# KPPY 76

Kyoung-Tark Kim, Masashi Shinohara, Jisu Jeong, Sun-Young Nam, Byung Hee An

Apr 30, 2016. 11:00am-6:00pm at PNU

Schedule | ||

11:00am - 11:50 |
Kyoung-Tark Kim Shanghai Jiao Tong University | Harmonic index designs in the binary Hamming schemes |

12pm - 1:25 | Lunch | |

1:30 - 2:20 |
Masashi Shinohara Shiga University | Maximum Euclidean sets that contain few non-congruent triangles |

2:30 - 3:20 |
Jisu Jeong KAIST | Invitation to Fixed-Parameter Algorithm |

3:30 - 4:20 |
Sun-Young Nam KIAS | Combinatorial models for shifted Littlewood-Richardson coefficients, |

4:30 - 5:20 |
Byung Hee An POSTECH | f-vectors of Gelfand-Zetlin polytopes |

## Abstracts

Kyoung-Tark Kim

Harmonic index designs in the binary Hamming schemes

Harmonic index designs in the binary Hamming schemes

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.

Masashi Shinohara

Maximum Euclidean sets that contain few non-congruent triangles

Maximum Euclidean sets that contain few non-congruent triangles

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.

Jisu Jeong

Invitation to Fixed-Parameter Algorithm

Invitation to Fixed-Parameter Algorithm

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.

Sun-Young Nam

Combinatorial models for shifted Littlewood-Richardson coefficients,

Combinatorial models for shifted Littlewood-Richardson coefficients,

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.

Byung Hee An

f-vectors of Gelfand-Zetlin polytopes

f-vectors of Gelfand-Zetlin polytopes

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