Math 128
Current Problems in Combinatorics
Last updated June 17, 2015 12:56:59 EDT

Each student is expected to give an in-class presentation on a topic of their choice. You can work individually, but for the longer papers it is recommended that you work in small groups. Each of you should speak for about 30 minutes, although this is quite flexible. Below is a list of suggested topics (more will be added as the course progresses). You are more than welcome to find a topic not listed here; for example, you can present a combinatorics paper reasonably related to the course from a journal of your choice. Just check with me first.

Scheduled presentations

• Jenny, Sorting and preimages of pattern classes. Wednesday, May 16.
• Dan, Sorting with two stacks in parallel. Thursday, May 18, at 1:15 in Kemeny 120. (Note: this is the Combinatorics Seminar time).
• Elizabeth, Simple permutations and pattern restricted permutations. Monday, May 21.
• Zach, Counting Reduced Decompositions. Monday, May 21. Here is a link to the applet.
• Kassie, Counting permutations with given cycle structure and descent set. Wednesday, May 23.
• Natasha, Search games. Wednesday, May 23.
• Nathan, Infinite antichains on permutations. Friday, May 25.
• Jonathan, Simple permutations and algebraic generating functions. Friday, May 25.
• Megan, Permutation tableaux. Wednesday, May 30.

A few suggested topics for student presentations

• [V.R. Pratt, Computing permutations with double-ended queues, parallel stacks and parallel queues, Proc. ACM Symp. Theory of Computing 5 (1973), 268–277.]
This is a classic paper that characterizes permutations that can be sorted by a dequeue (i.e., a double ended queue, where one can push or pop from either end). These permutations are characterized as those avoiding an infinite set A of patterns. The set A was the first published example of an infinite antichain in the pattern containment psoet. The enumeration of dequeue-sortable permutations remains unsolved.
Here is what Peter Doyle said about this paper: "I've been reading this paper by Vaughan Pratt. I love the way he writes! It reminds me of the way I write. And yes, nothing gives me more pleasure than rereading stuff I've written, and admiring the author's sense of humor. Well, almost nothing."

• [D. Spielman and M. Bona, An infinite antichain on permutations, Electron. J. Combin. 7 (2000), #N2.]
This short paper gives another construction of an infinite antichain in the pattern containment poset.

• [A. Claesson and H. Ulfarsson, Sorting and preimages of pattern classes, preprint, available here]
This is one of the most recent papers about sorting and pattern-avoiding permutations. Among other things, it deals with West-k-stack-sortable permutations, which, defined by Julian West in 1990, are those permutations that can be sorted by k passes through a stack whose operations are forced (by the algorithm for sorting 231-avoiding permutations).

• [V. Vatter and S. Waton, On points drawn from a circle, Electron. J. Combin. 18 (2011), P223].
This paper characterizes in terms of pattern avoidance the set of permutations that can be obtained from drawing points on a circle. It also enumerates these permutations by length.

• [M. Albert, M. Atkinson, Simple permutations and pattern restricted permutations, Discrete Math. 300 (2005), 1-15.]
This paper proves that every subclass of the separable permutations has an algebraic generating function.

• [Z. Stankova, Frobidden subsequences, Discrete Math. 132 (1994), 291-316.]
This paper proves that the patterns 4132 and 3142 are Wilf-equivalent.

• [I. Le, Wilf classes of pairs of permutations of length 4, Electron. J. Combin. 12 (2005), #R25.]
This paper, which is the result of an REU program, finds the generating function for the class Av(1342,2143) that you used in Problem Set 2.

• [M. Albert, M. Elder, A. Rechnitzer, P. Westcott and M. Zabrocki, On the Stanley–Wilf limit of 4231-avoiding permutations and a conjecture of Arratia, Adv. Appl. Math. 36 (2006), 96-105.]
This paper shows that the growth rate of the pattern 1324 is greater than 9.47, thus disproving a conjecture of Arratia that the growth rate of any pattern of length k is at most (k-1)^2.

• [M. Albert, M. Atkinson, V. Vatter, Counting 1324,4231-avoiding permutations, Electron. J. Combin. 16 (2009), #R135.]
This paper enumerates permutations avoiding both 1324 and 4231 by giving their generating function, which is rational.

• [V. Vatter, Small permutation classes, Proceedings of the LMS 103 (2011), 879–921.]
This paper studies permutation classes with smal growth rates, characterizing all the possible growth rates below a certain value.

• [E. Steingrimsson and L. Williams, Permutation tableaux and permutation patterns, J. Combin. Theorey Ser A 114 (2007), 211-234.]

• [E. Steingrimsson and B. Tenner, The Mobius function of the permutation pattern poset, J. Comb. 1 (2010), 39-52.] Last updated June 17, 2015 12:56:59 EDT