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

Abstract

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 $K_{2n}$.