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Partitions with special conditions

Source: 2015 Korean Mathematical Olympiad P4

November 1, 2015
combinatoricspartitionsbijection

Problem Statement

For positive integers n,k,ln, k, l, we define the number of ll-tuples of positive integers (a1,a2,⋯al)(a_1,a_2,\cdots a_l) satisfying the following as Q(n,k,l)Q(n,k,l).
(i): n=a1+a2+⋯+aln=a_1+a_2+\cdots +a_l
(ii): a1>a2>⋯>al>0a_1>a_2>\cdots > a_l > 0.
(iii): ala_l is an odd number.
(iv): There are kk odd numbers out of aia_i.
For example, from 9=8+1=6+3=6+2+19=8+1=6+3=6+2+1, we have Q(9,1,1)=1Q(9,1,1)=1, Q(9,1,2)=2Q(9,1,2)=2, Q(9,1,3)=1Q(9,1,3)=1.
Prove that if n>k2n>k^2, ∑l=1nQ(n,k,l)\sum_{l=1}^n Q(n,k,l) is 00 or an even number.
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