Applications Of Continued Fractions In Combinatorics Of Set Partitions

Combinatorics is a discipline that studies discrete structures,involving geometry,algebra,and analytical ideas.It is closely related to mathematical branches such as graph theory,abstract algebra,probability theory,and mathematical logic.The application of combinatorics is very extensive,and the fields of information science,intelligent science,especially computer science,all involve the theory and methods of combinatorics.The study of set partition statistics is one of the hot problems in combinatorics,mainly thanks to Milne’s work.In 1980,P.Flajolet,an academician of the French Academy of Sciences,used continuous fractions to study combinatorial objects,which provided a powerful tool for the study of partition statistics.Accordingly,countless researchers have popularized and applied the combinatorial aspects of continued fractions.In particular,in 2006,Kasraoui and Zeng studied the distribution properties of set partition statistics—crossings and nestings,which further developed the idea of continuous fractions.This paper mainly studies the application of continuous combinatorics in set partition,and has two contributions:(1)Involution proof of symmetric distribution of overlap number and covering number on set partition;(2)The refined Motzkin-Catalan identities are discovered and proved,and the results of Lin and Kim are generalized.The organization of the paper mainly includes the following three parts:The first part studies the theory of combinatorial aspects of continued fractions.First,the combinatorial basis covered in this paper is introduced.Then,we introduce and explain the theory of combinatorial aspects of continued fractions meticulously.And supplement and improve the proof of the relevant theorem.In particular,we improve the reversibility part of Flajolet’s bijective proof,which lays a foundation for studying the properties of set partition statistics.And supplement and improve the proof of the relevant theorem.Finally,using this theory,we obtain the continued fraction expansions of the corresponding ordinary generating functions.The second part studies the property of set partition statistics.Inspired by the work of Kasraoui and Zeng,we propose the symmetric distribution of the numbers of overlaps and coverings.By establishing involution on partition,we give the combinatorial proof of symmetric distribution.Using the theory in the first part,we obtain the continued fraction expansion of corresponding generating function to further verify the correctness of the conclusion.Finally.we generalize the symmetry property of overlaps and coverings to the case of k distance.The third part studies the application of continued fractions to combinatorial identities.Firstlythe classical conclusion in the combination is verified by using the continuous fractional method.We then give a new proof for the generalized Motzkin-Catalan equation.Finally,on this basis,we extend and propose a completely new regeneralized Motzkin-Catalan equation.