## KU Combinatorics Seminar Fall 2015

The Combinatorics Seminar meets on Friday in Snow 408 at 3-4pm.

Friday 8/28
Organizational meeting

Friday 9/4
Jeremy Martin
Parking Functions

Abstract: A parking lot consists of $$N$$ parking spots in a row along a one-way street, followed by a steep cliff. One at a time, each of $$N$$ cars enter the lot and tries to park in its favorite spot; if that spot is taken, the car car either parks in the next open spot, or drives off the cliff. If $$p(i)$$ is the favorite spot of the $$i$$th car, how many functions $$p$$ successfully park all the cars? The answer may surprise you, and it has applications to hyperplane arrangements, spanning trees, and chip-firing games on graphs.

Friday 9/11
Jeremy Martin
Parking Functions II

Friday 9/18
General discussion

Friday 9/25
Jie Huang (University of Nebraska, Kearney)
Modular Catalan Numbers

Abstract: There are many naturally defined sets which are enumerated by the Catalan number $$C_n$$. The elements of these sets are called Catalan objects. Well-known sets of Catalan objects include binary trees with $$n$$ internal nodes and parenthesizations of $$x_0*x_1*\cdots*x_n$$ where the $$x_i$$'s are general fixed complex numbers, and * is an arbitrary binary operation. For example of parenthesizations, $$C_3 = 5 = |S|$$, where $$S := \{a*(b*(c*d)) , a*((b*c)*d) , (a*b)*(c*d) , (a*(b*c))*d , ((a*b)*c)*d)\}$$. If * is not arbitrary, then some of the parenthesizations may not be distinct. For example, if * is +, then every element of $$S$$ is the same and $$|S| = 1$$. If * is $$-$$, then the 1st and 4th members of $$S$$ are both $$a-b+c-d$$, so $$|S|=4$$. In this talk, we will see a family of binary operations naturally indexed by positive integers $$k$$ which generalize + and $$-$$. (The operation is addition for $$k=1$$, and it is subtraction for $$k=2$$.) By the construction of this operation, the number $$N$$ of distinct parenthesizations of $$x_0*x_1*\cdots*x_n$$ satisfies $$1\leq N\leq C_n$$. Fixing k gives a sequence $$C_{n,k}$$ of numbers which enumerate restricted sets of Catalan objects. In particular, $$C_{n,k}$$ is the number of binary trees with n internal nodes that avoid having a certain subtree (the forbidden subtree is determined by $$k$$). We will discuss combinatorial properties of the numbers $$C_{n,k}$$ and other objects enumerated by $$C_{n,k}$$. This is joint work with my colleague Nickolas Hein, who got a B.A. and M.A. in Mathematics from KU.

Friday 10/2
No seminar

Friday 10/9
Robert Winslow
Coverings of Rectangle Pairs in the Integer Lattice

Friday 10/16
No seminar

Friday 10/23
Ken Duna
Generalized Chip-Firing Games

Friday 10/30
Bennet Goeckner
Decomposing Cohen-Macaulay Complexes

Friday 11/6
Jonathan Montaño
Integral Closure of Lex-Segment Ideals

Friday 11/13
Joseph Doolittle
Random Walks on Trees and Their Relation to Circuits

Friday 11/20
TBA

Friday 11/27
No seminar (Thanksgiving)

Friday 12/4
Martha Yip (University of Kentucky)
Generalized Kostka polynomials

Abstract: Kostka numbers appear in several areas of mathematics; for example, in representation theory, they appear as multiplicities in the decomposition of permutation modules into Specht modules, and in combinatorics, they enumerate the number of semistandard tableaux. Interesting combinatorics also arise from the study of various generalizations of the Kostka numbers, such as the charge statistic in the q-analogue of Lascoux and Schutzenberger.

We study a two-parameter generalization of the Kostka numbers in connection with Macdonald polynomials. In the case of Kostka numbers indexed by partitions of three rows or less, we give a combinatorial formula for computing these in terms of alcove walks. This is joint work with A. Ram and M. Yoo.

Friday 12/11
Stop Day

For seminars from previous semesters, please see the KU Combinatorics Group page. Last updated Thu 11/26/15