This is a simple set of Python 2.7 functions designed for use in the SageMath prompt or SageMath programs. These functions were designed to aid in testing theories for the research paper:

        Sequences in Dihedral Groups with Distinct Partial Products
               
              M. A. Ollis, Marlboro College

partsift.py requires elements of the following python and sage packages to work:

• itertools
• sympy
• sage.calculus
• sage.combinat

This funcion builds a polynomial from inputs r (even number or zero) and s (odd number) according to the form described in this PDF.

The function returns either only the polynomial as a Sage symbolic expression (when degree = "false", the default setting), or a list in the following format [polynomial (as a symbolic expression), degree of the polynomial (as an integer)].

This function takes a polynomial built from the form outlined by the previous function, and finds all monomials present in the expanded form of the polynomial that fit the following criteria:

• The monomial, when created from the built polynomial, has a non-zero coefficient.
• The degree of the monomial is the same as the degree of the polynomial.
• The can divide every possible x term with a degree of r and every possible y term with a degree of s. In other words, the highest exponent on any x variable in the monomial is r-1 and the highest exponent on any y variable in the monomial is s-1.

The function has a variety of options for returns, chosen by the form and sort_type variables. By default, it returns a list of monomials ordered by the distance of their coefficient from zero. Setting form = "tup" or form = "tuple" will change the output to a list of tuples containing (monomial*coefficient, coefficient) and setting sort_type = "value" will order the list by the values of coefficients.

This function executes the main purpose of this module. Quite simply, it combines the previous two functions. Input is r and s as with build_polynomial, and the function returns a list of monomials in the same fashion as find_monomials. sift also takes form and sort_type arguments in the same manner as find_monomials.

This module can be installed into your copy of Sage simply by copying partsift.py into Sage's version of Python. The location is as follows: */sagemath/local/lib/python2.7 with * being the path to where Sage is installed on your computer. Once the file is there, you can use these functions in your Sage programming at any time with import partsift.