Physical Sciences and Mathematics Commons^™

The Period Of The Coefficients Of The Gaussian Polynomial[N+33], Arturo J. Martinez

Theses and Dissertations

Definition 1. For any N, the central coefficient(s) of [N+33] is denoted by C0(N) and the coefficient that is x ''away" from the central coefficient(s) of [N+33] is denoted by Cx(N).

In [1] the following result is proved:

Theorem 2. The central coefficient(s) of the Gaussian polynomial [N+33] are described by the generating function

[Special characters omitted]

This generating function has period 4.

The main goal of this thesis is to generalize Theorem 0.2 by way of proving the following conjecture:

Conjecture 3. For any x the …

Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, Walter Mcfarland Bridges

LSU Doctoral Dissertations

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit …