We consider the number of various partitions of n with parts separated by parity and prove combinatorially several inequalities between these numbers. For example, we show that for \(n\ge 5\) we have \(p_{\text {od}}^{\text {eu}}(n)<p_{\text {ed}}^{\text {ou}}(n)\) , where \(p_{\text {od}}^{\text {eu}}(n)\) is the number of partitions of n with odd parts distinct and even parts unrestricted and all odd parts less than all even parts and \(p_{\text {ed}}^{\text {ou}}(n)\) is the number of partitions of n with even parts distinct and odd parts unrestricted and all even parts less than all odd parts. We also prove a conjectural inequality of Fu and Tang involving partitions with parts separated by parity with restrictions on the multiplicity of parts.
