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

Beck introduced two partition statistics NT(r,m,n) and Mω(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 Mω(r,m,n) for some small m have been established. In this paper, we prove an identity on NT(r,8,n) and Mω(r,4,n) which confirm a conjecture given by Mao.