Kostant partition function

introduction

Kostant’s partition function counts the number of ways to represent a particular vector (weight) as a nonnegative integral sum of positive roots of a Lie algebra.
For a given weight the q-analog of Kostant’s partition function is a polynomial where the coefficient of $q^k$ is the number of ways the weight can be written as a nonnegative integral sum of exactly $k$ positive roots.

Kostant’s partition function was introduced and studied by F.A. Berezin and I.M. Gelfand (Proc. Moscow Math. Soc. 5 (1956), 311-351) for the case $g=sl(n)$, and by B. Kostant (Trans. Amer. Math. Soc., 93 (1959), 53-73) for arbitrary semi–simple finite dimensional Lie algebra $g$

Kolman, B., and R. Beck. “Computers in Lie Algebras. I: Calculation of Inner Multiplicities.” SIAM Journal on Applied Mathematics 25, no. 2 (September 1, 1973): 300–312. doi:10.1137/0125032.

Flow polytopes and the Kostant partition function
Harris, Pamela E., Erik Insko, and Mohamed Omar. “The $q$-Analog of Kostant’s Partition Function and the Highest Root of the Classical Lie Algebras.” arXiv:1508.07934 [math], August 31, 2015. http://arxiv.org/abs/1508.07934.
Panyushev, Dmitri I. “On Lusztig’s $q$-Analogues of All Weight Multiplicities of a Representation.” arXiv:1406.1453 [Math], June 5, 2014. http://arxiv.org/abs/1406.1453.
lecouvey, Cedric. “Kostka-Foulkes Polynomials Cyclage Graphs and Charge Statistic for the Root System $C_{n}$.” arXiv:math/0310370, October 23, 2003. http://arxiv.org/abs/math/0310370.
Lansky, Joshua M. “A Q-Analog of Freudenthal’s Weight Multiplicity Formula.” Indagationes Mathematicae 11, no. 1 (January 1, 2000): 87–94. doi:10.1016/S0019-3577(00)88576-6.
Broer, Bram. “Line Bundles on the Cotangent Bundle of the Flag Variety.” Inventiones Mathematicae 113, no. 1 (n.d.): 1–20. doi:10.1007/BF01244299.
Gupta, R. K. “Characters and the q-Analog of Weight Multiplicity.” Journal of the London Mathematical Society s2-36, no. 1 (August 1, 1987): 68–76. doi:10.1112/jlms/s2-36.1.68.
Kato, Shin-ichi. “Spherical Functions and A q-analogue of Kostant's weight multiplicity formula.” Inventiones Mathematicae 66, no. 3 (n.d.): 461–68. doi:10.1007/BF01389223.