A bijective proof of infinite variated Good's inversion
type: Item
instance of: scholarly article
title: A bijective proof of infinite variated Good's inversion
zbMATH Open document ID: 0810.05070
author: Richard Ehrenborg
author: Miguel A. Méndez
published in: Advances in Mathematics
publication date: 1995-04-04T00:00:00Z
review text: 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 propert…
Mathematics Subject Classification ID: 05E99
Mathematics Subject Classification ID: 05A40
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 611217
zbMATH Keywords: combinatorial identities
zbMATH Keywords: bijective proofs
zbMATH Keywords: plethystic composition
zbMATH Keywords: formal series
zbMATH Keywords: species
zbMATH Keywords: \(c\)-monoid
zbMATH Keywords: Good's multivariated inversion formula
zbMATH Keywords: Polya's plethysm
zbMATH Keywords: infinite variated multivariable umbral calculus
zbMATH Keywords: plethystic transfer formula
zbMATH Keywords: plethystic Lagrange inversion formula
Wikidata QID: Q114234244
reviewed by: U. Kaljulaid
MaRDI profile type: Publication
OpenAlex ID: W1973467451
DOI: 10.1006/AIMA.1994.1011
description: scientific article; zbMATH DE number 611217