A combinatorial interpretation of Bessel polynomials and their first derivatives as ordered hit polynomials (Q2768121)

The authors consider the hit polynomial of the path \(P_{2n}\) embedded in the graph \(K_{2n}\) and denote the graph so formed \(K_{2n}|P_{2n}\). Here \(K_{2n}\) stands for the complete graph on \(2n\) vertices labelled by the integers \(\{1,\dots,2n\}\) and \(P_m\) the path on \(m\) vertices \((2\leq m\leq 2n)\). They give a combinatorial interpretation of the Bessel polynomials [see \textit{E. Grosswald}, ``Bessel polynomials'' (1978; Zbl 0416.33008)] through that hit polynomial. The given main results of the paper are: i) The \(n\)-th Bessel polynomial \(\theta_n(x)\) \((n\geq 1)\) equals the ordered hit polynomial of \(K_{2n}|P_{2n}\); ii) The first derivative \(\theta_n'(x)\) of \(\theta_n(x)\), equals the ordered hit polynomial of \(K_{2n}|P_{2n-1}\). The authors also define homogeneous Bessel polynomials and present a combinatorial interpretation of them. Some examples are finally given.