Finite analogs of partition bias related to hook length two and a variant of Sylvester's map

In this paper, we count the total number of hooks of length two in all odd partitions of $n$ and all distinct partitions of $n$ with a bound on the largest part of the partitions. We generalize inequalities of Ballantine, Burson, Craig, Folsom and Wen by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of $n$ in presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester's map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part.