Blecher–Knopfmacher's fixed-point inequality for integer partitions

Let $n$ be a positive integer, and let $p(n)$ denote the number of partitions of $n$. A partition $\lambda=(\lambda_1,\ldots,\lambda_j)$ of $n$, written $\lambda\vdash n$, has a fixed point if $\lambda_i=i$ for some index $i$. Let $f(n)$ be the number of partitions of $n$ with a fixed point and $g(n)$ the number without a fixed point. Blecher–Knopfmacher's conjecture. For all $n>2$, there are more partitions of $n$ without a fixed point than partitions of $n$ with a fixed point; equivalently,

$$g(n)>f(n).$$

The paper resolves this open question by proving the inequality and relating fixed points to Frobenius symbols, Dyson's crank, the mex, and other partition statistics.