Combinatorics

Symbolic method (combinatorics)

In combinatorics, the symbolic method is a technique for counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. The method is mostly associated with Philippe Flajolet and is detailed in Part A of his book with Robert Sedgewick, Analytic Combinatorics, 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 Bernoulli, Euler, Arthur Cayley, Schröder, Ramanujan, Riordan, Knuth, , 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 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 Foata and Schützenberger on permutations, Bender and Goldman on prefabs, and Joyal on combinatorial species. Note that this symbolic method in enumeration is unrelated to "Blissard's symbolic method", which is just another old name for umbral calculus. 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 limit laws, to random generation, all of them being suitable to automatization via computer algebra. (Wikipedia).

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Combinatorial Identities via both Algebraic and Combinatorial Proof [Discrete Math Class]

This video is not like my normal uploads. This is a supplemental video from one of my courses that I made in case students had to quarantine. This is a follow up to previous a video introducing combinatorial objects (in particular k-permutations and k-subsets) and a video about the sum and

From playlist Discrete Mathematics Course

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Philippe Flajolet, founder of Analytic Combinatorics (2012)

January 16, 2012 (09:15 AM PST - 10:15 AM PST) Speaker(s): Mireille Bousquet-Melou (Université de Bordeaux I) Analytic combinatorics is a modern basis for the quantitative study of combinatorial structures (such as words, trees, paths, graphs...), with applications to the study of their

From playlist Mathematics

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Pablo Shmerkin: Additive combinatorics methods in fractal geometry - lecture 1

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive

From playlist Combinatorics

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Peter Varju: Additive combinatorics methods in fractal geometry - lecture 3

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive

From playlist Combinatorics

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Peter Varju: Additive combinatorics methods in fractal geometry - lecture 2

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive

From playlist Combinatorics

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Peter Varju: Additive combinatorics methods in fractal geometry - lecture 1

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive

From playlist Combinatorics

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Pablo Shmerkin: Additive combinatorics methods in fractal geometry - lecture 3

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive

From playlist Combinatorics

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Pablo Shmerkin: Additive combinatorics methods in fractal geometry - lecture 2

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive

From playlist Combinatorics

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Supercongruences for Apery-like numbers by Brundaban Sahu

Program Workshop on Additive Combinatorics ORGANIZERS: S. D. Adhikari and D. S. Ramana DATE: 24 February 2020 to 06 March 2020 VENUE: Madhava Lecture Hall, ICTS Bangalore Additive combinatorics is an active branch of mathematics that interfaces with combinatorics, number theory, ergod

From playlist Workshop on Additive Combinatorics 2020

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Algebraic combinatorics: applications to statistical mechanics and complexity theory - Greta Panova

Short proofs are hard to find (joint work w/ Toni Pitassi and Hao Wei) - Ian Mertz Computer Science/Discrete Mathematics Seminar II Topic: Short proofs are hard to find (joint work w/ Toni Pitassi and Hao Wei) Speaker: Ian Mertz Affiliation: University of Toronto Date: December 5, 2017 F

From playlist Mathematics

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Introduction to Continuous Combinatorics I: the semidefinite method of flag... - Leonardo Coregliano

Computer Science/Discrete Mathematics Seminar II Topic: Introduction to Continuous Combinatorics I: the semidefinite method of flag algebras Speaker: Leonardo Coregliano Affiliation: Member, School of Mathematics Date: November 02, 2021 The field of continuous combinatorics studies lar

From playlist Mathematics

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Alin Bostan: Computer algebra for lattice path combinatorics

Classifying lattice walks in restricted lattices is an important problem in enumerative combinatorics. Recently, computer algebra has been used to explore and to solve a number of difficult questions related to lattice walks. We give an overview of recent results on structural properties a

From playlist Combinatorics

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Doron Zeilberger: Using symbolic dynamical programming in lattice paths combinatorics

CIRM HYBRID EVENT Recorded during the meeting "Lattice Paths, Combinatorics and Interactions" the June 25, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians

From playlist Combinatorics

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Rudi Mrazović - An asymptotic for the Hall-Paige conjecture (CMSA Combinatorics Seminar)

Rudi Mrazović presents "An asymptotic for the Hall-Paige conjecture", 17th March 2021 (CMSA Combinatorics Seminar). http://combinatorics-australasia.org/seminars.html

From playlist CMSA Combinatorics Seminar

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The Large Sieve (Lecture 2) by Satadal Ganguly

Program Workshop on Additive Combinatorics ORGANIZERS: S. D. Adhikari and D. S. Ramana DATE: 24 February 2020 to 06 March 2020 VENUE: Madhava Lecture Hall, ICTS Bangalore Additive combinatorics is an active branch of mathematics that interfaces with combinatorics, number theory, ergod

From playlist Workshop on Additive Combinatorics 2020

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Hypergraph matchings and designs – Peter Keevash – ICM2018

Combinatorics Invited Lecture 13.10 Hypergraph matchings and designs Peter Keevash Abstract: We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of d

From playlist Combinatorics

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Introduction to Combinatory Logic – #SoME2

This is Alexander Farrugia's and Giorgio Grigolo's submission to the second 3blue1brown Summer of Math Exposition. #some2 #mathematics #combinators #logic Music: Icelandic Arpeggios – DivKid

From playlist Summer of Math Exposition 2 videos

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The Large Sieve (Lecture 3) by Satadal Ganguly

Program Workshop on Additive Combinatorics ORGANIZERS: S. D. Adhikari and D. S. Ramana DATE: 24 February 2020 to 06 March 2020 VENUE: Madhava Lecture Hall, ICTS Bangalore Additive combinatorics is an active branch of mathematics that interfaces with combinatorics, number theory, ergod

From playlist Workshop on Additive Combinatorics 2020

Related pages

Graph (discrete mathematics) | Set theory | Labelled enumeration theorem | Cycle index | Stirling numbers and exponential generating functions in symbolic combinatorics | Arthur Cayley | Umbral calculus | Enumerative combinatorics | Combinatorics | Combinatorial species | Algebra | Sequence | Daniel Bernoulli | Asymptotic distribution | Disjoint union | Multiset | Stirling numbers of the second kind | Recurrence relation | Analytic Combinatorics | Philippe Flajolet | Set (mathematics) | Union (set theory) | Pólya enumeration theorem | Embedding | Cartesian product | Stirling numbers of the first kind | George Pólya | Random permutation statistics | Srinivasa Ramanujan | Computer algebra | Leonhard Euler | Generating function | Recursion | Combinatorial class