KPPY 76

In this talk I present our recent research on 'harmonic index designs' in the binary Hamming association schemes $H(d, 2)$. First, I will talk about the 'addition formula' in arbitrary commutative association schemes. From the nonnegativity feature due to the addition formula, we can perform Delsarte's linear programming method for the size of harmonic index designs in any $Q$-polynomial association scheme. After the careful selection of a test function for linear programming, we can define the notion of 'tight' designs. I will tell you our philosophy for choosing a test function as well. Most of our research are focused on the nonexistence of some tight harmonic index designs in the binary Hamming schemes. I will introduce the subject from very elementary conception. The audiences do not have to know special preparatory information. This is joint work with Eiichi Bannai, Etsuko Bannai, Takuya Ikuta, and Yan Zhu.

For a subset of $X$ of $\mathbb{R}$, we define $A_2(X)$ as a set of all distances of two distinct points of $X$. $X$ is called a $k$-distance set if $|A_2(X)| = k$. In other words, $X$ is a $k$-distance set if the set of all segments of two points of $X$ has exactly $k$ congruent classes. In this talk, we generalize of this concept. We also regard a collinear three points as a triangle in this talk. We define $A_3(X)$ as a set of all non-congruent three points subsets of $X$. $X$ is called a $k$-triangle set if $|A_3(X)| = k$. In this talk, we classify maximum $2$-triangle sets in $\mathbb{R}^d$, and determine maximum cardinalities of $k$-triangle for $d \leq 4$. This is a joint work with Professor Mitsugu Hirasaka.

Fixed-Parameter Algorithm is an algorithm that can solve a given problem in time $f(k)n^c$ where $f$ is a function, $k$ is a parameter, $c$ is a constant, and $n$ is the size of the input. Here is an example. VERTEX COVER problem asks the minimum size of a vertex cover of an input graph. $k$-VERTEX COVER problem is a decision problem that asks whether an input graph has a vertex cover of size $k$. There exists an Fixed-Parameter Algorithm that can solve $k$-VERTEX COVER problem in time $O(2kn)$. In this talk, I will give many examples about Fixed-Parameter Algorithm so that everyone knows a basis concept of it.

Since combinatorial definitions of Schur P-functions are given by weight generating functions of three kinds of shifted tableaux, it seems to be very natural to prove the positivity of shifted Littlewood-Richardson(LR) coefficients in terms of such tableaux. Indeed, it was known that the followings are combinatorial models for shifted LR coefficients corresponding to three kinds of shifted tableaux: (a) Littlewood-Richardson-Stembridge tableaux due to Stembridge, (b) $\lambda$-good semistandard decomposition tableaux due to Cho, and (c) shifted Littlewood-Richardson decomposition tableaux due to Grantcharov, Jung, Kang, Kashiwara, and Kim. The purpose of this talk is to briefly introduce the above three combinatorial models and then provide explicit bijections among them.

We provide the differential equations on the generating functions for $f$-vectors of Gelfand-Zetlin polytopes.