# Enumerative combinatorics

Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions. The simplest such functions are closed formulas, which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck of n cards is f(n) = n!. The problem of finding a closed formula is known as algebraic enumeration, and frequently involves deriving a recurrence relation or generating function and using this to arrive at the desired closed form. Often, a complicated closed formula yields little insight into the behavior of the counting function as the number of counted objects grows. In these cases, a simple asymptotic approximation may be preferable. A function is an asymptotic approximation to if as . In this case, we write (Wikipedia).

Christian Krattenthaler - Determinants and Pfaffians in Enumerative Combinatorics (2011)

Slides for this talk: http://www.mat.univie.ac.at/~kratt/vortrag/combdet.pdf Abstract: In this talk I shall explain why many enumerative combinatorialists are fascinated by determinants — obviously from a strongly biased personal perspective. The particular sources where determinants ari

From playlist Mathematics

Algebraic and Convex Geometry of Sums of Squares on Varieties (Lecture 1) by Greg Blekherman

PROGRAM COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS: Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE & TIME: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is t

Cyril Banderier - Enumeration and Generation of Young Tableaux with Walls: the Density Method

We consider a generalization of Young tableaux in which we allow some consecutive pairs of cells with decreasing labels, conveniently visualized by a ”wall” between the corresponding cells. This leads to new classes of recurrences, and to a surprisingly rich zoo of generating functions (al

Large deviations for random hives and the spectrum of the sum of two random.. by Hariharan Narayanan

PROGRAM COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS: Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE & TIME: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is t

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

Circular Fence Posets and Associated Polytopes with Unexpected Symmetry by Mohan Ravichandran

PROGRAM COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS: Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE & TIME: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is t

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

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 2/10

23 janvier 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, or

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 1/10

23 janvier 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, or

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 3/10

30 janvier 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, or

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 4/10

30 janvier 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, or

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

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 5/10

6 février 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, ori

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 9/10

27 février 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, or

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 10/10

27 février 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, or

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 6/10

06 février 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, or

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 8/10

13 février 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, or

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

Enumerative Combinatorics of Maps - CEB T1 2017 - Bousquet-Mélou - 7/10

13 février 2017 2017 - T1 - Combinatorics and interactions - CEB Trimester Mireille Bousquet-Mélou (Labri) ENUMERATIVE COMBINATORICS OF MAPS Keywords: planar maps, recursive structures, bijections with trees. Maps equipped with an additional structure (forests, self-avoiding walks, or