This material has been published in Trans. Amer. Math. Soc. 354 (2002), 3835-3853, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by the American Mathematical Society. This material may not be copied or reposted without explicit permission.

G. E. Andrews, Christian Krattenthaler, L. Orsina and P. Papi

ad-Nilpotent b-ideals in sl(n) having a fixed class of nilpotence: Combinatorics and enumeration

(19 pages)

Abstract. We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n+1,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 Cn. 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.

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