Transitive factorisations into transpositions and holomorphic mappings on the sphere
Authors:
I. P. Goulden and D. M. Jackson
Journal:
Proc. Amer. Math. Soc. 125 (1997), 51-60
MSC (1991):
Primary 05A15; Secondary 05E99, 58C10, 70H20
DOI:
https://doi.org/10.1090/S0002-9939-97-03880-X
MathSciNet review:
1396978
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We determine the number of ordered factorisations of an arbitrary permutation on $n$ symbols into transpositions such that the factorisations have minimal length and such that the factors generate the full symmetric group on $n$ symbols. Such factorisations of the identity permutation have been considered by Crescimanno and Taylor in connection with a class of topologically distinct holomorphic maps on the sphere. As with Macdonaldâs construction for symmetric functions that multiply as the classes of the class algebra, essential use is made of Lagrange inversion.
- Michael Crescimanno and Washington Taylor, Large $N$ phases of chiral ${\rm QCD}_2$, Nuclear Phys. B 437 (1995), no. 1, 3â24. MR 1354345, DOI https://doi.org/10.1016/0550-3213%2894%2900561-R
- JĂłzsef DĂ©nes, The representation of a permutation as the product of a minimal number of transpositions, and its connection with the theory of graphs, Magyar Tud. Akad. Mat. KutatĂł Int. Közl. 4 (1959), 63â71 (English, with Russian and Hungarian summaries). MR 115936
- I. M. GelâČfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, BirkhĂ€user Boston, Inc., Boston, MA, 1994. MR 1264417
- I. P. Goulden, A differential operator for symmetric functions and the combinatorics of multiplying transpositions, Trans. Amer. Math. Soc. 344 (1994), no. 1, 421â440. MR 1249468, DOI https://doi.org/10.1090/S0002-9947-1994-1249468-3
- I.P.Goulden, J. L. Harer and D.M.Jackson, The virtual Euler characteristic of the moduli spaces of real and complex algebraic curves (preprint).
- I. P. Goulden and D. M. Jackson, Symmetric functions and Macdonaldâs result for top connexion coefficients in the symmetric group, J. Algebra 166 (1994), no. 2, 364â378. MR 1279263, DOI https://doi.org/10.1006/jabr.1994.1157
- I. P. Goulden and D. M. Jackson, Combinatorial enumeration, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. With a foreword by Gian-Carlo Rota; Wiley-Interscience Series in Discrete Mathematics. MR 702512
- J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), no. 3, 457â485. MR 848681, DOI https://doi.org/10.1007/BF01390325
- D. M. Jackson, Counting cycles in permutations by group characters, with an application to a topological problem, Trans. Amer. Math. Soc. 299 (1987), no. 2, 785â801. MR 869231, DOI https://doi.org/10.1090/S0002-9947-1987-0869231-9
- D. M. Jackson, Some combinatorial problems associated with products of conjugacy classes of the symmetric group, J. Combin. Theory Ser. A 49 (1988), no. 2, 363â369. MR 964394, DOI https://doi.org/10.1016/0097-3165%2888%2990062-3
- I. G. Macdonald, Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. MR 553598
Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 05A15, 05E99, 58C10, 70H20
Retrieve articles in all journals with MSC (1991): 05A15, 05E99, 58C10, 70H20
Additional Information
I. P. Goulden
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
MR Author ID:
75735
Email:
ipgoulden@math.uwaterloo.ca
D. M. Jackson
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
MR Author ID:
92555
Email:
dmjackson@dragon.uwaterloo.ca
Received by editor(s):
July 20, 1995
Communicated by:
Jeffry N. Kahn
Article copyright:
© Copyright 1997
American Mathematical Society