lecture Note Series 218, Cambridge University Press, 1995, 103--118.

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