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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions
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by C. Krattenthaler and T. W. Müller PDF
Trans. Amer. Math. Soc. 362 (2010), 2723-2787 Request permission


Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,\dots ,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=\sigma _1\sigma _2\cdots \sigma _d$ of a Coxeter element $c$ of $W$, such that $\sigma _i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,\dots ,d$, and such that the factorisation is “minimal” in the sense that the sum of the ranks of the $T_i$’s, $i=1,2,\dots ,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author in [“Topics in Discrete Mathematics,” M. Klazar et al. (eds.), Springer–Verlag, Berlin, New York, 2006, pp. 93–126] and [Séminaire Lotharingien Combin. 54 (2006), Article B54l]. The type $A_n$ decomposition numbers have been computed by Goulden and Jackson in [Europ. J. Combin. 13 (1992), 357–365], albeit using a somewhat different language. We explain how to extract the type $B_n$ decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. 24 (2000), 22–56] on map enumeration. Our formula for the type $D_n$ decomposition numbers is new. These results are then used to determine, for a fixed positive integer $l$ and fixed integers $r_1\le r_2\le \dots \le r_l$, the number of multi-chains $\pi _1\le \pi _2\le \dots \le \pi _l$ in Armstrong’s generalised non-crossing partitions poset, where the poset rank of $\pi _i$ equals $r_i$ and where the “block structure” of $\pi _1$ is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type $D_n$ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong’s $F=M$ Conjecture in type $D_n$, thus completing a computational proof of the $F=M$ Conjecture for all types. It also allows one to address another conjecture of Armstrong on maximal intervals containing a random multi-chain in the generalised non-crossing partitions poset.
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Additional Information
  • C. Krattenthaler
  • Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna, Austria
  • MR Author ID: 106265
  • T. W. Müller
  • Affiliation: School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London E1 4NS, United Kingdom
  • Received by editor(s): June 30, 2008
  • Received by editor(s) in revised form: December 10, 2008
  • Published electronically: December 17, 2009
  • Additional Notes: The first author’s research was partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 2723-2787
  • MSC (2000): Primary 05E15; Secondary 05A05, 05A10, 05A15, 05A18, 06A07, 20F55, 33C05
  • DOI:
  • MathSciNet review: 2584617