On the set partitions that require maximum sorts through the aba-avoiding stack

Recently, Xia introduced a deterministic variation _ of Defant and Kravitz's stack-sorting maps for set partitions and showed that any set partition p is sorted by ^N (p) ₀₁₀, where N (p) is the number of distinct alphabets in p. Xia then asked which set partitions p are not sorted by ₀₁₀^N (p) -1. In this note, we prove that the minimal length of a set partition p that is not sorted by ₀₁₀^N (p) -1 is 2N (p). Then we show that there is only one set partition of length 2N (p) and {N (p) + 1 2} + 2N (p) 2 set partitions of length 2N (p) +1 that are not sorted by ₀₁₀^N (p) -1.