Enumerative combinatorics
Constructive combinatorics
Inductive and injective proofs of log concavity results
Discrete Mathematics
Generating functionology
Combinatorics and total positivity
Journal of Combinatorial Theory Series A
Combinatorial proof of the log-concavity of the sequence of matching numbers
Journal of Combinatorial Theory Series A
On the unimodality and combinatorics of Bessel numbers
Discrete Mathematics - The 2000 Com2MaC conference on association schemes, codes and designs
On the total positivity of restricted Stirling numbers
European Journal of Combinatorics
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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 nth 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.