The aim of this conference is to allow participants an opportunity to discuss recent developments and new ideas in algebraic and enumerative combinatorics and related areas such as representation theory, algebraic geometry and statistical physics. Topics will include (but are not limited to): plane partitions, P-partitions, alternating sign matrices, perfect matchings, and symmetric functions.

Masao Ishikawa (Okayama University), Kento Nakada (Okayama University), Yasuhide Numata (Shinshu University), Soichi Okada (Nagoya University), Takeshi Suzuki (Okayama University), Hiroyuki Tagawa (Wakayama University), Itaru Terada (University of Tokyo), Hiro-Fumi Yamada (Kumamoto University).

**Roger Behrend** (Cardiff University)

**Mihai Ciucu** (Indiana University)

**Shuhei Kamioka** (Kyoto University)

**Jang Soo Kim** (Sungkyunkwan University)

**Hiroshi Naruse** (Yamanashi University)

**Greta Panova** (University of Pennsylvania)

**Tom Roby** (University of Connecticut)

These are photographs of Okayama station, Okayama University and Kourakuen Garden. Okayama University, located near downtown Okayama, is just two kilometers from Okayama railway station. Okayama is a city located approximately 200 km west to Osaka, which is a major transportation hub for the Chugoku and Shikoku areas. This enables you to access Okayama easily from any other major cities in Japan as Tokyo or Osaka.

The conference program/schedule is now available – click here to view the PDF file. The following invited speakers will give 2 talks: Roger Behrend, Mihai Ciucu, Shuhei Kamioka, Jang Soo Kim, and Greta Panova. The slides of the talks are available by clicking the titles.

Soichi Okada | Opening Adrress | 09:20 - 09:30 |
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Jang Soo Kim | Combinatorics of the Selberg integral | 09:30 - 10:30 |

Greta Panova | Hook formulas for skew shapes | 10:50 - 11:50 |

Soichi Okada | d-Complete posets and hook formulas |
13:20 - 14:20 |

Yasuhide Numata | On a framework for Hillman-Grassl algorithms | 14:40 - 15:40 |

Hiro-Fumi Yamada | On Sato's "On Hirota's Bilinear Equations" | 16:00 - 17:00 |

Name | Title | Time |
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Hiroshi Naruse | Hook formula for skew shape on d-complete poset |
09:30 - 10:30 |

Mihai Ciucu | Lozenge tilings with gaps in a 90 degree wedge domain | 10:50 - 11:50 |

Conference Lunch at Terrace Cafe | 12:00 - 13:00 |

Roger Behrend | The combinatorics of alternating sign matrices | 13:20 - 14:20 |

Shuhei Kamioka | Nice formulas for plane partitions | 14:40 - 15:40 |

Ryo Tabata | Limiting behavior of immanants | 16:00 - 17:00 |

Name | Title | Time |
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Jang Soo Kim | Hook length property of d-complete posets via q-integrals |
09:30 - 10:30 |

Greta Panova | Hook formulas for skew shapes | 10:50 - 11:50 |

Itaru Terada | On an involution on the set of Littlewood-Richardson tableaux | 12:10 - 13:10 |

Conference Photo | 13:10 - 13:20 | |

Problem Session and Free Discussion | 14:30 - 17:00 | |

Banquet | 18:30 - 20:30 |

Name | Title | Time |
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Tom Roby | Paths to understanding birational rowmotion | 09:30 - 10:30 |

Mihai Ciucu | Lozenge tilings with gaps in a 90 degree wedge domain | 10:50 - 11:50 |

Roger Behrend | The combinatorics of alternating sign matrices | 13:20 - 14:20 |

Shuhei Kamioka | Nice formulas for plane partitions | 14:40 - 15:40 |

Motoki Takigiku | A factorization formula for K-k-Schur functions |
16:00 - 17:00 |

Name | Title | Time |
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Masao Ishikawa | (q,t)-hook formula for Tailed Insets |
09:30 - 10:30 |

Ayumu Hoshino | Tableau Formulas for One-Row Macdonald Polynomials | 10:50 - 11:50 |

Fumihiko Nakano | Generalized carries process and riffle shuffles | 12:10 - 13:10 |

Masao Ishikawa | Closing address | 13:10 - 13:20 |

In these talks, I will discuss various topics related to the combinatorics of alternating sign matrices. These will include alternating sign matrix symmetry classes, alternating sign matrix statistics, descending plane partitions, totally symmetric self-complementary plane partitions, alternating sign triangles, fully packed loop configurations, the alternating sign matrix polytope, and the alternating sign matrix poset.

We consider a triangular gap of side two in a 90 degree angle on the triangular lattice with mixed boundary conditions: a constrained, zig-zag boundary along one side, and a free lattice line boundary along the other. We study the interaction of the gap with the corner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its three images in the sides of the angle. This provides evidence for a unified way of understanding the interaction of gaps with the boundary under mixed boundary conditions, which we present as a conjecture. Our conjecture is phrased in terms of the steady state heat flow problem in a uniform block of material in which there are a finite number of heat sources and sinks. This new physical analogy is equivalent in the bulk to the electrostatic analogy we developed in previous work, but arises as the correct one for the correlation with the boundary.

