Stirling Numbers Of The Second Kind And Their Generalizations

Enumerative combinatorics is an important part in combinatorics and counts the number of elements of a finite set S.Stirling numbers of the second kind are the primary combinatorial numbers.They play an important and specific role in enumerative problem.Stirling numbers of the second kind are the numbers of k-partitions on[n].The study of Stifling numbers of the second kind is quite system.They are unimodal,log-concave,log-convex and P(?)lya-frequency sequences and so on.We get new numbers from restricting elements in every subset.So the generalizations of Stirling numbers of the second kind arise in combinatorics.This article is devoted to the basic properties of these new numbers.The main frame is as follows.The first chapter contributes to the basic properties of Stirling numbers of the second kind. First of all,we introduce recurrence relations and generating functions of Stirling numbers of the second kind.Then,we give Stirling matrix of the second kind and the asymptotic formula for Stirling numbers of the second kind.After that,we conclude the properties of Bell numbers and Bell polynomials.The second part investigates the generalizations of Stirling numbers of the second kind in detail.We study the properties of r-Stirling numbers of the second kind.Then we show log-convexity of r-Bell numbers and q-log-convexity of r-Bell polynomials.We also show operator on the r-Stirling sequences of the second kind that preserve log-convexity and r-Stirling sequences of the second kind are PF sequences.At last,we give the properties of Bessel numbers,associal r-Stirling numbers of the second kind and Catalan numbers.