Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern restricted permutations. The objective of this paper is not only a confirmation of Hong and Li's conjecture and Burstein's first conjecture, but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted permutation case, and the $\mathsf{asc}$ statistic concerned in the restricted inversion sequence case, which yield a new equidistribution result.
翻译:最近,Hong和Li对四长模式避免反向序列进行了系统研究,特别是,他们推测,OEIS A218225号条目可以列出0021美元的避免反向序列数。同时,Burstein建议,同一序列也可能计算出三套模式限制的变相。本文的目的不仅是确认Hong和Li的推测和Burstein的第一个猜想,而且还有两个更微妙的产生功能特性,分别是限制性变换案所涉的美元和限制性变换序列案所涉的美元和美元(mathsfsf)和美元(sasc)数据,从而产生新的均匀分布结果。