Topics in the Generation of Ideals of Posets

We study the generation of the fixed ideals of finite partial orders with particular attention in finding Gray codes for their generation. Pruesse and Ruskey conjecture that the graph J(P,k), which contains as vertices the k-ideals of the poset P, with an edge between vertices that differ by a swap, has a Hamiltonian path. The conjecture is true for series-parallel posets and interval orders. We prove the conjecture also holds for the fence posets, but that the conjecture is false for the 3-ideals of the crown poset with six elements. We also provide an infinite family of posets for which the conjecture does not hold. We study the Whitney numbers of fence posets to show a different but related conjecture of Pruesse and Ruskey also holds for fences and crowns with a small number of exceptions. We study Hamiltonian cycles in the Johnson graph (n,k) with certain properties about adjacent elements which we name t-full Hamiltonian cycles. We show their connection to the recently proved Middle Levels Theorem and how to construct them. We use t-full Hamiltonian cycles to show that, for k <= n-2 the graph J(Cr(2n), k) has a Hamiltonian cycle. We also apply 2-full Hamiltonian cycles to show that (P,3) has a Hamiltonian path for any height two poset P. Further, we introduce t-full Hamiltonian connected paths, and show that 1-full Hamiltonian connected paths exist in J(n,k). We introduce a weaker version of t-full Hamiltonian paths called pair-adjacent paths, and give algorithms for their construction. We show how these algorithms can generate the 3-ideals for crown posets with more than 6 elements. Finally, we study two applications of the generation of fixed-sized ideals. We formulate the information set decoding method in terms of error-correcting codes with a poset metric, and show how the probability of success in the guessing phase of such algorithms is minimized for anti-chain posets. We generalize ordered covering arrays (OCA) for general posets, and give elementary constructions of OCAs for level-regular rooted tree posets.

Name thesis:degree_name

Doctor of Philosophy (Ph.D.)

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Applied Mathematics

Grantor dc:publisher

Carleton University

Year dc:date.issued

2025

Author dc:creator

• Powers, Mackenzie William

Statement dc:rights

• Copyright © 2025 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, distribution to students, research and scholarship. Theses may only be shared by linking to the Carleton University Institutional Repository and no part may be copied without proper attribution to the author; no part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner.

Language dc:language.iso

en

DOI

10.22215/etd/2025-16456

OAI identifier oai:identifier

oai:carleton.scholaris.ca:20.500.14718/43599