Partitions, the partition function p(n), and hook lengths of their Ferrers–Young diagrams are important objects in combinatorics, number theory, representation theory. For positive integers n t, we study $$p_t^\mathrm{e}(n)$$ (resp. $$p_t^\mathrm{o}(n)$$ ), partitions with an even odd) t-hooks. We limiting behavior ratio $$p_t^\mathrm{e}(n)/p(n)$$ , which also gives $$p_t^\mathrm{o}(n)/p(n)$$ s...