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Articles 1 - 4 of 4
Full-Text Articles in Physical Sciences and Mathematics
Staircase Packings Of Integer Partitions, Melody Arteaga
Staircase Packings Of Integer Partitions, Melody Arteaga
Mathematics, Statistics, and Computer Science Honors Projects
An integer partition is a weakly decreasing sequence of positive integers. We study the family of packings of integer partitions in the triangular array of size n, where successive partitions in the packings are separated by at least one zero. We prove that these are enumerated by the Bell-Like number sequence (OEIS A091768), and investigate its many recursive properties. We also explore their poset (partially ordered set) structure. Finally, we characterize various subfamilies of these staircase packings, including one restriction that connects back to the original patterns of the whole family.
Mixing Measures For Trees Of Fixed Diameter, Ari Holcombe Pomerance
Mixing Measures For Trees Of Fixed Diameter, Ari Holcombe Pomerance
Mathematics, Statistics, and Computer Science Honors Projects
A mixing measure is the expected length of a random walk in a graph given a set of starting and stopping conditions. We determine the tree structures of order n with diameter d that minimize and maximize for a few mixing measures. We show that the maximizing tree is usually a broom graph or a double broom graph and that the minimizing tree is usually a seesaw graph or a double seesaw graph.
The Rook-Brauer Algebra, Elise G. Delmas
The Rook-Brauer Algebra, Elise G. Delmas
Mathematics, Statistics, and Computer Science Honors Projects
We introduce an associative algebra RBk(x) that has a basis of rook-Brauer diagrams. These diagrams correspond to partial matchings on 2k vertices. The rook-Brauer algebra contains the group algebra of the symmetric group, the Brauer algebra, and the rook monoid algebra as subalgebras. We show that the basis of RBk(x) is generated by special diagrams si, ti (1 <= i < k) and pj (1 <= j <= k), where the si are the simple transpositions that generated the symmetric group Sk, the ti are the "contraction maps" which generate the …
Extremal Random Walks On Trees, Meng Wang
Extremal Random Walks On Trees, Meng Wang
Mathematics, Statistics, and Computer Science Honors Projects
We study random walks on trees, where we iteratively move from one vertex to a randomly chosen adjacent vertex. We study two quantities arising in random walks: the hitting time and the mixing time. The hitting time is the expected number of steps to walk between a chosen pair of vertices. The mixing time is the expected number of steps before the distribution of the current state is proportional to its degree. For a fixed tree size, we prove that the star is the unique minimizing structure and the path is the unique maximizing structure for both quantities.