A bijective proof of infinite variated Good's inversion
The idea to prove combinatorial identities by bijective arguments seems to be as old as combinatorics itself. G.-C. Rota initiated to systemize such bijective arguments. After a while, again, this idea was demonstrated to be fruitful in R. Stanley's bijective proof of Waring's formula and in proofs of classical identities for symmetric polynomials. \textit{G.-C. Rota} et al. [Combinatorics '84 Proc. Int. Conf. Finite Geom. Comb. Struct. Bari/Italy 1984, Ann. Discrete Math. 30, 107-113 (1986; Zbl 0615.18003)] developed a systematic theory of bijective proofs based on Joyal's species. Notice that in algebraic geometry there was grown up the idea of giving topologies without indicating the underlying space. And in the sixties, due to A. Grothendieck's pioneering work, it become widely known that categories provide means for such an approach. \textit{A. Joyal's} idea [Ad. Math. 42, 1-82 (1981; Zbl 0491.05007)] is to define combinatorial structures equivariantly, i.e. without a committed choice of the underlying set. Namely, species is defined as an endofunctor on the category of finite sets and their bijections. So, e.g., the species `tree' is the functor \(T\) assigning to every finite set \(S\) the set \(T(S)\) of all trees whose vertex-set coincides with \(S\). \textit{O. Nava} and \textit{G.-C. Rota} [Adv. Math. 58, 61-88 (1985; Zbl 0634.05010)] did a step further using partitionals---functors from the category of partitions on finite sets to the category of finite sets---for giving a natural combinatorial interpretation to the plethystic composition of formal series. In the present paper the authors further develop the second author and \textit{O. Nava's} [J. Comb. Theory, Ser. A 64, No. 1, 102-129 (1993; Zbl 0787.05095)] generalization of Joyal's theory of species so interpreting operations with formal series in multivariable case in terms of operations with colored sets. Using the notion of \(c\)-monoid, i.e. of left cancellable monoid with indivisibility property for its identity element, the authors introduce the `Verschiebungsoperator' on formal series and so it becomes possible to define a general plethystic composition of formal series and to find its combinatorial interpretation---through (parallel) plethystic composition of colored species. They generalize also Good's multivariated inversion formula to the infinite variated case and deduce from it the Lagrange inversion formula for generalized plethysm. From this, in turn, they derive the inversion formula for Polya's plethysm previously studied by G. Labelle. Using the infinite variated multivariable umbral calculus developed by \textit{W. Chen} [Ph. D. Thesis, M.I.T., 1991]---via notions of plethystic delta operator and its plethystic basic sequence---the authors prove in the general setting of colorations the equivalence of the plethystic transfer formula and the plethystic Lagrange inversion formula. In the reviewer's opinion, nowadays this interesting stream in combinatorics the authors of this paper write on, if having in mind something like the Weil hypotheses, is a fair play not only for combinatorics itself. It stands quite prophetic for some parts of algebra and computer science also.
infinite variated multivariable umbral calculus
scientific article; zbMATH DE number 611217