Proofs of some conjectures of Chan–Mao–Osburn on Beck’s partition statistics

Recently, George Beck introduced two partition statistics NT(m, j, n) and $$M_{\omega }(m,j,n)$$ , which denote the total number of parts in n with rank congruent to m modulo j ones crank respectively. Andrews proved a congruence on 5, was conjectured by Beck. Very recently, Chan, Mao Osburn established Andrews–Beck type congruences posed several conjectures involving . Some those were Chern Mao. In this paper, we confirm remaining three Chan–Mao–Osburn due We also present new n).