Combinatorics, Probability and Computing
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Let A"1,...,A"h be finite non-empty subsets of a field K and let s"k(x"1,...,x"h) be the elementary symmetric polynomial of degree k in h indeterminates. Here we present some estimates for the cardinality of the sets of the images of all h-tuples of A"1x...xA"h by the polynomial s"k, with and without the restriction that the elements of the h-tuples are pairwise distincts.