$3$ boxes have some balls, for example box $A$ has $1$ balls, box $B$ has $2$ balls, box $C$ has $3$ balls
If only care the generating function of box $A$, can it use box $A$ and non box $A$ to count?
How to count? polynomial count?
Update: the number of balls in each box is determined by the number after throw a die, for box A, i throw a die, if get 1 then i put a ball in box A, for box B, if throw a die and get 2 then i put two balls in box B, etc..the goal is to concern the distribution in box A in a group of boxes
Remark: use z represent the number of appearance of balls in box A