On the number of even and odd strings along the overpartitions of n

摘要

Recently, Andrews, Chan, Kim, and Osburn introduced the even strings and the odd strings in the overpartitions. We show that their conjecture A_k(n) ≥ B_k(n) holds for large enough positive integers n, where A_k(n) (resp. B_k(n)) is the number of odd (resp. even) strings along the overpartitions of n. We introduce m-strings and show how this new combinatorial object is related with another positivity conjecture of Andrews, Chan, Kim, and Osburn. Finally, we confirm that the positivity conjecture is also true for large enough integers.