C sequential optimization numbers

TL;DR

This paper introduces C sequential optimization numbers, a new concept related to Stirling numbers of the first kind, providing new properties, bounds, and probabilistic insights.

Contribution

It defines C sequential optimization numbers, establishes their properties, and connects them to Stirling numbers, offering new bounds and probabilistic results.

Findings

C sequential optimization numbers generalize Stirling numbers of the first kind.

Derived explicit upper bounds for C sequential optimization numbers.

Proved the probability concentration of Stirling numbers in O(log n) is nearly 100%.

Abstract

This work establishes a definition that is more basic than the previous ones, for the Stirling numbers of first kind, which is a sufficient but not necessary condition for the previous definition. Based on this definition and a combinatorial problem, we discover C sequential optimization numbers, where C is a k+1-tuple vector. For C= (0,1), we prove that C sequential optimization numbers are the unsigned Stirling numbers of first kind. We can deduce the properties of C sequential optimization numbers by following the properties of the Stirling numbers of first kind and we give specific examples such as the recurrence formula and an instance of C sequential optimization numbers. We also give specific new properties such as an explicit upper bound of them. We prove the probability that the unsigned Stirling numbers of first kind are concentrated in O(logn) is nearly 100%.