Orbits of Pairs in Abelian Groups

This page contains the data files and sage programs which are connected with the paper Orbits of Pairs in Abelian Groups by C. P. Anilkumar and Amritanshu Prasad.

Sage code

The program for generating the $n_\lambda(q)$'s is pairs.sage This function provides the function pairs, whose input is a partition, and output is a polynomial in `q`. For example:

      
	sage: load "http://www.imsc.res.in/~amri/pairs.sage"
	sage: pairs(Partition([2,1]))
	q^2 + 5*q + 5

This module provides another useful function. The function TruncatePartition reduces the multiplicity of each part of a partition that exceeds two to two (its use is illustrated later):

      
	sage: TruncatePartition(Partition([3, 3, 3, 2, 2, 1]))
	[3, 3, 2, 2, 1]

The polynomials $n_\lambda(q)$:

For each partition $\lambda$, $n_\lambda(q)$ is the number of $G_\lambda$-orbits in $M_\lambda\times M_\lambda$. A list of the polynomials $n_\lambda(q)$ for partitions $\lambda$ where no part is repeated more than two times is available as a text file and as a sage dictionary. By Corollary 24 of the paper, if $\lambda$ has a part appearing with multiplicity greater than two, its multiplicity can be reduced to two without changing $n_\lambda(q)$.

      
	sage: D = load('http://www.imsc.res.in/~amri/pairs/nlambda.sobj')
	Attempting to load remote file:
	http://www.imsc.res.in/~amri/pairs/nlambda.sobj
	Loading: [........]
	sage: D[Partition([2,1])]
	q^2 + 5*q + 5

Partitions where parts repeat more than two times are not keys. Attempting

      
	sage: D[Partition([2,2,2])]

will give rise to a KeyError. Instead, one should load the file pairs.sage and use the function TruncatePartition

      
	sage: load "http://www.imsc.res.in/~amri/pairs.sage"
	sage: D[TruncatePartition(Partition([2,2,2]))] 
	q^2 + 3*q + 5

Checking with the Kac-Stanley formula

Checking of our results via the Kac-Stanley formula was achieved using checks.sage. The following example verifies the equation (46) of the paper for $n=3$:

      
	sage: load "http://www.imsc.res.in/~amri/pairs/checks.sage"
	sage: check(3)
	True