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

George E. Andrews, Christian Krattenthaler, Luigi Orsina, Paolo Papi

Research output: Contribution to journalArticlepeer-review

22 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)3835-3853
Number of pages19
JournalTransactions of the American Mathematical Society
Volume354
Issue number10
DOIs
StatePublished - Oct 2002

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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