Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity

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≥ 5 we have podᵉᵘ(n)<pedᵒᵘ(n), where podᵉᵘ(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 pedᵒᵘ(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.