Given a simple Lie algebra $mathfrak{g}$, Kostants weight $q$-multiplicity formula is an alternating sum over the Weyl group whose terms involve the $q$-analog of Kostants partition function. For $xi$ (a weight of $mathfrak{g}$), the $q$-analog of Ko
stants partition function is a polynomial-valued function defined by $wp_q(xi)=sum c_i q^i$ where $c_i$ is the number of ways $xi$ can be written as a sum of $i$ positive roots of $mathfrak{g}$. In this way, the evaluation of Kostants weight $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a highest weight representation of $mathfrak{g}$. In this paper, we give closed formulas for computing weight $q$-multiplicities in a highest weight representation of the exceptional Lie algebra $mathfrak{g}_2$.
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The $q$-analog of Kostants weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the $q$-analog of Kostants partition function. This formula, when evaluated at $q=1$, gives the multiplicit
y of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra $mathfrak{sl}_4(mathbb{C})$ and give closed formulas for the $q$-analog of Kostants weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the $q$-analog of Kostants partition function by counting restricted colored integer partitions. These formulas, when evaluated at $q=1$, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostants weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of $mathfrak{sl}_4(mathbb{C})$, which are associated to the Weyl alternation sets. This work answers a question posed in 2019 by Harris, Loving, Ramirez, Rennie, Rojas Kirby, Torres Davila, and Ulysse.
Let $G=(V,E)$ be a finite connected graph along with a coloring of the vertices of $G$ using the colors in a given set $X$. In this paper, we introduce multi-color forcing, a generalization of zero-forcing on graphs, and give conditions in which the
multi-color forcing process terminates regardless of the number of colors used. We give an upper bound on the number of steps required to terminate a forcing procedure in terms of the number of vertices in the graph on which the procedure is being applied. We then focus on multi-color forcing with three colors and analyze the end states of certain families of graphs, including complete graphs, complete bipartite graphs, and paths, based on various initial colorings. We end with a few directions for future research.
It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy $F_n =F_{n-1}+F_{n-2}$ for $ngeq 3$, $F_1 =1$ and $F_2 =2$. In this paper, for any $n,minmathbb{N}$ we c
reate a sequence called the $(n,m)$-bin sequence with which we can define a notion of a legal decomposition for every positive integer. These sequences are not always positive linear recurrences, which have been studied in the literature, yet we prove, that like positive linear recurrences, these decompositions exist and are unique. Moreover, our main result proves that the distribution of the number of summands used in the $(n,m)$-bin legal decompositions displays Gaussian behavior.
