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Project Description

My project hopes to unite two different areas of study: competition graphs and permutation patterns. The structures we encounter in the competition graphs can help us count and classify certain permutations. We can also use permutations as a tool to examine the enumerative aspects of competition graphs.

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Weekly Log

Week 1: June 2 - June 6

• Familiarized myself with literature in competition graphs
• Read papers on permutation patterns
• Several simple observations relating the permutations and their competition graphs
• Verified an existing combinatorial result using competition graphs
• Gave the first presentation (see below)

Week 2: June 9 - June 13

• Started by characterizing competition graphs of permutations in S_n(123) and S_n(132)
• Verified results from literature regarding S_n(132)
• Proved identity between Motzkin numbers and Catalan numbers using competition graphs
• Wrote programs to help me count competition graphs

Week 3: June 16 - June 20

• Computed and proved some formulas for counts of special competition graphs
• m copies of K_n
• Paths of length 3 (P₃)
• Stars (complete bipartite graph K_{1, 3})
• Goals for next week: try to generalize formulas to larger classes of graphs

Week 4: June 23 - June 27

• Conjectured a recurrence relation that would compute graph counts very quickly
• Counted graphs restricted to S_n(123) and S_n(132)
• Found bijections between S_n(123) and S_n(132) based on paths (P₃) and stars (K_{1, 3})
• Proved the bijections
• Generalized the bijections to P_m and K_{1, m} as well as S_n

Week 5: June 30 - July 4

• Investigating a new counting technique for K_{1, 3} that involves inserting extra terms
• Derived insertion rules for the above approach
• Generalized the insertion approach for K_{1, m}
• Proved the recurrence conjectured last week

Week 6: July 7 - July 11

• Revisited the characterization of competition graphs of permutations in S_n(123) and S_n(132)
• Made conjecture regarding the above problem, proved half of the conjecture
• Solidified some of the proofs I wrote up earlier
• Started drafting the final report

Week 7: July 14 - July 18

• Further discussions and new ideas on the characterization of competition graphs of permutations in S_n(123) and S_n(132) with my mentor
• Finished drafting the final report
• Editing and revision on the final report
• Gave the second presentation (see below)

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Presentations

• First Presentation (Introducing the Problem)
• Second Presentation (Results)

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Additional Information

Here are some other sites of interest:

• The OEIS, an online database of integer sequences
• PermLab, an incredibly useful tool for generating data sets of different permutation classes.
• Dr. Brian Nakamura, my project mentor.
• The REU Website