$ad$-nilpotent $\mathfrak b$-ideals in $sl(n)$ having a fixed class of nilpotence: combinatorics and enumeration
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- by George E. Andrews, Christian Krattenthaler, Luigi Orsina and Paolo Papi PDF
- Trans. Amer. Math. Soc. 354 (2002), 3835-3853 Request permission
Abstract:
We study the combinatorics of $ad$-nilpotent ideals of a Borel subalgebra of $sl(n+1,\mathbb C)$. 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 $ad$-nilpotent ideals and Dyck paths. Finally, we propose a $(q,t)$-analogue of the Catalan number $C_n$. These $(q,t)$-Catalan numbers count, on the one hand, $ad$-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.References
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Additional Information
- George E. Andrews
- Affiliation: Department of Mathematics, The Pennsylvania State University, 215 McAllister Building, University Park, Pennsylvania 16802
- MR Author ID: 26060
- Email: andrews@math.psu.edu
- Christian Krattenthaler
- Affiliation: Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
- MR Author ID: 106265
- 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
- MR Author ID: 322097
- Email: papi@mat.uniroma1.it
- 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 - © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 3835-3853
- MSC (2000): Primary 17B20; Secondary 05A15, 05A19, 05E15, 17B30
- DOI: https://doi.org/10.1090/S0002-9947-02-03064-7
- MathSciNet review: 1926854