Some Statistics On Combinatorial Objects

Combinatorial statistics on permutations have been one of the most active research areas in enumerative combinatorics.Researchers further extended the statistics on permutation to other combinatorial structures,and obtained a series of abundant research results.The problem of pattern avoidance on combinatorial structures has been extensively studied in recent decades.This paper is mainly concerned with the equidistribution of several statistics on Fishburn matrices and permutations avoiding certain pattern.To this end,we aim to establish a bijection between these two combinatorial objects.In Chapter 1,we mainly introduces the research background,preliminaries and our main results.In Chapter 2,we mainly provide a bijection between pattern avoiding permutations and ascent sequences,and show that the bijection verifies the equidistribution of two 4-tuples of statistics.In Chapter 3,we mainly establish a bijection between ascent sequences and Fishburn matrices,and show that the bijection verifies the equidistribution of two 4-tuples of statistics.