Main theorem via recursion relations

Define the counting function of of symmetric polynomials by

$$\chi_{n,k}(q)=\sum_{i=0}^\infty a_{i,k} q^i,\qquad a_{i,k}=\sharp\{\la, \;\;\la_1\leq k,\;\;\vert\la\vert=i\}.$$

The number $a_{i,k}$ counts polynomials of total degree $i$, such that degree in any variable is at most $k$.

Theorem 6  

$$\chi_{k,n}(q)={n+k\choose k}_q.$$

Exercise:Prove the theorem using Pascal identity $($3$)$. $\;\Box$