Combinatorial proofs of inverse relations and log-concavity for Bessel numbers

Let the Bessel number of the second kind B ( n , k ) be the number of set partitions of [ n ] into k blocks of size one or two, and let the Bessel number of the first kind b ( n , k ) be the coefficient of x n − k in  − y n − 1 ( − x ) , where y n ( x ) is the n th Bessel polynomial. In this paper, we show that Bessel numbers satisfy two properties of Stirling numbers: The two kinds of Bessel numbers are related by inverse formulas, and both Bessel numbers of the first kind and those of the second kind form log-concave sequences. By constructing sign-reversing involutions, we prove the inverse formulas. We review Krattenthaler’s injection for the log-concavity of Bessel numbers of the second kind, and give a new explicit injection for the log-concavity of signless Bessel numbers of the first kind.