The starting point for our analysis is an exact formula we prove for the number of lozenge tilings of certain trapezoidal regions with mixed boundary conditions, which is equivalent to a new, multi-parameter generalization of a classical plane partition enumeration problem (that of enumerating symmetric, self-complementary plane partitions).

We present explicit formulas for the Macdonald polynomials of types C_n in the one-row case. In view of the combinatorial structure, we call them "tableau formulas". For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series. We remark that the correlation functions of the deformed W algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables. This talk is based on our paper,Feigin,Hoshino, Noumi,Shibahara and Shiraishi, Tableau Formulas for One-Row Macdonald Polynomials of Types C_n and D_n.

Okada presented a conjecture on (q,t)-hook formula for generald-complete posets in the paper, Soichi Okada, (q, t)-Deformations of multivariate hook product formulae, J. Algebr. Comb. (2010) 32, 399–416. We consider the Tailed Inset case, and reduce the conjectured identity to an indentity of the Macodonald polynomials rephrasing Okada's (q,t)-weights via Pieri coefficients of the Macodonald polynomials. Joint work with Frederic Jouhet (University of Lyon I).

Plane partitions are so nice that there are nice generating functions for plane partitions which can be factored, such as MacMahon’s formula and its variants. In this talk a close connection between plane partitions and an integrable system, the discrete two-dimensional (2D) Toda molecule, is clarified. The main theorem is: each non-vanishing solution to the discrete 2D Toda molecule gives a nice (weighted) generating function for boxed (reverse) plane partitions. As an example a new weighted generating function for boxed reverse plane partitions of arbitrary shape which generalizes MacMahon’s formula and a trace generating function of Gansner type is shown.

In 1944, Selberg evaluated a multivariate integral, which generalizes Euler's beta integral. In 1980, Askey conjectured aq-integral version of the the Selberg integral, which was proved independently by Habsieger and Kadell in 1988. In this talk, we focus on the combinatorial aspects of the Selberg integral. First, we review the following fact observed by Igor Pak: evaluating the Selberg integral is essentially the same as counting the linear extensions of a certain poset. Consideringq-integrals over order polytopes, we give a combinatorial interpretation for Askey's q-Selberg integral. We also find a connection between the Selberg integral and Young tableaux. As applications we enumerate Young tableaux of various shapes.

The hook length formula ford-complete posets states that the P-partition generating function for them is given by a product in terms of hook lengths. We give a new proof of the hook length formula usingq-integrals. The proof is done by a case-by-case analysis consisting of two steps. First, we express the P-partition generating function for each case as aq-integral and then we evaluate theq-integrals. Severalq-integrals are evaluated using partial fraction expansion identities and others are verified by computer. This is joint work with Meesue Yoo.

Carries process is a Markov chain of carries in addingnnumbers. We consider a generalization of that, studied the transition probability matrix, and its relation to combinatorics. The results include : (1) the stationary distribution is proportional to the decent statistics of colored permutation group (2) left eigenvector matrix is equal to the Foulkes character table ofG(p, n)(3) Stirling-Frobenius number appears in the right eigenvector matrix. (4) Discussion on the generalized riffle shuffles whose descent process is equally distributed to the carries process. This is a joint work with Taizo Sadahiro (Tsuda College).

We use equivariant K-theory of flag variety to represent a hook formula for generating function of reverse plane partitions on skew shaped-complete poset in terms of excited diagrams. This is a joint work with Soichi Okada.

The classical Hillman-Grassl algorithm induces a bijection from reverse plane partitions to multisets of hooks. We introduce a framework such that the analogues of the algorithm work. We apply the framework to swivels-free $d$-complete posets. This is based on the joint work with K. Nakada and S. Okamura.

d-Complete posets are a class of posets introduced by Proctor as a generalization of shapes (Young diagrams), shifted shapes (shifted Young diagrams) and rooted trees, and Peterson and Proctor obtained a hook product formula ford-complete posets. In the first half of this talk, I will give an introduction tod-complete posets and hook product formulas for them. In the second half, which is based on a joint work with H. Naruse, I present a skew hook formula ford-complete posets. This formula generalizes both Naruse's skew hook formula and aq-hook formula for skew shapes given by Morales-Pak-Panova.

We will show several combinatorial and aglebraic proofs of this formula, leading to a bijection between SSYTs or reverse plane partitions of skew shape and certain integer arrays that gives twoq-analogues of the formula. These formulas can also be proven via non-intersecting lattice paths interpretations, for example connecting Dyck paths and alternating permutations.

