The independence of the candidates generated in the existing interleaved partitioning partial transmit sequences (IP-PTS) is first investigated. The result shows that these candidates are not fully independent, which leads to performance inferior to the adjacent partitioning PTS (AP-PTS) in which all the generated candidates are mutually independent. To improve the performance of the IP-PTS in peak-to-average power ratio reduction, a novel type of phase sequences for IP-PTS is proposed that can be used to produce fully independent candidates so that IP-PTS can achieve similar performance to AP-PTS, while keeping low complexity.