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number of good sequences

Source: Iran 3rd round 2011-combinatorics exam-p5

September 4, 2011
functionalgebralinear equationcombinatorics proposedcombinatorics

Problem Statement

Suppose that nn is a natural number. we call the sequence (x1,y1,z1,t1),(x2,y2,z2,t2),.....,(xs,ys,zs,ts)(x_1,y_1,z_1,t_1),(x_2,y_2,z_2,t_2),.....,(x_s,y_s,z_s,t_s) of Z4\mathbb Z^4 good if it satisfies these three conditions:
i) x1=y1=z1=t1=0x_1=y_1=z_1=t_1=0.
ii) the sequences xi,yi,zi,tix_i,y_i,z_i,t_i be strictly increasing.
iii) xs+ys+zs+ts=nx_s+y_s+z_s+t_s=n. (note that ss may vary).
Find the number of good sequences.
proposed by Mohammad Ghiasi
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