We will also show how excited diagrams give asymptotic results for the number of skew Standard Young Tableaux in various regimes of convergence for both partitions. Multivariate versions of the hook formula with consequences to exact product formulas for certain skew SYTs and lozenge tilings with multivariate weights, which also appear to have interesting behavior in the limit. Joint work with A. Morales and I. Pak

Birational rowmotionis an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a posetP, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers, and birational rowmotion toY-systems of type $A_m \times A_n$ described in Zamolodchikov periodicity. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengthsrandsisr+s+2(first proved by D.~Grinberg and the second author), as well as a proof of the birational analogue ofhomomesy along filesfor such posets.

The Littlewood-Richardson rule is one of the most important properties in the representation theory of the symmetric group, which can describe the products of Schur functions using Young diagrams in a combinatorial way. The Littlewood-Richardson rule is also applied to the expansion of immanants, generalizations of the determinant and the permanent, by principal minors. In this talk, we discuss limiting behavior of immanants and some results obtained by applying this rule.

We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{\mu\le\lambda} g^{(k)}_{\mu}$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by $\widetilde{g}^{(k)}_{\lambda}$. As an application of this, we also give a $k$-rectangle factorization formula $\widetilde{g}^{(k)}_{R_t\cup\lambda}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda}$ where $R_t=(t^{k+1-t})$, analogous to that of $k$-Schur functions $s^{(k)}_{R_t\cup\lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}$.

We interpret a bijection, constructed by O. Azenhas, between the Littlewood--Richardson (LR) tableaux of shape $\lambda/\mu$ and weight $\nu$ on one hand and those of shape $\lambda/\nu$ and weight $\mu$ on the other, as a correspondence between the irreducible components of algebraic varieties $\mathcal{G}_{\mu\nu}^\lambda$ and $\mathcal{G}_{\nu\mu}^\lambda$, which are actually isomorphic, but the irreducible components of $\mathcal{G}_{\mu\nu}^\lambda$ are naturally labelled by the LR tableaux of shape $\lambda/\mu$ and weight $\nu$ while those of $\mathcal{G}_{\nu\mu}^\lambda$ by those of shape $\lambda/\nu$ and weight $\mu$. These varieties are closely related with the Hall polynomials indexed by the same triples of partitions.

To be more explicit, Azenhas' bijection consists of iterating a certain deletion operation starting from a given LR tableau $T$ of shape $\lambda/\mu$ and $\nu$ to produce a sequence of LR tableaux $T=T^{(n)},T^{(n-1)},\dots,T^{(1)},T^{(0)}=\varnothing$ in which each $T^{(r)}$ has outer shape with exactly $r$ rows, and while doing so recording how the inner shape shrinks in the form of another LR tableau with $\mu$ and $\nu$ swiched. Our interpretation is obtained by revealing what is represented by these intermediate LR tableaux, in particular $T^{(n-1)}$ obtained in the first step.

In 1980 Mikio Sato published a hand-written note, which contains some tables on the KdV equations. I have been trying to read out information from these tables for a long time. And recently I finally found what Sato wanted to tell us on the KdV equations. I will report the meaning of Sato's tables, and view the KdV equations as the orthogonality relations of the symmetric group characters.

The conference will be held in Graduate School of Natural Science and Technology Building No.1, 2nd Floor. It is N24 building in Campus map. The access map to Okayama is here and Tsushima campus map is here. This photo is the entrance door of N24 building.

We will organize Conference Lunch on February 20 (Tue).

We will also organize the the following banquet:

Date: | February 21 (Wed) 18:30 - 20:30 |

Place: | Recent Culture Hotel, 4F, Room Venetia |

Dinner: | Japanese cuisine |

Price: | 5,500JPY (including beer, Japanese sake and nonalcoholic drinks) |

The conference take place at the 2nd floor of Graduate School of Natural Science and Technology, Building No.1.

Take Bus 47 from Okayama Station West Exit to Okayama University West Gate. |
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Walk to N24 building in Campus map. |
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You can walk or Take Bus 47. |
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If you get out of Shinkansen exit, turn right and go through the central hall of the station. After you pass the taxi pool, you go right and take the escalator or stairs to go down to the ground. Find the Bus Stop No.22 and take the Okaden Bus No.47 whose final destination is Okayama University of Science. The time table is here. Get off the bus at the bus stop named "Okadai Nishi Mon". Go into the West Gate of Okayama University. You will see the University Library ahead and Faculty of Science Building on the right. You turn right before the library and walk straight. Finally you will find Graduate School of Natural Science and Technology, Building No.1. Get into the entrance door and go up to the 2nd floor by stairs. The conference will be held in the large lecture room.

Recent Culture Hotel (The closest hotel to Okayama University)

Okayama Royal Hotel

Via Inn Okayama

Hotel Granvia Okayama

Tsushima College Residence (Please contact us if you would like to stay the residence).

Okaden Bus (Take Bus 47 to come to Tsushima Campus.)

Ryobi Bus

Tramway

Algebraic Combinatorics related to Young diagrams and Statistical Physics (2012)

Masao Ishikawa's Homepage

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