Smarandache sums of products

This paper deals with the sums of products of first n natural numbers, taken r at a time. Many interesting results about the summations are obtained. Ramasubramanian has already made some work in this direction. This paper is an extension of his work. In next part, the sums of odd and even natural numbers are discussed, and also of natural numbers, not necessarily beginning with one. After that, properties of sequences, arising out of these sums are obtained. Interestingly, the numbers thus obtained are Stirlings numbers

Telescoping Sums, Permutations, and First Occurrence Distributions

Telescoping sums very naturally lead to probability distributions on ${\mathbb Z}^+$. But are these distributions typically cosmetic and devoid of motivation? In this paper we give three examples of "first occurrence" distributions, each defined by telescoping sums, and that each arise from concrete questions about the structure of permutations.Comment: 13 page

Waiting Time Distribution for the Emergence of Superpatterns

Consider a sequence X_1, X_2,... of i.i.d. uniform random variables taking values in the alphabet set {1,2,...,d}. A k-superpattern is a realization of X_1,...,X_t that contains, as an embedded subsequence, each of the non-order-isomorphic subpatterns of length k. We focus on the non-trivial case of d=k=3 and study the waiting time distribution of tau=inf{t>=7: X_1,...,X_t is a superpattern}Comment: 17 page