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ad-nilpotent $\frak b$-ideals in sl(n) having a fixed class of nilpotence: combinatorics and enumeration

George E. Andrews·Christian Krattenthaler·Luigi Orsina·Paolo Papi
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Abstract

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of $sl(n+1,\Bbb C)$. 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.

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George E. Andrews, Christian Krattenthaler, Luigi Orsina, Paolo Papi. 2000-04-17. ad-nilpotent $\frak b$-ideals in sl(n) having a fixed class of nilpotence: combinatorics and enumeration. https://arxiv.org/abs/math/0004107

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