Symbolic method (combinatorics)

In <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, the symbolic method is a technique for <a href="/facts/Enumerative_combinatorics/AtqWU2bj">counting combinatorial objects</a>. It uses the internal structure of the objects to derive formulas for their <a href="/facts/Generating_function/K7UVjnjL">generating functions</a>. The method is mostly associated with <a href="/facts/Philippe_Flajolet/8C9NqUlD">Philippe Flajolet</a> and is detailed in Part A of his book with <a href="/facts/Robert_Sedgewick_(computer_scientist)/f1lY2MFL">Robert Sedgewick</a>, <a href="/facts/Analytic_Combinatorics/Uu9sF1dp">Analytic Combinatorics</a>, while the rest of the book explains how to use complex analysis in order to get asymptotic and probabilistic results on the corresponding generating functions.
During two centuries, generating functions were popping up via the corresponding recurrences on their coefficients (as can be seen in the seminal works of <a href="/facts/Daniel_Bernoulli/NRs5Ne5r">Bernoulli</a>, <a href="/facts/Leonhard_Euler/z2fsbjgj">Euler</a>, <a href="/facts/Arthur_Cayley/NBd7QkeE">Arthur Cayley</a>, <a href="/facts/Ernst_Schr%C3%B6der_(mathematician)/GsUT8tID">Schröder</a>, 
<a href="/facts/Srinivasa_Ramanujan/HBPr4bzd">Ramanujan</a>, <a href="/facts/John_Riordan_(mathematician)/kKrpSbPW">Riordan</a>, <a href="/facts/Donald_Knuth/GozNzDM6">Knuth</a>, Comtet [fr], etc.).
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 <a href="/facts/George_P%C3%B3lya/cYgGdvJu">Pólya</a>, 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 <a href="/facts/Dominique_Foata/rg2xQbRI">Foata</a> and <a href="/facts/Marcel-Paul_Sch%C3%BCtzenberger/aF62Lo9H">Schützenberger</a> on permutations, 
Bender and Goldman on prefabs, and <a href="/facts/Andr%C3%A9_Joyal/Xc0dKuXB">Joyal</a> on <a href="/facts/Combinatorial_species/qge5Q0no">combinatorial species</a>.
Note that this symbolic method in enumeration is unrelated to "Blissard's symbolic method", which is just another old name for <a href="/facts/Umbral_calculus/hN5Ph72G">umbral calculus</a>.
The symbolic method in combinatorics constitutes the first step of many analyses of combinatorial structures, 
which can then lead to fast computation schemes, to asymptotic properties and <a href="/facts/Asymptotic_distribution/dN7jVHqm">limit laws</a>, to <a href="/facts/Random_generation/I94vq0AQ">random generation</a>, all of them being suitable to automatization via <a href="/facts/Computer_algebra/hjun1KAX">computer algebra</a>.

Symbolic method (combinatorics) open-in-new

Symbolic method (combinatorics)