Nonnegative Integral Solutions to Linear Equations

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 H_n(0) = 1, H_n(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

$${H_3}\left( r \right) = \left( {\mathop 4\limits^{r + 4} } \right) + \left( {\mathop 4\limits^{r + 3} } \right) + \left( {\mathop 4\limits^{r + 2} } \right).$$

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