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.