By E. M. Kleinberg
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3C] Write a program which lists an partitions of n a) with an odd number of parts; b) with an even number of parts; and c) into distinct, odd parts. Formulate a conjecture and prove it. 18. [3C] Write a program which lists all partitions of n a) into parts which are congruent to 1 or 4 mod 5; b) into parts whose differences are at 1east two; and c) into distinct parts, where each even partis > twice the number of odd parts. Investigate your data. If a) is rep1aced by 2 or 3 mod 5, can you fmd an appropriate b)'?
Vi-l• j}. Second, j :s; max{vi' v2, . , vi_1} by the RG condition on vi, so max{vi' v2, ... , vi-l• j} = max{vi' v2, ... , vi_1} = ui. St n -rn positions. 2) dm,t = t dm-1,1 + dm-1,1+1 because we may place either t + 1 in the n - rn+ 1 position, leaving rn - 1 positions to fill, with largest value now t + 1; or we may place 1, 2, ... , t in the n - rn+ 1 position, leaving rn- 1 positions to fùl, with largest value still t 21 ALGORTIHM 12: begin Rank Restricted Growth Function u, +- 1 for i +- 2 to n do if ui-l > vi-l then else R+-0 for i +- n downto 1 do t+- ui R +-R+~i,t"(vi -1) Rank(v)+- R end.
We have just shown that the roots of S0 (x) and S0 _ 1(x) inter/ace: between any two consecutive roots of one polynomial there is exactly one root of the other polynomial.