2025/10/29 by Tang, Michael
#05A05 #05E05 #16T30 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.25524
A permutation statistic is substring-compatible if its value on a permutation determines its value on every substring of that permutation. We construct the substring coalgebra of such a statistic, an analog of the shuffle algebra of a shuffle-compatible statistic introduced by Gessel and Zhuang. Furthermore, we show that for substring-compatible statistics that also satisfy a weak form of shuffle compatibility, the shuffle algebra and substring coalgebra can be combined to yield a Hopf algebra. Finally, we conjecture that the only nontrivial permutation statistics that are both shuffle-compatible and substring-compatible are the descent set, the peak set, and the valley set, and we describe our progress towards proving this conjecture.