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

Andrews, George; Krattenthaler, Christian; Orsina, Luigi; Papi, Paolo

doi:10.1090/S0002-9947-02-03064-7

$ad$-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.

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