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Orbits of the Dissected Polygons of the Generalized Catalan Numbers

Auger, Joseph Thomas
http://rave.ohiolink.edu/etdc/view?acc_num=akron1302701692

Abstract Details

2011, Master of Science, University of Akron, Mathematics.
Catalan numbers occur regularly throughout many different areas of mathematics. They date back all the way to Euler, and today there are literally hundreds of different realizations of the Catalan number. We can extend the idea of a Catalan number to a generalized Catalan number by adding and extra parameter. In particular, one realization of the generalized Catalan numbers is the number of ways to dissect a regular polygon with (p-1)k+2 sides into k disjoint p+1-gons using (k-1) non-intersecting diagonals. The action of the dihedral group partitions these dissections into some number of orbits. Our main goal of this paper is to find a way to count these orbits for given p and k. In order to do this, we introduce the notion of p-ary trees. We show that the number of diagonalizations of our polygon is equivalent to the number of p-ary trees with k source nodes. Finally, we use Burnside's Lemma as a way to count the number of orbits for a given tree core and extract formulas for the number of orbits for many tree cores that exist for large enough p and k.
James Cossey, Dr. (Advisor)
Jeffrey Riedl, Dr. (Committee Member)
Stefan Forcey, Dr. (Committee Member)
89 p.
Mathematics
Catalan; Burnside's Lemma; trees; dissected polygons

Recommended Citations

Citations

  • Auger, J. T. (2011). Orbits of the Dissected Polygons of the Generalized Catalan Numbers [Master's thesis, University of Akron]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=akron1302701692

    APA Style (7th edition)

  • Auger, Joseph. Orbits of the Dissected Polygons of the Generalized Catalan Numbers. 2011. University of Akron, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=akron1302701692.

    MLA Style (8th edition)

  • Auger, Joseph. "Orbits of the Dissected Polygons of the Generalized Catalan Numbers." Master's thesis, University of Akron, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=akron1302701692

    Chicago Manual of Style (17th edition)

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© 2011, all rights reserved.
This open access ETD is published by University of Akron and OhioLINK.