Combinatorics of bessel numbers

Abstract

The Bessel numbers are associated with set partitions each of whose block has either one or two elements. The paper is devoted to the properties of the Bessel numbers, which have natural similarities with the Stirling numbers of the second kind. We give the generating function for the Bessel numbers, define the dual Bessel numbers and its signless form, which are reminiscent of the Stirling numbers of the first kind and its signless form. Further, we combinatorially prove the inversion formulas and orthogonality formulas for Bessel numbers and dual Bessel numbers. Both Bessel numbers and signless dual Bessel numbers form log-concave sequences. We show the log-concavity of Bessel numbers by the Krattenthaler`s injection and that of signless dual Bessel numbers by constructing a new injection. We also give a combinatorial proof of the sum of the coefficients of a certain polynomial associated with Bessel numbers.