Symbolic method (combinatorics): Difference between revisions

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It was then slowly realized that the generating functions were capturing many other facets of the initial discrete combinatorial objects, and that this could be done in a more direct formal way: The recursive nature of some combinatorial structures

translates, via some isomorphisms, into noteworthy identities on the corresponding generating functions.

Following the works of [[George_Pólya|Pólya]], further advances were thus done in this spirit in the 1970s with generic uses of languages for specifying combinatorial classes and their generating functions, as found in workworks by [[Dominique Foata|Foata]] and [[Marcel-Paul Schützenberger|Schützenberger]] <ref name="fs">{{cite journal|last1=Foata|first1=D.|last2=Schützenberger|first2=M.|title=Théorie géométrique des polynômes Eulériens|journal=Lectures Notes in Math.|date=1970|volume=138}}</ref> on permutations,

Bender and Goldman on prefabs <ref>{{cite journal|last1=Bender|first1=E.A.|last2=Goldman|first2=J.R.|title=Enumerative uses of generating functions|journal=Indiana Univ. Math. J.|date=1971|volume=20|pages=753–764}}</ref>, and [[André Joyal|Joyal]] on [[combinatorial species]] <ref>{{cite journal|last1=Joyal|first1=André|title=Une théorie combinatoire des séries formelles|journal=Adv. Math.|date=1981|volume=42|pages=1–82|ref=joy}}</ref>.