9시 40분 - 10시 00분 : 입장 및 토의

10시 - 11시 : 류미수

11시 - 12시 : 허지선

12시 - 13시 30분 : 점심

13시 30분 - 14시 30분 : 남하얀

14시 30분 -15시 30분 : 조현수

15시 30분 - 15시 40분 : 휴식

15시 40분 - 16시 40분 : 강순이

강순이
Title : Alder-Andrews type partition inequality
Abstract : In 1956, Alder conjectured that $q_d(n)-Q_d(n)\geqq 0$, where $q_d(n)$ and $Q_d(n)$ are the number of partitions of $n$ into parts differing by at least $d$ and the number of partitions of $n$ into parts which are congruent to $\pm$1 (mod $d$+3), respectively. It took more than 50 years to complete the proof and the first breakthrough was made by Andrews in 1971, who proved that the conjecture holds for $d= 2^r-1$ ($r\geqq 4$). In this talk, we prove two analogous partition inequalities following Andrew's method. One of them generalizes the second Rogers-Ramanujan identity, which is the number of partitions of $n$ into parts differing by at least $d$ with the smallest part at least $2$ is greater than or equal to that of partitions of $n$ into parts congruent to $\equiv \pm 2 \pmod{d+3}$ excluding $d+1$ when $d=2^{r}-2$ ($r\geqq 2$, $r\neq 4$) . We also show the asymptotic behavior of the difference of the two partition numbers.

남하얀
Title : Recent studies of core partitions
Abstract : In this talk, we study the concept of a simultaneous core partition and talk about a history of studies of core partitions. This talk consists of three parts: the number of simultaneous core partitions, the size of core partitions, and the inequalities related to core partitions. At the end of each section, the speaker will present her recent work.

류미수
Title : Enumeration of standard barely set-valued tableaux of shifted shapes
Abstract : In this work, we prove the CDE property of the trapezoidal shifted shapes by counting standard barely set-valued tableaux via q-integral method. A standard barely set-valued tableau of shape $\lambda$ is a filling of the Young diagram $\lambda$ with integers $1,2,\dots,|\lambda|+1$ such that the integers are increasing in each row and column, and every cell contains one integer except one cell that contains two integers. Counting standard barely set-valued tableaux is closely related to proving Young's lattice has the coincidental down-degree expectations (CDE) property. Using $q$-integral techniques we give a formula for the number of standard barely set-valued tableaux of arbitrary shifted shapes. We then prove a conjecture of Reiner, Tenner and Yong on the CDE property of trapezoidal shifted shape $(n,n-2,\dots,n-2k)$. This is joint work with Jang Soo Kim and Michael Schlosser.

조현수
Title : Path interpretation of self-conjugate (s,s+d,...,s+pd)-core partition
Abstract : A partition is called an s-core if it has no box of hook length s. In 2002, Anderson found the number of (s,t)-core partitions for relatively prime positive integers s and t. Motivated by Anderson, there has been considerable interest in simultaneous core partitions recently. Especially, researchers are concerned with simultaneous core partitions whose cores line up with an arithmetic progression. In this talk, we give a lattice path interpretation of self-conjugate (s,s+d,...,s+pd)-core partitions for relatively prime positive integers s and d. We also enumerate self-conjugate (s,s+d,s+2d)-core partitions and self-conjugate (s,s+d,s+2d,s+3d)-core partitions. This is a joint work with JiSun Huh.

허지선
Title : On the number of equivalence classes arising from partition involutions
Abstract : Involutions have played important roles in many research areas including the theory of partitions. In this talk, we give relations between the number of equivalence classes in the set of partitions arising from an involution and the number of partitions satisfying a certain parity condition. We examine the number of equivalence classes arising from the conjugations on ordinary partitions, overpartitions, and partitions with distinct odd parts. We also consider other types of involutions on partitions into distinct parts, partitions with restricted crank, and Frobenius symbols. This is joint work with Byungchan Kim.