Abstract view
Inequivalent Transitive Factorizations into Transpositions
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Published:2001-08-01
Printed: Aug 2001
I. P. Goulden
D. M. Jackson
F. G. Latour
Abstract
The question of counting minimal factorizations of permutations into
transpositions that act transitively on a set has been studied extensively
in the geometrical setting of ramified coverings of the sphere and in the
algebraic setting of symmetric functions.
It is natural, however, from a combinatorial point of view to ask how such
results are affected by counting up to equivalence of factorizations, where
two factorizations are equivalent if they differ only by the interchange of
adjacent factors that commute. We obtain an explicit and elegant result for
the number of such factorizations of permutations with precisely two
factors. The approach used is a combinatorial one that rests on two
constructions.
We believe that this approach, and the combinatorial primitives that have
been developed for the ``cut and join'' analysis, will also assist with the
general case.
MSC Classifications: |
05C38, 15A15, 05A15, 15A18 show english descriptions
Paths and cycles [See also 90B10] Determinants, permanents, other special matrix functions [See also 19B10, 19B14] Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] Eigenvalues, singular values, and eigenvectors
05C38 - Paths and cycles [See also 90B10] 15A15 - Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 05A15 - Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 15A18 - Eigenvalues, singular values, and eigenvectors
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