Open Accessππ-nilpotent π-ideals in π°π©(π«) having a fixed class of nilpotence: combinatorics and enumeration
Author(s) -
George E. Andrews,
Christian Krattenthaler,
Luigi Orsina,
Paolo Papi
Publication year - 2002
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/s0002-9947-02-03064-7
Subject(s) - algorithm , type (biology) , mathematics , artificial intelligence , computer science , geology , paleontology
We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n + 1, β). 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