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Articles 1 - 6 of 6
Full-Text Articles in Mathematics
An Alternate Proof For The Top-Heavy Conjecture On Partition Lattices Using Shellability, Brian Macdonald, Josh Hallam
An Alternate Proof For The Top-Heavy Conjecture On Partition Lattices Using Shellability, Brian Macdonald, Josh Hallam
Honors Thesis
A partially ordered set, or poset, is governed by an ordering that may or may not relate
any pair of objects in the set. Both the bonds of a graph and the partitions of a set are
partially ordered, and their poset structure can be depicted visually in a Hasse diagram. The
partitions of {1, 2, ..., n} form a particularly important poset known as the partition lattice
Πn. It is isomorphic to the bond lattice of the complete graph Kn, making it a special case
of the family of bond lattices.
Dowling and Wilson’s 1975 Top-Heavy Conjecture states that …
Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis
Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis
Rose-Hulman Undergraduate Mathematics Journal
Dominion is a deck-building card game that simulates competing lords growing their kingdoms. Here we wish to optimize a strategy called Big Money by modeling the game as a Markov chain and utilizing the associated transition matrices to simulate the game. We provide additional analysis of a variation on this strategy known as Big Money Terminal Draw. Our results show that player's should prioritize buying provinces over improving their deck. Furthermore, we derive heuristics to guide a player's decision making for a Big Money Terminal Draw Deck. In particular, we show that buying a second Smithy is always more optimal …
Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett
Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett
Journal of Humanistic Mathematics
Mathematics is often presented as a neatly polished finished product, yet its development is messy and often full of mis-steps that could have been avoided with hindsight. An experience with a puzzle illustrates this conflict. The puzzle asks for the probability that a group of four and a group of two are seated adjacently within a hundred seats, and is solved using combinatorics techniques.
Counting Conjugates Of Colored Compositions, Jesus Omar Sistos Barron
Counting Conjugates Of Colored Compositions, Jesus Omar Sistos Barron
Honors College Theses
The properties of n-color compositions have been studied parallel to those of regular compositions. The conjugate of a composition as defined by MacMahon, however, does not translate well to n-color compositions, and there is currently no established analogous concept. We propose a conjugation rule for cyclic n-color compositions. We also count the number of self-conjugates under these rules and establish a couple of connections between these and regular compositions.
Slₖ-Tilings And Paths In ℤᵏ, Zachery T. Peterson
Slₖ-Tilings And Paths In ℤᵏ, Zachery T. Peterson
Theses and Dissertations--Mathematics
An SLₖ-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher k by constructing a bijection between SLₖ-tilings and certain pairs of bi-infinite strips of vectors in ℤᵏ called paths. The key ingredient in the proof is the relation to Plucker friezes and …
Enumeration Of Lattice Paths With Restrictions, Vince White
Enumeration Of Lattice Paths With Restrictions, Vince White
Electronic Theses and Dissertations
Lattice path enumeration, through the lens of Catalan numbers, plays a crucial role in combinatorics. This thesis delves into enumerations of some of the most common lattice paths – north-east paths, up-down paths, and Dyck paths – with restrictions applied. The first restriction is counting north-east lattice paths that only cross the diagonal line, y=x, once. The second form of lattice paths with restrictions is up-down paths that cross the x-axis exactly once and fall to a fixed depth of k. While working through this module, a novel proof for a known integer sequence was used, then applied to generate …