A *corner* of an integer partition is a location at where a box can be added to its Ferrers diagram to give a new partition. 

E.g. the partition $\{1,1,1\}$ has two corners, and $\{1,2\}$ has three corners.

Let $p(n,r)$ denote the number of integer partitions of $n$ with $r$ corners.

Let $P(x,t) = \sum_{n,r\geq0} p(n,r)x^nt^r = t + t^2x + 2t^2x^2 + (2t^2+t^3)x^3 + \ldots$ be the generating function for these numbers.

1) Is there an elementary (**EDIT:** i.e. probabilistic) proof that $P(x,1-x) = 1$?

Thus for $x \in [0,1)$, the map $\lambda \to x^{|\lambda|}(1-x)^{\#corners (\lambda)}$ defines a probability distribution on the set of partitions of all integers.

2) Is it well known that $P(x,t) = \prod_{k=1}^{\infty} \frac{1-x^{k-1}(1-t)}{1-x^k} = \frac{(1-t,x)_\infty}{(x,x)_\infty}$?

Note: In [Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, and Toufik Mansour, "Counting corners in partitions"](https://www.researchgate.net/publication/273286302_Counting_corners_in_partitions), the following formula (Theorem 2.2) is given

$P(x,t) = t + t^2 \sum_{j\geq 1} \frac{x^j}{1-x^j} \prod_{i=1}^{j-1} \frac{1-(1-t)x^i}{1-x^i}$