From sets to functions: Three elementary examples

In the paper “On the foundations of combinatorial theory III. Theory of binomial enumeration” [7-j, Mullin and Rota developed the theory of polynomials of binomial type (definitiou below). They observed that in many cases these sequences of polynomials and the linear relations between them, given by the so-called connection constants, could be treated set-theoretically. But the problem of extending this approach to all sequences of bincmial type remains open. In this note--with a view towards understanding the tools required for such a set-theoretic alpproach-we will treat completely three sequences of binomial type and their connection constants: the power sequence, lower, and upper factorials. None of our Formulae are new, but sc:me of the underlying combinatorics is. Specifically, it will be seen that each pair of the above sequences is associated with a partially ordered set and certain ‘functinns’ (actually special cases of Mullin and Rota’s reluctant functions). Thes? the two iDverse connection constants formulae for the given sequence pair will be obtained in one case by summing over the poset, land in the other by differentiating i.e. Miibius inversion. Although these polynomial identities are proved only for positive integral values of thL variable x, it immediately follows that they hold for all x E @.