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Séminaire Lotharingien de Combinatoire, B44h (2000), 7 pp.

# Peter Kirschenhofer and Oliver Pfeiffer

#
On a Class of Combinatorial Diophantine Equations

**Abstract.**
We give a combinatorial proof for a second order recurrence for the polynomials
*p*_{n}(*x*), where *p*_{n}(*k*) counts
the number of integer-coordinate lattice points
**x** = (*x*_{1},...,x_{n}) with
||**x**|| = *\sum_{i=1}^n* |*x*_{i}| <= *k*.
This is the main step to get finiteness results on the number of solutions of
the diophantine equation
*p*_{n}(*x*) = *p*_{m}(*y*)
if *n* and *m* have different parity. The combinatorial approach also allows
to extend the original diophantine result to
more general combinatorial situations.

Received: June 28, 2000; Accepted: December 14, 2000.

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