What is the generating function of partitions of $n$ into $a_1, \ldots, a_r$ where $a_1 \le \ldots \le a_r$, $a_1 = 1$ and $a_i - a_{i-1} \le 1$ for $2 \le i \le r$?

Hint: find a bijection between set of these partitions and some set of partitions with a well-known generating function.

I've tried to find a bijection using Ferrers diagrams but with no result.

  • $\begingroup$ Is $a_i$ integer? $\endgroup$ – Zhaohui Du Aug 31 '19 at 11:24
  • $\begingroup$ Yes, it is. I added tag 'discrete-mathematics' to the question. $\endgroup$ – pblass Aug 31 '19 at 13:36
  • $\begingroup$ What if you take into account how many times you use $1,$ then how many times you have $2,$ etc etc. i.e., $b_i = |\{j:a_j=i\}|,$ notice that $\sum i*b_i=n$ $\endgroup$ – Phicar Aug 31 '19 at 13:44
  • $\begingroup$ I moved my answer to math.stackexchange.com/questions/3317618/… $\endgroup$ – Robert Z Aug 31 '19 at 14:17