-nilpotent -ideals in having a fixed class of nilpotence: combinatorics and enumeration

Authors:
George E. Andrews, Christian Krattenthaler, Luigi Orsina and Paolo Papi

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3835-3853

MSC (2000):
Primary 17B20; Secondary 05A15, 05A19, 05E15, 17B30

Published electronically:
June 10, 2002

MathSciNet review:
1926854

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the combinatorics of -nilpotent ideals of a Borel subalgebra of . We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between -nilpotent ideals and Dyck paths. Finally, we propose a -analogue of the Catalan number . These -Catalan numbers count, on the one hand, -nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.

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Additional Information

**George E. Andrews**

Affiliation:
Department of Mathematics, The Pennsylvania State University, 215 McAllister Building, University Park, Pennsylvania 16802

Email:
andrews@math.psu.edu

**Christian Krattenthaler**

Affiliation:
Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

Email:
KRATT@Ap.Univie.Ac.At

**Luigi Orsina**

Affiliation:
Dipartimento di Matematica, Istituto G. Castelnuovo, Università di Roma “La Sapienza", Piazzale Aldo Moro 5, 00185 Roma, Italy

Email:
orsina@mat.uniroma1.it

**Paolo Papi**

Affiliation:
Dipartimento di Matematica, Istituto G. Castelnuovo, Università di Roma “La Sapienza", Piazzale Aldo Moro 5, 00185 Rome, Italy

Email:
papi@mat.uniroma1.it

DOI:
https://doi.org/10.1090/S0002-9947-02-03064-7

Keywords:
\emph{ad}-nilpotent ideal,
Lie algebra,
order ideal,
Dyck path,
Catalan number,
Chebyshev polynomial

Received by editor(s):
April 25, 2000

Published electronically:
June 10, 2002

Additional Notes:
The first author was partially supported by National Science Foundation Grant DMS 9870060.

The second author was partially supported by the Austrian Science Foundation FWF, grant P13190-MAT

The fourth author’s research was partially supported by EC’s IHRP Programme, grant HPRN-CT-2001-00272

Article copyright:
© Copyright 2002
American Mathematical Society