MacMahon's partition analysis. VII: Constrained compositions (Q2782403)

[For other parts see Zbl 0924.11087, Zbl 0977.05004, Zbl 0979.05008, Zbl 0921.05006, Zbl 0972.05004, Zbl 1027.05005, Zbl 0992.05017, and Zbl 0995.05007.]NEWLINENEWLINENEWLINEThe authors apply P. A. MacMahon's partition analysis to a set of problems extending the generating function for the homogeneous symmetric functions but with certain constraints. One such constraint would be to move from the case of compositions (unordered sums) that derives from the simplest case described above to that for partitions, that is, where the summands are non-increasing. Another such problem is to generate triples of integers \(a_1\), \(a_2\), \(a_3\) so that they form the sides of a non-degenerate triangle listed in non-increasing order, that is, where \(a_2+ a_3> a_1\). All examples given are those where the number of MacMahon's Diophantine inequalities is the same as the number of integer variables. One problem, a two-variable system of Diophantine inequalities, appeared as Problem B3 of the Putnam Mathematical Competition in 2000. This is then generalized. The software Mathematica is used in some of the calculations.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00024].