Problem

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

Tags: function, algebra, linear equation, combinatorics proposed, combinatorics



Suppose that $n$ is a natural number. we call the sequence $(x_1,y_1,z_1,t_1),(x_2,y_2,z_2,t_2),.....,(x_s,y_s,z_s,t_s)$ of $\mathbb Z^4$ good if it satisfies these three conditions: i) $x_1=y_1=z_1=t_1=0$. ii) the sequences $x_i,y_i,z_i,t_i$ be strictly increasing. iii) $x_s+y_s+z_s+t_s=n$. (note that $s$ may vary). Find the number of good sequences. proposed by Mohammad Ghiasi