Harmonic numbers associated with inversion numbers in terms of determinants

Abstract

It has been known that some numbers, including Bernoulli, Cauchy, and Euler numbers, have such correspondingnumbers in terms of determinants of Hessenberg matrices. There exist inversion relations between the originalnumbers and the corresponding numbers. In this paper, we introduce the numbers related to harmonic numbers in determinants.We also give several of their arithmetical and/or combinatorial properties and applications. These conceptscan be generalized in the case of hyperharmonic numbers.