Generating function with dependencies

Question

I'm trying to find the exponential generating function for the numbers of ways to distribute $r$ distinct objects into five different boxes when $b_1 < b_2 \leq 4$, where $b_i$ denotes the number of objects in the $i$-th box. I know how to find a exponential generating function in general, but I don't understand how to handle the constraint that $b_1 < b_2$. It seems this would change what I could use to generate $b_1$ for each possible $b_2$. Any hint on how to deal with this complication would be appreciated.

Answer 1

Here's a rather thorough hint: Since you've stated you know how to find the exponential generating function in general, I won't explain that part. To handle the constraint, however, consider combining the first two boxes into one bigger box -- giving us 4 boxes in all. That is, the possible values are integers from $1$ to $7$ inclusive (we still want to satisfy the constraint though). Then look at each case: if $b_2=1$, then $b_1 = 0$, if $b_2=2,$ then $b_1$ is $0$ or $1$, and so forth.

Keep in mind what the coefficients of the terms in the exponential generating function mean: they are the number of ways of distributing $k$ distinct objects, where $k$ is from the $\frac{x^k}{k!}$ term. So, for example, if we have $3$ balls in our combined box, the coefficient of the term $\frac{x^k}{k!}$, for $k = 3$, should be $\binom{3}{0} + \binom{3}{1}$.