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 works by [[Dominique Foata|Foata]] and [[Marcel-Paul Schützenberger|Schützenberger]]<ref name="fs">{{cite journal|last1=Foata|first1=Dominique|authorlink1=Dominique Foata|last2=Schützenberger|first2=Marcel-P.|authorlink2=Marcel-Paul Schützenberger|title=Théorie géométrique des polynômes Eulériens|journal=Lectures Notes in Mathematics|series=Lecture Notes in Mathematics|date=1970|volume=138|doi=10.1007/BFb0060799|isbn=978-3-540-04927-2|doi-access=free|arxiv=math/0508232}}</ref> on permutations,

Bender and Goldman on prefabs,<ref>{{cite journal|last1=Bender|first1=Edward A.|last2=Goldman|first2=Jay R.|title=Enumerative uses of generating functions|journal=Indiana University Mathematics Journal|date=1971|volume=20|issue=8|pages=753–764|doi=10.1512/iumj.1971.20.20060|doi-access=free}}</ref> and [[André Joyal|Joyal]] on [[combinatorial species]].<ref>{{cite journal|last1=Joyal|first1=André|authorlink1=André Joyal|title=Une théorie combinatoire des séries formelles|journal=[[Advances in Mathematics]]|date=1981|volume=42|pages=1–82|ref=joy|doi=10.1016/0001-8708(81)90052-9|doi-access=free}}</ref>

Note that this symbolic method in enumeration is unrelated to "Blissard's symbolic method", which is just another old name for [[umbral calculus]].