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.
ad-nilpotent b-Ideals in sl(n) having a fixed class of nilpotence: Combinatorics and enumeration / G. E., Andrews; C., Krattenthaler; Orsina, Luigi; Papi, Paolo. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 354:10(2002), pp. 3835-3853. [10.1090/s0002-9947-02-03064-7]
ad-nilpotent b-Ideals in sl(n) having a fixed class of nilpotence: Combinatorics and enumeration
ORSINA, Luigi;PAPI, Paolo
2002
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
