Location

Room 1909

Session Format

Paper Presentation

Research Area Topic:

Natural & Physical Sciences - Mathematics

Co-Presenters, Co- Authors, Co-Researchers, Mentors, or Faculty Advisors

Faculty Advisor: Hua Wang

Abstract

Pattern packing concerns finding an optimal permutation that contains the maximum number of occurrences of a given pattern and computing the corresponding packing density. In many instances such an optimal permutation can be characterized directly and the number of occurrences of the pattern in interest may be enumerated explicitly. In more complicated patterns a direct characterization may be more challenging, however computational results for long permutations can help provide an indirect characterization of the general form of an optimal permutation. Much work has been done on the study of pattern packing in layered patterns, as the optimal permutation of a layered pattern is easily characterized. It has been shown that there always exists a layered optimal permutation of a given layered pattern. Because all length three-patterns and all but two (under equivalence) length-four patterns are layered, this result solves the pattern packing problem for many simple patterns. A broader class of permutations called colored permutations is formed by assigning permuted elements a color from a corresponding color set. We explore the consequences of colored permutations and patterns on the pattern packing problem. Through examining the novel concept of colored blocks within a colored pattern or permutation, we present analogous results on optimal colored permutations of patterns containing two or three colored blocks. We also conjecture an extended result for patterns containing more than three colored blocks. The results we present encompass a broader class of patterns than the analogous layered patterns, with limitations first arising when a colored block contains a consecutively monochromatic non-layered pattern. From numerical observations that colored patterns are refinements of their associated non-colored patterns, we also present an explicit relationship between packing densities of colored patterns and their consecutively monochromatic constituents. This result is also conjectured to hold for any colored pattern containing any number of colored blocks.

Keywords

Permutation, Pattern, Pattern packing, Packing density

Presentation Type and Release Option

Presentation (Open Access)

Start Date

4-24-2015 9:30 AM

End Date

4-24-2015 10:30 AM

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Apr 24th, 9:30 AM Apr 24th, 10:30 AM

Packing Densities of Colored and Non-Colored Patterns

Room 1909

Pattern packing concerns finding an optimal permutation that contains the maximum number of occurrences of a given pattern and computing the corresponding packing density. In many instances such an optimal permutation can be characterized directly and the number of occurrences of the pattern in interest may be enumerated explicitly. In more complicated patterns a direct characterization may be more challenging, however computational results for long permutations can help provide an indirect characterization of the general form of an optimal permutation. Much work has been done on the study of pattern packing in layered patterns, as the optimal permutation of a layered pattern is easily characterized. It has been shown that there always exists a layered optimal permutation of a given layered pattern. Because all length three-patterns and all but two (under equivalence) length-four patterns are layered, this result solves the pattern packing problem for many simple patterns. A broader class of permutations called colored permutations is formed by assigning permuted elements a color from a corresponding color set. We explore the consequences of colored permutations and patterns on the pattern packing problem. Through examining the novel concept of colored blocks within a colored pattern or permutation, we present analogous results on optimal colored permutations of patterns containing two or three colored blocks. We also conjecture an extended result for patterns containing more than three colored blocks. The results we present encompass a broader class of patterns than the analogous layered patterns, with limitations first arising when a colored block contains a consecutively monochromatic non-layered pattern. From numerical observations that colored patterns are refinements of their associated non-colored patterns, we also present an explicit relationship between packing densities of colored patterns and their consecutively monochromatic constituents. This result is also conjectured to hold for any colored pattern containing any number of colored blocks.