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

Let the Bessel number of the second kind B(n,k)B(n,k) be the number of set partitions of [n][n] into kk blocks of size one or two, and let the Bessel number of the first kind b(n,k)b(n,k) be the coefficient of xn−kxn−k in −yn−1(−x)−yn−1(−x), where yn(x)yn(x) is the nnth 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.