Fall 2015

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

Please contact Jeremy Martin if you are interested in speaking.

**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