# Algebraic Combinatorics Seminar

 This is the webpage for the Algebraic Combinatorics Seminar at KIAS. The seminar is run by Jang Soo Kim, Seungjin Lee, Hwanchul Yoo, and Meesue Yoo. If you have questions or if you want to receive emails about upcoming seminars, please contact Jang Soo Kim (kimjs@kias.re.kr). Click [Algebraic Combinatorics Seminar] to subscribe our seminar calinder in iCal format.

• July 22 (Monday), 2013
• 2:30 PM - 3:30 PM in KIAS 1424
• Speaker: Meesue Yoo (KIAS)
• Title: Macdonald polynomials and the space of diagonal harmonics
• Abstract : We see the construction of Macdonald polynomials as a q,t-twisting of Schur functions and see how Garsia and Haiman tried to realize them as characters of certain bigraded Sn-modules to prove the positivity of q,t-Kostka polynomials. As a byproduct of this approach, they obtained positivities of various combinatorial objects such as q,t-Catalan numbers. We introduce related conjectures which are still open. This talk is a preliminary talk of Prof. Kyungyong Lee's.
• July 22 (Monday), 2013
• 4:00 PM - 5:00 PM in KIAS 1424
• Speaker: Kyungyong Lee (Wayne State University)
• Title: Toward the canonical bijection for q,t-Catalan objects
• Abstract : The q,t-Catalan numbers arise in a wide range of subjects. In fact, for each positive integer n, the n-th q,t-Catalan number(s) can be defined in various ways. We introduce algebro-geometric q,t-Catalan numbers, commutative algebraic ones, representation theoretic ones, symmetric function theoretic ones, topological ones, and at least four combinatorial ones. After decades of intensive study, it was eventually proved that all these definitions are equivalent, which implies that combinatorial q,t-Catalan polynomials are symmetric. Despite this result, it is still an important open problem to find a bijective proof for the symmetry. We present a very promising approach to this problem.
• July 29 (Monday), 2013
• We have Pseudo Sage Day!
• 10:00 AM - 12:00 PM (Sage tutorial) and 2:00 PM - 5:00 PM (Coding help) in KIAS 1424
• Speaker: Travis Scrimshaw (UC Davis)
• July 30 (Tuesday), 2013
• 1:00 PM - 3:00 PM in KIAS 1423
• Speaker: Hwanchul Yoo (KIAS)
• Title: On "Permutohedra, associahedra, and beyond" (1)
• Abstract: We will review Postnikov's paper, "Permutohedra, associahedra, and beyond". http://arxiv.org/abs/math/0507163
• August 12 (Monday)
• 11:00 AM - 12:00 PM in KIAS 1503 (This is a colloquium talk.)
• Speaker: Richard Stanley (MIT)
• Title: Alternating permutations
• A permutation (a_1,a_2,...,a_n) of 1,2,...,n is called alternating if (a_1 > a_2 < a_3 > a_4 <...). The number of alternating permutations of 1,2,...,n is denoted $E_n$ and is called an Euler number. The most striking result about alternating permutations is the generating function $$\sum_{n\geq 0}E_n\frac{x^n}{n!} = \sec x+\tan x,$$ found by D\'esir\'e Andr\'e in 1879. We will discuss this result and how it leads to the subject of combinatorial trigonometry.'' We will then survey some further aspects of alternating permutations, including some other objects that are counted by $E_n$, the use of the representation theory of the symmetric group to count certain classes of alternating permutations, and the distribution of the length of longest alternating subsequence of a random permutation.
• August 12 (Monday)
• 2:00 PM - 3:00 PM in KIAS 1424
• Speaker: Dennis Stanton (University of Minnesota)
• Title: Open problems in positivity
• Abstract: I will talk about a variety of open positivity problems in combinatorics. The related topics are lattice paths, integer partitions, and q-analogues and (q,t)-analogues.
• August 12 (Monday)
• 4:00 PM - 5:00 PM in KIAS 1209
• Speaker: Suho Oh (University of Michigan)
• Title : Triangulations and Matching Ensemble.
• Abstract : Studying triangulations of product of simplices is an interesting problem that has ties to many different fields. We show that one can use a collection of matchings with certain axiom system, to completely describe the triangulations of product of simplices. This is in joint work with Hwanchul Yoo.
