arXiv subjects

William M. Y. Goh

Publications and source records attributed to William M. Y. Goh.

5 recordsLinked to original sources

arXiv2007

Partition Polynomials: Asymptotics and Zeros

Let $F_n(x)$ be the partition polynomial $\sum_{k=1}^n p_k(n) x^k$ where $p_k(n)$ is the number of partitions of $n$ with $k$ parts. We emphasize the computational experiments using degrees up to $70,000$ to discover the asymptotics of these polynomials. Surprisingly, the asymptotics of $F_n(x)$ have two scales of orders $n$ and $\sqrt{n}$ and in three different regimes inside the unit disk. Consequently, the zeros converge to network of curves inside the unit disk given in terms of the dilogarithm.

math.CO↗

arXiv2007

Polynomials associated with Partitions: Polynomials associated with Partitions: Their Asymptotics and Zeros

Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting behavior of their zeros as sets and densities.

math.CO↗

arXiv2004

On the Zero Attractor of the Euler Polynomials

We study the limiting behavior of the zeros of the Euler polynomials. When linearly scaled, they approach a definite curve in the complex plane related to the Szego curve which governs the behavior of the roots of the Taylor polynomials associated to the exponential function. Further, under a conformal transformation, the scaled zeros are uniformly distributed.

math.CO↗

arXiv2001

Average number of distinct part sizes in a random Carlitz composition

A composition of an integer is called Carlitz if adjacent parts are different. Several characteristics of random Carlitz compsitions have been studied recently by Knopfmacher and Prodinger. We complement their work by establishing the asymptotics of the average number of distinct part sizes in a random Carlitz composition.

math.CO↗

arXiv2001

S-partitions

This note reports on the number of s-partitions of a natural number n. In an s-partition each cell has the form $2^k-1$ for some integer k. Such partitions have potential applications in cryptography, specifically in distributed computations of the form $a^n$ mod m. The main contribution of this paper is a correction to the upper bound on the number of s-partitions presented by Bhatt. We will give a precise asymptotics for the number of such partitions for a given integer n.

math.CO↗