Counting integer solutions for a system of inequalitiesI wish to enumerate the number of solutions of the system of equations and inequalities for 3 non-negative integer unknowns x , y , z ≥ 0: ( a, b specified) x + y + z = a x + y &gt; b y + z &gt; b Is there an elegant way of finding the number of solutions or must I use an exhaustive numerical algorithm?

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Counting integer solutions for a system of inequalitiesI wish to enumerate the number of solutions of the system of equations and inequalities for 3 non-negative integer unknowns   x  ,  y  ,  z  ≥  0: (  a,  b specified)                    x        +        y        +        z                            =        a                            x        +        y                            &amp;gt;        b                            y        +        z                            &amp;gt;        b            Is there an elegant way of finding the number of solutions or must I use an exhaustive numerical algorithm?

Tianna Deleon · Accepted Answer

We may suppose that   b  ∈  {  −  1  ,  0  ,  1  ,  …  ,  a  −  1  }  ..We consider the plane   (  x  ,  z  ), your bounds are   x  +  z  ⩽  a,  0  ⩽  x  ,  z  &amp;lt;  a  −  b. Now fix   x, then   z varies from   0 to   min  (  a  −  x  ,  a  −  b  −  1  ). So, for   x  ⩽  b there are   a  −  b ways to choose   z, and for   a  ⩾  x  ⩾  b  +  1 there exist   a  −  x  +  1 ways to choose   z. Totally we get   (  b  +  1  )  (  a  −  b  )  +  (  a  −  b  )  (  a  −  b  +  1  )  /  2  =  (  a  −  b  )  (  a  +  b  +  3  )  /  2 solutions.

Counting integer solutions for a system of inequalities I wish to enumerate the number of solutions

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