140 research outputs found
Counting factorizations of Coxeter elements into products of reflections
In this paper, we count factorizations of Coxeter elements in well-generated
complex reflection groups into products of reflections. We obtain a simple
product formula for the exponential generating function of such factorizations,
which is expressed uniformly in terms of natural parameters of the group. In
the case of factorizations of minimal length, we recover a formula due to P.
Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex
case. For the symmetric group, our formula specializes to a formula of D. M.
Jackson.Comment: 38 pages, including 18 pages appendix. To appear in Journal of the
London Mathematical Society. v3: minor changes and corrected references; v2:
added extended discussion on the definition of Coxeter element
Crossings and nestings in set partitions of classical types
In this article, we investigate bijections on various classes of set
partitions of classical types that preserve openers and closers. On the one
hand we present bijections that interchange crossings and nestings. For types B
and C, they generalize a construction by Kasraoui and Zeng for type A, whereas
for type D, we were only able to construct a bijection between non-crossing and
non-nesting set partitions. On the other hand we generalize a bijection to type
B and C that interchanges the cardinality of the maximal crossing with the
cardinality of the maximal nesting, as given by Chen, Deng, Du, Stanley and Yan
for type A. Using a variant of this bijection, we also settle a conjecture by
Soll and Welker concerning generalized type B triangulations and symmetric fans
of Dyck paths.Comment: 22 pages, 7 Figures, removed erroneous commen
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