# Generating function of 'monotonic' partitions [duplicate]

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.

• Is $a_i$ integer? – Zhaohui Du Aug 31 '19 at 11:24
• Yes, it is. I added tag 'discrete-mathematics' to the question. – pblass Aug 31 '19 at 13:36
• 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 $\sum i*b_i=n$ – Phicar Aug 31 '19 at 13:44
• I moved my answer to math.stackexchange.com/questions/3317618/… – Robert Z Aug 31 '19 at 14:17