Distribution Properties for t-Hooks in Partitions

Hooks and Powers of Parts in Partitions

Many textbooks contain material on partitions. Two standard references are [A] and [S]. A partition of a natural integer n with parts λ1, . . . , λk is a finite decreasing sequence λ = (λ1 ≥ λ2 ≥ · · · ≥ λk > 0) of natural integers λ1, . . . , λk > 0 such that n = ∑k i=1 λi. We denote by |λ| the content n of λ. Partitions are also written as sums: n = λ1 + · · ·+ λk and one uses also the (abusi...

Generalizations of the Skew t-Normal Distribution and their Properties

In this paper we consider several generalizations of the skew t-normal distribution, and some of their properties. Also, we represent several theorems for constructing each generalized skew t-normal distribution. Next, we illustrate the application of the proposed distribution studying the ratio of two heavy metals, Nickel and Vanadium, associated with crude oil in Shadgan wetland in the south-...

T-partitions and S-complete T-partitions of a Graph

Hooks and Shifts 1 Running head: HOOKS AND SHIFTS Hooks and Shifts in Instrumented Mathematics Learning

This paper presents and analyzes vignettes selected from empirical data gathered in a designbased research study investigating the microgenesis of proportional reasoning through technology-mediated embodied interaction. 22 Grades 4-6 students participated in individual or paired semi-structured clinical interviews in which they engaged in problem-solving activities involving remote manipulation...

On the Maximum Number of k-Hooks of Partitions of n

Let αk(λ) denote the number of k-hooks in a partition λ and let b(n, k) be the maximum value of αk(λ) among partitions of n. Amdeberhan posed a conjecture on the generating function of b(n, 1). We give a proof of this conjecture. In general, we obtain a formula that can be used to determine b(n, k). This leads to a generating function formula for b(n, k). We introduce the notion of nearly k-tri...