A Combinatorial Interpretation of Bessel Polynomials and their First Derivatives as Ordered Hit Polynomials

Consider the hit polynomial of the path \(P_{2n}\) embedded in the complete graph \(K_{2n}\). We give a combinatorial interpretation of the \(n\)-th Bessel polynomial in terms of a modification of this hit polynomial, called the ordered hit polynomial. Also, the first derivative of the \(n\)-th Bessel polynomial is shown to be the ordered hit polynomial of \(P_{2n-1}\) embedded in \(K_{2n}\).