On the Joint Distributions of Increasing and Decreasing Successions of Arbitrary Multisets

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...

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Bibliographic Details
Published in:Methodology and computing in applied probability Vol. 27; no. 3; p. 57
Main Author: Kong, Yong
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2025
Springer Nature B.V
Subjects:
Business and Management
Economics
Electrical Engineering
Life Sciences
Mathematics and Statistics
Statistical methods
Statistics
60C05
Random permutations of multisets
68R05
Generating functions
68R15
Combinatorial probability
Exact enumeration problems
System of recurrences
Increasing and decreasing successions
Increasing and decreasing 2-sequences
05A05
05A15
62E15
ISSN:1387-5841, 1573-7713
Online Access:Get full text
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Summary: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.
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ISSN:1387-5841
1573-7713
DOI:10.1007/s11009-025-10186-2