Workshop on Algebraic and Enumerative Combinatorics.

Date

2019 Jan. 15th (Tue.) -- 17th (Thu.)

Venue

Shinshu University, Matsumoto, Japan.

Romms are the following:

[Local infomation (in English)] [Local infomation (in Japanese)]

Schedule (Tentative)

2019 Jan. 15th (Tue.)

14:30 -- 15:30
Jang Soo KIM (Sungkyunkwan Univ.) / 김 장수 (성균관대학교)
Lecture hall tableaux
We introduce lecture hall tableaux, which are fillings of a skew Young diagram satisfying certain conditions. Lecture hall tableaux generalize both lecture hall partitions and anti-lecture hall compositions, and also contain reverse semistandard Young tableaux as a limit case. We show that the coefficients in the Schur expansion of multivariate little $q$-Jacobi polynomials are generating functions for lecture hall tableaux. Using a Selberg-type integral we show that the moment of multivariate little $q$-Jacobi polynomials, which is equal to a generating function for lecture hall tableaux of a Young diagram, has a product formula. We also explore various combinatorial properties of lecture hall tableaux. This is joint work with Sylvie Corteel.
16:15 -- 17:15
Sun-mi YUN (Sungkyunkwan Univ.) / 윤 선미 (성균관대학교)
The Characteristic Polynomial of the Alternating Permutation Poset.
For the symmetric group $S_n$, a permutation w can be considered as a word of simple transpositions $s_i = (i, i+1)$. We define some order of permutations with respect to the minimal length of the word expression of each permutation. Then $S_n$ with this order is a poset. We consider the set of alternating permutations with the same order, which is also a poset. We call it the alternating permutation poset. It turns out that its characteristic polynomial has a nice form. In this talk we prove this property.
Social dinner

2019 Jan. 16th (Wed.)

10:00 -- 11:00
Ayaka ISHIKAWA (Yokohama National Univ.) / 石川 彩香 (横浜国立大学)
The enumeration of unlabeled rooted trees using Young tableaux (Young tableaux を用いた根付き木の数え上げ)
$n$頂点ラベル付き木の個数はケイリーの公式より $n^{n-2}$で与えられる. また, $n$頂点ラベル付き根付き木の個数は, 根の取り方が 頂点の個数分あることから $n^{n-1}$ と導ける. しかし, ラベルが 付いていない場合は, 根の有無にかかわらず 木の個数を与える 明示公式は得られていない. 今回は, 根付き木を Young tableaux の列に対応させること で, "ある条件" を満たす根付き木の個数を Young tableaux の数え上げを用いて明示公式 で与える.
Most of the tree enumeration formulas are generating functions or recurrence formulas. In this talk, we show the explicit formula for the number of unlabeled rooted trees with a certain condition. The formula is described in terms of Young tableaux.
11:45 -- 12:45
Hiroshi NARUSE (Yamanashi Univ.) / 成瀬 弘 (山梨大学)
Some combinatorial properties of dual factorial Schur functions.
For type A case, factorial Schur function and its dual are studied by A.Molev. In this talk I will mainly report for type C results. There are also several similarities between factorial Schur Q function and its dual. For example divided differences, excited Young diagrams, generating functions, and Pfaffian formulas. The Hopf algebra structure and dual graded graph point of view are important. Type D case, the K-theory and beyond will also be mentioned.
Lunch
14:30 -- 15:30
Meesue YOO (Sungkyunkwan Univ.) / 유 미수 (성균관대학교)
Schur expansion of LLT polynomials related to certain graphs.
LLT polynomials are a family of symmetric functions introduced by Lascoux, Leclerc and Thibon in 1997 which naturally arise in the description of the power-sum plethysm operators on symmetric functions. Grojnowski and Haiman proved that they are Schur positive using Kazhdan--Lisztig theory, but there is no Known combinatorial formula for the Schur coefficients. In this talk, we utilize linear relations introduced by Lee to prove some combinatorial formulas for the Schur coefficients of LLT polynomials, when the LLT polynomials are indexed by certain diagrams related to particular type of graphs. This is a joint work with Jisun Huh and Sun-Young Nam.
16:15 -- 17:15
Jihyeug JANG (Sungkyunkwan Univ.) / 장 지혁 (성균관대학교)
Combinatorial proof of two constant term identities
These constant term identities was introduced in 〈A COMBINATORIAL MODEL FOR COMPUTING VOLUMES OF FLOW POLYTOPES〉. We prove these identities using labeled dyck path and parenthesizing. Chains of noncrossing partitions play an important role in our proof.

2019 Jan. 17th (Thu.)

10:00 -- 11:00
Soichi OKADA (Nagoya Univ.) / 岡田 聡一 (名古屋大学)
Birational rowmotion and Coxeter-motion on minuscule posets.
Birational rowmotion is a discrete dynamical system associated with a finite poset $P$, which is a birational lift of the combinatorial rowmotion acting on order ideals of $P$. Grinberg--Roby and Musiker--Roby establishes nice properties such as periodicity and file homomesy for the birational rowmotion on a product of two chains. In this talk we extend their results to the birational rowmotion on minuscule posets. Also we introduce another dynamical system, called birational Coxeter-motion, and study their properties.
11:45 -- 12:45
Akiko YAZAWA (Shinshu Univ.) / 矢澤 明喜子 (信州大学)
The Hessians of Kirchhoff polynomials.
Let us consider the weighted generating function for the spanning trees. These are called Kirchhoff polynomials. We consider the Hessian matrices of Kirchhoff polynomials. In this presentation, we show some Hessians of the polynomials do not vanish by calculating the eigenvalues. As an application, we show the strong Lefschetz property for some Artinian Gorenstein algebras associated to the graphic matroids.

Banquet

If you would like to attend the banquet, please let us know by email ( nu at math.shinshu-u.ac.jp ) before Jan 1st 2019.

The baquet will be held on 15th (Tue.) from 18:30 to 20:00 at Matsumoto Karaage Center [松本からあげセンター] on the 4th floor of Matsumoto Station. The fee is 4500JPY (3000JPY for a sutdent). Please pay at the workshop on Tuesday.

Misc

Speakers can use projectors and whiteboards. Please see this page for the detail.

The room will be open from the morning.

Some notepads and pens are available in the conference room.

Organizer

Yasuhide Numata (Shinshu Univ.) and Meesue Yoo (Sungkyunkwan Univ.) are organizers of this workshop. If you have any query for this workshop, please ask Yasuhide NUMATA (Shinshu University, nu at math.shinshu-u.ac.jp) by e-mail.

Acknowledgements

This workshop is partly supported by the following: