# Nonnegative Integral Solutions to Linear Equations

• Richard P. Stanley
Chapter
Part of the Progress in Mathematics book series (PM)

## Abstract

The first topic will concern the problem of solving linear equations in nonnegative integers. In particular, we will consider the following problem which goes back to MacMahon. Let
$${H_n}\left( r \right)\,\,: = \,number\,of\,n \times n\,\mathbb{N} - matrices\,having\,line\,sums\,r\,,$$
where a line is a row or column, and an ℕ-matrix is a matrix whose entries belong to ℕ. Such a matrix is called an integer stochastic matrix or magic square. Keeping r fixed, one finds that Hn(0) = 1, Hn(1) = n!, and Anand, Dumir and Gupta [A-D-G] showed that
$$\sum\limits_{n \geqslant 0} {\frac{{{H_n}\left( 2 \right){x^n}}} {{{{(n!)}^2}}} = \frac{{{e^{x/2}}}} {{\sqrt {1 - x} }}}$$
. See also Stanley [St5, Ex. 6.11]. Keeping n fixed, one finds that H1(r) = 1, H2(r) = r+1, and MacMahon [MM, Sect. 407] showed that
$${H_3}\left( r \right) = \left( {\mathop 4\limits^{r + 4} } \right) + \left( {\mathop 4\limits^{r + 3} } \right) + \left( {\mathop 4\limits^{r + 2} } \right).$$
.