Extending a Recent Result of Santos on Partitions into Odd Parts

In a recent note, Santos proved that the number of partitions of n using only odd parts equals the number of partitions of n of the form p1 + p2 + p3 + p4 + ... such that p1 ≥ p2 ≥ p3 ≥ p4 ≥ ··· ≥ 0 and p1 ≥ 2p2 + p3 + p4 + .... Via partition analysis, we extend this result by replacing the last inequality with p1 ≥ k2p2+k3p3+k4p4+..., where k2, k3, k4,... are nonnegative integers. Several applications of this result are mentioned in closing