40 research outputs found
Five Guidelines for Partition Analysis with Applications to Lecture Hall-type Theorems
Five simple guidelines are proposed to compute the generating function for
the nonnegative integer solutions of a system of linear inequalities. In
contrast to other approaches, the emphasis is on deriving recurrences. We show
how to use the guidelines strategically to solve some nontrivial enumeration
problems in the theory of partitions and compositions. This includes a
strikingly different approach to lecture hall-type theorems, with new
-series identities arising in the process. For completeness, we prove that
the guidelines suffice to find the generating function for any system of
homogeneous linear inequalities with integer coefficients. The guidelines can
be viewed as a simplification of MacMahon's partition analysis with ideas from
matrix techiniques, Elliott reduction, and ``adding a slice''
The Fractal and The Recurrence Equations Concerning The Integer Partitions
This paper introduced a way of fractal to solve the problem of taking count
of the integer partitions, furthermore, using the method in this paper some
recurrence equations concerning the integer partitions can be deduced,
including the pentagonal number theorem
Lecture Hall Theorems, q-series and Truncated Objects
We show here that the refined theorems for both lecture hall partitions and
anti-lecture hall compositions can be obtained as straightforward consequences
of two q-Chu Vandermonde identities, once an appropriate recurrence is derived.
We use this approach to get new lecture hall-type theorems for truncated
objects. We compute their generating function and give two different
multivariate refinements of these new results : the q-calculus approach gives
(u,v,q)-refinements, while a completely different approach gives odd/even
(x,y)-refinements. From this, we are able to give a combinatorial
characterization of truncated lecture hall partitions and new finitizations of
refinements of Euler's theorem
Composition of Integers with Bounded Parts
In this note, we consider ordered partitions of integers such that each entry is no more than a fixed portion of the sum. We give a method for constructing all such compositions as well as both an explicit formula and a generating function describing the number of k-tuples whose entries are bounded in this way and sum to a fixed value g
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