The primitive partition identities up to a largest part of 27 have been computed using a specialized algorithm that is much faster than a general-purpose Buchberger procedure. See Haus, Köppe, Weismantel: A Primal All-Integer Algorithm Based on Irreducible Solutions, Math. Programming, Series B, 96 (2003), no. 2, pp. 205-246.
The computer program that we used is available as a single C++ source file, hilbert.cc. It is free software and distributed under the GNU General Public License.
It was written in 1998/1999, and updated in 2002, so as to work with
current C++ compilers.
It is now available as part of 4ti2
We provide pre-computed tables of primitive partition identities in the following binary format.
The first byte of the file is the order n of the table. The rest of the file consists of the vectors representing primitive partition identities. The order of the vectors is arbitrary. Because for each vector, its negative is also a solution, only the vector whose greatest nonzero component is positive is stored. Each vector is stored as n signed bytes (negative numbers being encoded in 2's complement).
For each of the provided tables, we show the number of vectors and the computation time on a Sun Fire 480 R (UltraSparc III+, 1050 MHz).