The $(s, s+d, \dots, s+pd)$-core partitions and rational Motzkin paths
For a positive integer $t$, a partition $\lambda$ is a $t$-core partition if it has no box of hook length $t$. The study of simultaneous core partitions began with the work of Anderson. In this talk, we first propose an $(s+d, d)$-abacus for $(s,s+d,\dots,s+pd)$-core partitions, and then establish a bijection between the $(s,s+d, \dots, s+pd)$-core partitions and rational Motzkin paths of type $(s+d, -d)$. This bijection not only gives a lattice path interpretation of the $(s,s+d, \dots, s+pd)$-core partitions but also counts them with a closed formula. Also we enumerate $(s, s+1, \dots, s+p)$-core partitions with $k$ corners and self-conjugate $(s,s+1,\dots, s+p)$-core partitions.
On the partitions into squares whose reciprocal sum to one
We show that for all positive integers $n >1223$, there is a partition of $n$ into squares whose reciprocals sum to 1. This is a joint work with J.-Y. Kim, C.-G. Lee and P.-S. Park.
Parity bias in partitions
In this talk, we will examine parity bias in integer partitions and discuss how $q$-series transformations, combinatorial models, and asymptotic analysis work together to study their arithmetic. This talk is based on a joint work with Byungchan Kim and Jeremy Lovejoy.
Identities for partitions and overpartitions with even smallest parts
In this talk I will discuss identities for partitions and overpartitions with even smallest parts. These resemble classical partition identities except that a minimum number of the smallest parts of the partition must be even, this number being determined by the number of parts in certain congruence classes. The proofs involve revisiting Alladi and Gordon's treatment of Schur's partition theorem and using partial staircases or partial generalized staircases in place of classical staircases. The overpartition case is joint work with Min-Joo Jang.
Singular overpartitions and partitions with prescribed hook differences
Singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row, were first introduced by Andrews in 2015. In his paper, Andrews investigated an interesting subclass of singular overpartitions, namely $(K,i)$-singular overpartitions for integers $K, i$ with $ 1\le i<K/2$. The definition of such singular overpartitions requires successive ranks, parity blocks, and anchors. The concept of successive ranks was extensively generalized to hook differences by Andrews, Baxter, Bressoud, Burge, Forrester, and Viennot in 1987. In this talk, employing hook differences, we generalize parity blocks.
Using this combinatorial concept, we define $(K,i,\alpha, \beta)$-singular overpartitions for positive integers $\alpha, \beta$ with $\alpha+\beta<K$, and then we show some connections between such singular overpartitions and ordinary partitions.