• August 26 (Monday)
• 2:30 PM - 5:00 PM (with a 30-min break) in KIAS 1424
• Speaker: Suyoung Choi (Ajou University)
• Title: The number of real toric varieties
• Abstract: A fundamental theorem in toric geometry is that there is a bisection between compact smooth toric varieties (say, toric manifolds) and complete non-singular fans. We note that a complete non-singular fan can be regarded as a pair of a simplicial complex $K$ and the data of rays $\lambda$. When M is a toric variety of complex dimension n, there is a canonical involution on M and its fixed points form a real subvariety of real dimension n, called a real toric variety. Similarly, real toric manifolds also can be described as a pair of $K$ and $\mathbb{Z}_2$ version of $\lambda$. We note that for a fixed $K$, there are finitely many real toric manifolds over $K$. Hence, the counting problem for real toric manifolds over a given $K$ should be interesting. In the talk, we discuss about the classification and enumeration of (real) toric manifolds over certain simplicial complexes including the boundary of a cross polytope and a simplicial complex with a few vertices. In addition, we also discuss about the topological classification of real toric varieties.
• August 30 (Friday)
• 4:00 PM - 5:00 PM in KIAS 1424
• Speaker: Hwanchul Yoo (KIAS)
• Title: On "Permutohedra, associahedra, and beyond" (2)
• Abstract: We will review Postnikov's paper, "Permutohedra, associahedra, and beyond". http://arxiv.org/abs/math/0507163
• September 3 (Tuesday)
• 4:00 PM - 5:00 PM in KIAS 1424
• Speaker: Seungjin Lee (KIAS)
• Title: Centrally symmetric polytopes with many faces
• Abstract: We study the convex hull of the symmetric moment curve $U_k(t)=(\cos t, \sin t, \cos 3t, \sin 3t, \ldots, \cos (2k-1)t, \sin (2k-1)t)$ in ${\mathbb R}^{2k}$ and provide deterministic constructions of centrally symmetric polytopes with a record high number faces. In particular, we prove the local neighborliness of the symmetric moment curve, meaning that as long as $k$ distinct points $t_1, \ldots, t_k$ lie in an arc of a certain length $\phi_k > \pi/2$, the points $U_k(t_1), \ldots, U_k(t_k)$ span a face of the convex hull of $U_k(t)$. In this talk, I will use the local neighborliness of the symmetric moment curve to construct $d$-dimensional centrally symmetric 2-neighborly polytopes with approximately $3^{d/2}$ vertices. This is in joint work with Alexander Barvinok and Isabella Novik.
• October 23 (Wed)
• 4:00 PM - 5:00 PM in KIAS 1209
• Speaker: Meesue Yoo (KIAS)
• Title: On "m-level rook placements"
• Abstract: We review the recent paper of Barrase, Loehr, Remmel and Sagan on m-level rook placements. We also consider the signed cycle counting rook numbers in the wreath product of a cyclic group and the permutation group.
• October 28 (Mon)
• 4:00 PM - 5:00 PM in KIAS 1209
• Speaker: Hwanchul Yoo (KIAS)
• Title: On "Permutohedra, associahedra, and beyond" (3)
• Abstract: We will review Postnikov's paper, "Permutohedra, associahedra, and beyond". http://arxiv.org/abs/math/0507163
• October 29 (Tue)
• 4:00 PM - 5:00 PM in KIAS 1423
• Speaker: Jang Soo Kim (KIAS)
• Title: The Selberg integral and Young books
• Abstract: The Selberg integral is an important integral first evaluated by Selberg in 1944. Stanley found a combinatorial interpretation of the Selberg integral in terms of probability. In this talk we find a connection between the Selberg integral and shifted Young tableaux of staircase shape and standard Young tableaux of square shape. We define 'Young books' which are a generalization of both of these shifted and standard Young tableaux and find a formula for the number of Young books. This is joint work with Suho Oh.
