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Full-Text Articles in Discrete Mathematics and Combinatorics
The Search For The Cyclic Sieving Phenomenon In Plane Partitions, William J. Asztalos
The Search For The Cyclic Sieving Phenomenon In Plane Partitions, William J. Asztalos
DePaul Discoveries
The efforts of this research project are best understood in the context of the subfield of dynamical combinatorics, in which one enumerates a set of combinatorial objects by defining some action to guide the search for underlying structures. While there are many examples with varying degrees of complexity, the necklace problem, which concerns the possible unique configurations of beads in a ring up to rotational symmetry, is a well-known example. Though this sort of approach to enumeration has been around for a century or more, activity in this area has intensified in the last couple of decades. Perhaps the most …
Covering Arrays For Equivalence Classes Of Words, Joshua Cassels, Anant Godbole
Covering Arrays For Equivalence Classes Of Words, Joshua Cassels, Anant Godbole
Undergraduate Honors Theses
Covering arrays for words of length t over a d letter alphabet are k × n arrays with entries from the alphabet so that for each choice of t columns, each of the dt t-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case, words are equivalent if they induce the same partition of a t element set. In the second case, words of the same weighted sum are equivalent. In both cases we produce logarithmic upper bounds …
On The Density Of The Odd Values Of The Partition Function, Samuel Judge
On The Density Of The Odd Values Of The Partition Function, Samuel Judge
Dissertations, Master's Theses and Master's Reports
The purpose of this dissertation is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo $2$. We provide a doubly-indexed, infinite family of conjectural identities in the ring of series $\Z_2[[q]]$, which relate $p(n)$ with suitable $t$-multipartition functions, and show how to, in principle, prove each such identity. We will exhibit explicit proofs for $32$ of our identities. However, the conjecture remains open in full generality. A striking consequence of these conjectural identities is that, under suitable …