Berenshtein and Zelevinskii provided an exhaustive list of pairs of weights (λ, µ) of simple Lie algebras g (up to Dynkin diagram isomorphism) for which the multiplicity of the weight µ in the representation of g with highest weight λ is equal to one. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity for subsets of these pairs of weights and show that, in these cases, the cardinality of these contributing sets is enumerated by (multiples of) Fibonacci numbers. We conclude by using these results to compute the associated q-multiplicity for the pairs of weights considered, and conjecture that in all cases the q-multiplicity of such pairs of weights is given by a power of q.

Supplemental File 1. The interaction plot produced by the two-way ANOVA test for the confidence survey given to MATH115 students. Notice that the three aspects of the factor Question are clumped together for each level for the factor Statistical Test (i.e. the main effect for Question was insignificant). Also, notice that students' confidence in all three aspects of ANOVA and regression tests were among the lowest for these students, while their confidence in all three aspects of a t-test were highest. The y-axis represents the mean rating from 1 to 5 for confidence (1 represents low confidence).

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Kostant's partition function counts the number of distinct ways to express a weight of a classical Lie algebra g as a sum of positive roots of g. We refer to each of these expressions as decompositions of a weight. Our main result considers an infinite family of weights, irrespective of Lie type, for which we establish a closed formula for the q-analog of Kostant's partition function and then prove that the (normalized) distribution of the number of positive roots in the decomposition of any of these weights converges to a Gaussian distribution as the rank of the Lie algebra goes to infinity. We also extend these results to the highest root of the classical Lie algebras and we end our analysis with some directions for future research.