dimacs logo

General Information

Kelly Toppin
CoRE 442
shippensburg University
Faculty Advisor:
Dr Aaron Jaggard
Permutation Statistics and Parallels Between Families of Permutations

Project Description

In this project, we'll focus on questions related to permutation statistics. These are nonnegative-integer-valued functions on permutations; classical examples of statistics on a permutation p include its descent number des(p) (the number of i such that p(i)>p(i+1)), the number of inversions inv(p) (the number of i < j such that p(i)>p(j)), and the major index maj(p) (the sum of the values i such that p(i)>p(i+1)). One important problem is the determination of the distribution of a statistic (or tuples of statistics)---i.e., the number of permutations for which the statistic takes a certain value---over all permutations of a given length. A number of interesting results have been obtained showing that certain statistics (or tuples of statistics) have the same distribution; see [1] for one recent example of this that also provides a way to decompose classical statistics into sums of other statistics.

If you are interested in this, feel free to e-mail me with questions or comments.

Project Log

Week 1
Familiarized self with problem.

Week 2
At military training.

Week 3
At military training.

Week 4
Read papers on "Permutation Statistics"

Week 5
Worked on a possible conjecture about permutation labeling. I also went on a group field trip to Telcordia

Week 6
Came up with a possible conjecture on this week. We went on the group field trip to Bell Labs

Week 7
Prepared and gave a summary presentation of my work on the project. Due to the fact that I am working on several conjectures which are not publish yet I can not display my final presentation. Thank You for your understanding

Week 8
Discussed the direction of the project with Dr. Jaggard. Working on proving my labeling conjecture and working on another conjecture including two other permutation statistics. Leaving on Friday