Myungjun Yu

Finding the largest size of a partition under certain restrictions has been an interesting subject to study. For example, it is proved by Olsson and Stanton that for two coprime integers [Formula: see text] and [Formula: see text], the largest size of an [Formula: see text]-core partition is [Formula: see text]. Xiong found a formula for the largest size of a [Formula: see text]-core partitions with distinct parts. In this paper, we find an explicit formula for the largest size of an [Formula: s

Given a prime $p$, let $P(t)$ be a non-constant monic polynomial in $t$ over the ring $\mathbb{Z}_{p}$ of $p$-adic integers. Let $X_{n}$ be an $n \times n$ random matrix over $\mathbb{Z}_{p}$ with independent entries that lie in any residue class modulo $p$ with probability at most $1 - ε$ for a fixed real number $0 < ε< 1$. We prove that as $n \rightarrow \infty$, the distribution of the cokernel $\mathrm{cok}(P(X_{n}))$ of $P(X_{n})$ converges to the distribution given by a finite produc

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text]. Suppose that [Formula: see text] has complex multiplication over [Formula: see text], i.e. [Formula: see text] is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number.

In this paper, we study the distribution of the cokernels of random $p$-adic matrices with fixed zero entries. Let $X_n$ be a random $n \times n$ matrix over $\mathbb{Z}_p$ in which some entries are fixed to be zero and the other entries are i.i.d. copies of a random variable $ξ\in \mathbb{Z}_p$. We consider the minimal number of random entries of $X_n$ required for the cokernel of $X_n$ to converge to the Cohen--Lenstra distribution. When $ξ$ is given by the Haar measure, we prove a lower bound

Given a positive integer $r$ and a prime power $q$, we estimate the probability that the characteristic polynomial $f_{A}(t)$ of a random matrix $A$ in $\mathrm{GL}_{n}(\mathbb{F}_{q})$ is square-free with $r$ (monic) irreducible factors when $n$ is large. We also estimate the analogous probability that $f_{A}(t)$ has $r$ irreducible factors counting with multiplicity. In either case, the main term $(\log n)^{r-1}((r-1)!n)^{-1}$ and the error term $O((\log n)^{r-2}n^{-1})$, whose implied constan

We extend an observation due to Stong that the distribution of the number of degree $d$ irreducible factors of the characteristic polynomial of a random $n \times n$ matrix over a finite field $\mathbb{F}_{q}$ converges to the distribution of the number of length $d$ cycles of a random permutation in $S_{n}$, as $q \rightarrow \infty$, by having any finitely many choices of $d$, say $d_{1}, \dots, d_{r}$. This generalized convergence will be used for the following two applications: the distribut

We study the distribution of the cokernels of random row-sparse integral matrices $A_n$ according to the determinantal measure from a structured matrix $B_n$ with a parameter $k_n \ge 3$. Under a mild assumption on the growth rate of $k_n$, we prove that the distribution of the $p$-Sylow subgroup of the cokernel of $A_n$ converges to that of Cohen--Lenstra for every prime $p$. Our result extends the work of A. Mészáros which established convergence to the Cohen--Lenstra distribution when $p \ge

We introduce the multi-Gaussian polynomial [Formula: see text], a multi-partition analogue of the Gaussian polynomial (also known as [Formula: see text]-binomial coefficient), as the generating function for certain restricted multi-color partitions. We study basic properties of multi-Gaussian polynomials and non-symmetric properties of [Formula: see text]. We also derive a Sylvester-type identity and its application.