ad-NILPOTENT b-IDEALS IN sl(n) HAVING A FIXED CLASS OF NILPOTENCE: COMBINATORICS AND ENUMERATION
نویسندگان
چکیده
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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