On the Joint Distributions of Increasing and Decreasing Successions of Arbitrary Multisets - Kong, Yong (Methodology and computing in applied probability, 2025)
The joint distribution of increasing and decreasing successions in arbitrary multisets has remained an open problem for a long time. Here, we present a systematic approach inspired by methods in statistical physics to solve this problem. Using the two-step approach to pattern distributions in random sequences previously developed by the author, we derive recurrence and explicit formulas for the generating functions of increasing and decreasing successions in multisets. From these generating functions, explicit formulas for the mean, variance, and covariance are obtained.