SCHURS PARTITION THEOREM, COMPANIONS, REFINEMENTS AND GENERALIZATIONS

Schur's partition theorem asserts the equality S(n) = S-1(n), where S( n) is the number of partitions of n into distinct parts = 1, 2 (mod 3) and S-1(n) is the number of partitions of n into parts with minimal d ifference 3 and no consecutive multiples of 3. Using a computer search Andrews found a companion result S(n) = S-2(n), where S-2(n) is the n umber of partitions of n whose parts e(i) satisfy e(i)-e(i+1) greater than or equal to 3, 2 or 5 according as e(i) = 1, 2 or 3 (mod 3). By m eans of a new technique called the method of weighted words, a combina torial as well as a generating function proof of both these theorems a re given simultaneously. It is shown that S-1(n) and S-2(n) are only t wo of six companion partition functions S-j(n), j = 1, 2, ..., 6, all equal to S(n). A three parameter refinement and generalization of thes e results is obtained.