We study F (n; m), the number of compositions of n in which repetition of parts is allowed, but exactly m distinct parts are used. We obtain explicit formulas, recurrence relations, and generating functions for F (n; m) and for auxiliary functions related to F . We also consider the analogous

A partition of the positive integer n into distinct parts is a decreasing sequence of positive integers whose sum is n, and the number of such partitions is denoted by Q(n). If we adopt the convention that Q(0) = 1, then we have the generating function

Abstract. We study the generating function for Q(n), the number of partitions of a natural number n into distinct parts. Using the arithmetic properties of Fourier coefficients of integer weight modular forms, we prove several theorems on the divisibility and distribution of Q(n) modulo primes p

We study ^ F (n; m) [ ^ G(n; m)], the number of compositions [partitions] of n in which repetition of parts is allowed, but the sum of distinct parts is m. We obtain generating functions for xed m as well as formulas and asymptotic estimates for the mean values of ^ F and ^ G. Partitions

The automatic discovery of distinctive parts for an ob-ject or scene class is challenging since it requires simulta-neously to learn the part appearance and also to identify the part occurrences in images. In this paper, we propose a simple, efficient, and effective method to do so. We ad

A program slice consists of the parts of a program that (potentially) affect the values computed at some point of interest, referred to as a slicing criterion. The task of computing program slices is called program slicing. The original definition of a program slice was presented by Weiser

. Bousquet-Melou and Eriksson showed that the number of partitions of n into distinct parts whose alternating sum is k is equal to the number of partitions of n into k odd parts, which is a refinement of a well-known result by Euler. We give a di#erent graphical interpretation of the bijection

We establish an explicit formula for the maximum value of the product of parts for partitions of a positive integer into distinct parts (sequence A034893 in the On-Line Encyclopedia of Integer Sequences).

We consider the problem of using a large unlabeled sample to boost performance of a learning algorithm when only a small set of labeled examples is available. In particular, we consider a setting in which the description of each example can be partitioned into two distinct views, motivated

We establish an explicit formula for the maximum value of the product of parts for partitions of a positive integer into distinct parts (sequence A034893 in the On-Line Encyclopedia of Integer Sequences). 1