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Let L be a sparse context-free language over an alphabet of t letters and let f L : N t → N be its Parikh counting function. We prove the following two results: 1. There exists a partition of N t into a finite family of polyhedra such that the function f L is a quasi-polynomial on each polyhedron of the partition. 2. There exists a partition of N t into a finite family of rational subsets such that the function f L is a polynomial on each set of the partition.doi:10.1016/j.tcs.2009.09.006 fatcat:svzkva7myff6hpfmq7zsbav7oy