A proof of a conjecture of Mao on Beck's partition statistics modulo 8

Beck introduced two partition statistics $NT(r,m,n)$ and $M_{\omega }(r,m,n)$,which denote the total number of parts in the partition of $n$ with rank congruent to $r$ modulo $m$ and the total number of ones in the partition of $n$ with crank congruent to $r$ modulo $m$, respectively. In recent years, a number of congruences and identities on $NT(r,m,n)$ and $M_{\omega }(r,m,n)$ for some small $m$ have been established.In this paper, we prove an identity on $NT(r,8,n)$ and $M_{\omega }(r,4,n)$ which confirm a conjecture given by Mao.