Inequalities on the total number of t-hooks of partitions

Let p_t^⋆(n) (resp. sc_t^⋆(n) and dd_t^⋆(n)) be the total number of t-hooks among the ordinary (resp. self-conjugate and doubled distinct) partitions of n. We combinatorially derive the generating functions for p_t^⋆(n), sc_t^⋆(n), and dd_t^⋆(n). By constructing combinatorial injections using these generating functions, we investigate the monotonicity of these three counting functions according to n. In particular, we show that p_t^⋆(n)≥p_t+1^⋆(n) except for n=t+1, sc_2k−1^⋆(n)≥sc_2k^⋆(n) for n>84k, k≥1, dd₁^⋆(n)≥dd₂^⋆(n) except for n=6,10, and dd_2k^⋆(n)≥dd_2k+1^⋆(n) except for (k,n)=(1,12),(2,10),(3,14).