We describe computations related to the problem considered in the paper “On the Asymptotic Behavior of some Counting Functions” (Maciej Radziejewski and Wolfgang A. Schmid, in preparation). We assume the reader is familiar with the notions of half-factoriality, sequences and block monoids, atoms, zero-sumfree sequences, and the cross number. Some background information can be found in our paper and in many other papers.
Let G be a finite abelian group with at least three elements. Let
In our paper and in an earlier paper by the first author it was conjectured (in
other terms) that every finite abelian group with at least three elements
The study of this problem was motivated by the investigations of the structure
of elements with factorizations of at most k distinct lengths in
algebraic number fields and in Krull monoids. A constant that naturally arises
in the description of such elements was called
The calculations were performed in the following order (the programs performing the different tasks are named on the left):
|parameters_C0||For a given n we find all the groups of order 3 ≤ |G| ≤ n, other than cyclic p-groups and cyclic groups of order pq (where we know the conjecture holds and we know the half-factorial subsets). For each group we compute the list of inclusion-maximal subsets satisfying the C0 condition of J. Sliwa — all up to group automorphy. These serve as parameters for further steps.|
|We compute all the inclusion-maximal half-factorial subsets contained in the C0-subsets computed before. That gives us the list of all inclusion-maximal half-factorial subsets (for each group) up to group automorphy (although there still may be multiple half-factorial subsets equivalent by group automorphy).|
|select_max_card||For each group we pick one half-factorial subset of maximum cardinality. It seems that the condition given in the conjecture holds for all half-factorial sets in each group (at least for |G| ≤ 50), while the conjecture only requires it “for at least one half-factorial subset of maximum cardinality”. If the condition is satisfied for the subset we picked, the conjecture is verified for the given group, otherwise we'll have to pick another subset.|
|psi1||For each group G and the half-factorial subset G0 selected for this group we find the set G0' of elements g in G such that all atoms and zero-sumfree sequences made of elements of G0 with sum -g have the same cross number. Then we output G0' and G0'' = G0' \ G0. If G0'' is nonempty, the conjecture is verified for the given group.|
Each program name above provides a link to an executable program file that you are welcome to download and try out (for use under MS Windows 95® or later, tested only under MS Windows XP®). Please note that these programs are command line tools with very little explanation. They can be run using the batch file calc_psi1.bat and the parameter file n.txt (which contains just one number: the maximum group order).
You may also like to browse the C++ source code if you are into programming and would like to study the method in detail.
The results and partial results are given below:
|C0.txt||Groups and C0 subsets. The neutral element is ommitted from the C0 subsets in order to make the subsequent calculations slightly faster — it is treated in a special way.|
Groups and half-factorial subsets. Again, the neutral element is always
Here is an uncompressed version
|Groups and half-factorial subsets of maximum cardinality (one subset for each group).|
|psi1.txt||Final results: for each group G the subsets G0, G0', and G0'' are given.|
The format of the results requires some explanation. Groups are specified as
vectors of dimensions in the cyclic p-group decomposition. Subsets are
given as vectors of element numbers — the elements within a group of order n
are numbered from 0 to
A short “revision history”:
The complete package (all the files referred to on this page as well as the page itself) can be found here (4.5 MB).
© 2004 Maciej Radziejewski. The programs and scripts available from this page come WITHOUT ANY WARRANTY WHATSOEVER. You can use them at your own risk. If you copy any of the programs or scripts, please keep them together with this web page file.
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