• November 29 (Friday)
• 4:00 PM - 5:00 PM in KIAS 1424
• Speaker: Uijin Jung (Ajou University)
• Title: Maps between shifts of finite type and Graph homomorphisms
• Abstract: Given a directed graph G, a shift of finite type defined by G is the set of all bi-infinite paths in G. The class of shifts of finite type forms a subclass of symbolic dynamical systems which are very well known. As are defined from directed graphs, shifts of finite type has nice structures which can be proved using combinatorial ways. Especially, the theory of resolving graph homomorphisms intertwines graph theory and symbolic dynamics. We will introduce maps between shifts of finite type induced from graph homomorphisms, investigate certain graph homomorphisms and extension of them, and will see how the existence of certain graph homomoprhism is related to the existence and extension of maps between two shifts of finite type.
• December 2 (Monday)
• 4:00 PM - 5:00 PM in KIAS 1424
• Speaker: Marie Albenque (LIX, Ecole polytechnique)
• Title: Unified construction between blossoming trees and planar maps
• Abstract: Bijections between decorated trees and planar maps are a precious tool to get some insight about the structure of planar maps (planar graphs with a fixed embedding). Many bijections have been obtained in the last 15 years and in this talk I"ll present a general framework that allows to get most of these constructions as special cases and to obtain also new results. This construction relies deeply on the theory of alpha-orientations introduced by Felsner. Since most of the families of maps can be characterized by such orientations, I will illustrate how this generic bijective method permits to get bijections between blossoming trees and for example Eulerian maps, bipolar orientations and simple triangulations and quadrangulations. This is joint work with Dominique Poulalhon.
• December 4 (Wednesday)
• 4:00 PM - 5:00 PM in KIAS 1503
• Speaker: Hanchul Park (Ajou University)
• Title: Nestohedra and cohomology of real toric manifolds
• Abstract: Among simple polytopes, nestohedra $P$ forms an interesting subfamily containing the permutohedra and the associahedra. From a view of toric topology, each of them supports a canonical real toric manifold $M(P)$. In this talk, we study the cohomology of $M(P)$ which is a combinatorial invariant of $P$, especially $P$ is obtained from a graph, called a graph associahedron. In that case, the invariant is computed using so-called a-numbers of the graph. Moreover, we see an example of $M(P)$ whose cohomology has odd torsion, which is an important example in toric topology. This work is jointly with Prof. Suyoung Choi.
• December 16 (Monday)
• 4:00 PM - 5:00 PM in KIAS 1424
• Speaker: Jae-Hoon Kwon (Sungkyunkwan University)
• Title : Plane partitions and Demazure characters
• Abstract : In this talk, we introduce a new representation theoretic interpretation of RSK algorithm and its application. We first show that RSK correspondence is a $\mathfrak{gl}_{\infty}$-crystal isomorphism between two realizations of the crystal graph of a maximal parabolic Verma module. Then a flagged version of the RSK correspondence is recovered in a natural way by computing a Demazure crystal graph of a maximal parabolic Verma module. As an application, we discuss a relation between a Demazure crystal and plane partitions with a bounded condition.
• December 17 (Tuesday)
• 4:00 PM - 5:00 PM in KIAS 1424
• Speaker: Byungchan Kim (Seoul National University of Science and Technology)
• Title : Positive Moments in Partitions
• Abstract : In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. The strict inequality between these moments will be discussed. Analogous moments for overpartitions will be also discussed. Several combinatorial model to explain these inequalities will be given.
• December 19 (Thursday)
• 4:00 PM - 5:00 PM in KIAS 1424
• Speaker: Masao Ishikawa (University of the Ryukyus)
• Title: A one parameter generalization of Pfaffians
• Abstract: It is well-known that the alternating sign matrices arise In the lambda-determinant, which is a one-parameter deformation of the Dodgson condensation formula by Mills, Robbins and Rumsey. We give an one-parameter anaology for Pfaffians. We prove that the resulting rational function is a polynomial with weights given by the crossings and nestings of perfect matchings and prove several identities and closed-form evaluations.
• December 30 (Monday)
• 4:00 PM - 5:00 PM in KIAS 1424
• Speaker: Jang Soo Kim
• Title: A $q$-Selberg integral and Schur functions
• Abstract: In this talk we introduce a $q$-Selberg integral and show that it is a generating function for the $m$th power of Schur functions with certain specialization. This is work in progress with Soichi Okada.