 Kostant partition function

The Kostant partition function $P(\nu )$ counts the ways to express a weight $\nu$ as a nonnegative integer combination of positive roots. By the Poincare-Birkhoff-Witt theorem, $P(\lambda -\mu )$ is the multiplicity of the weight $\mu$ in the Verma module $M(\lambda )$.

The Kostant partition function is a concept from the field of representation theory and algebraic combinatorics. It counts the number of ways to express a non-negative integer as a sum of certain weights associated with the roots of a Lie algebra, specifically in the context of semisimple Lie algebras.