Nonnegative Integral Solutions to Linear Equations

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


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


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Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Richard P. Stanley
    • 1
  1. 1.Mathematics Department, 2-375Massachusetts Institute of TechnologyCambridgeUSA

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