Communal Partitions of Integers

There is a well-known formula due to Andrews that counts the number of incongruent triangles with integer sides and a fixed perimeter. In this note, we consider the analagous question counting the number of k-tuples of nonnegative integers none of which is more than 1/(k−1) of the sum of all the integers. We give an explicit function for the generating function which counts these k-tuples in the case where they are ordered, unordered, or partially ordered. Finally, we discuss the application to algebraic geometry which motivated this question. Required Publisher's Statement The original version is available from the publisher at: http://www.westga.edu/~integers/cgi-bin/get.cgi This article is available at The Cupola: Scholarship at Gettysburg College: http://cupola.gettysburg.edu/mathfac/3 #A70 INTEGERS 11 (2011) COMMUNAL PARTITIONS OF INTEGERS Darren B Glass Department of Mathematics, Gettysburg College, Gettysburg, Pennsylvania [email protected] Received: 6/28/11, Revised: 8/26/11, Accepted: 11/22/11, Published: 12/7/11 Abstract There is a well-known formula due to Andrews that counts the number of incongruent triangles with integer sides and a fixed perimeter. In this note, we consider the analagous question counting the number of k-tuples of nonnegative integers none of which is more than 1 k−1 of the sum of all the integers. We give an explicit function for the generating function which counts these k-tuples in the case where they are ordered, unordered, or partially ordered. Finally, we discuss the application to algebraic geometry which motivated this question.There is a well-known formula due to Andrews that counts the number of incongruent triangles with integer sides and a fixed perimeter. In this note, we consider the analagous question counting the number of k-tuples of nonnegative integers none of which is more than 1 k−1 of the sum of all the integers. We give an explicit function for the generating function which counts these k-tuples in the case where they are ordered, unordered, or partially ordered. Finally, we discuss the application to algebraic geometry which motivated this question.