Combinatorics | Factorial and binomial topics
In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations. The combinations are represented as strictly decreasing sequences ck > ... > c2 > c1 ≥ 0 where each ci corresponds to the index of a chosen element in a given k-combination. Distinct numbers correspond to distinct k-combinations, and produce them in lexicographic order. The numbers less than correspond to all k-combinations of {0, 1, ..., n − 1}. The correspondence does not depend on the size n of the set that the k-combinations are taken from, so it can be interpreted as a map from N to the k-combinations taken from N; in this view the correspondence is a bijection. The number N corresponding to (ck, ..., c2, c1) is given by . The fact that a unique sequence corresponds to any non-negative number N was first observed by D. H. Lehmer. Indeed, a greedy algorithm finds the k-combination corresponding to N: take ck maximal with , then take ck−1 maximal with , and so forth. Finding the number N, using the formula above, from the k-combination (ck, ..., c2, c1) is also known as "ranking", and the opposite operation (given by the greedy algorithm) as "unranking"; the operations are known by these names in most computer algebra systems, and in computational mathematics. The originally used term "combinatorial representation of integers" was shortened to "combinatorial number system" by Knuth,who also gives a much older reference;the term "combinadic" is introduced by James McCaffrey (without reference to previous terminology or work). Unlike the factorial number system, the combinatorial number system of degree k is not a mixed radix system: the part of the number N represented by a "digit" ci is not obtained from it by simply multiplying by a place value. The main application of the combinatorial number system is that it allows rapid computation of the k-combination that is at a given position in the lexicographic ordering, without having to explicitly list the k-combinations preceding it; this allows for instance random generation of k-combinations of a given set. Enumeration of k-combinations has many applications, among which are software testing, sampling, quality control, and the analysis of lottery games. (Wikipedia).
Graphing the system of two linear inequalities with two horizontal line
👉 Learn how to graph a system of inequalities. A system of inequalities is a set of inequalities which are collectively satisfied by a certain range of values for the variables. To graph a system of inequalities, each inequality making up the system is graphed individually with the side of
From playlist Solve a System of Inequalities by Graphing
How to graph the system of linear inequalities of one horizontal and one vertical
👉 Learn how to graph a system of inequalities. A system of inequalities is a set of inequalities which are collectively satisfied by a certain range of values for the variables. To graph a system of inequalities, each inequality making up the system is graphed individually with the side of
From playlist Solve a System of Inequalities by Graphing
Graphing a system of linear inequalities
👉 Learn how to graph a system of inequalities. A system of inequalities is a set of inequalities which are collectively satisfied by a certain range of values for the variables. To graph a system of inequalities, each inequality making up the system is graphed individually with the side of
From playlist Solve a System of Inequalities by Graphing
How to graph and shade a system of linear inequalities
👉 Learn how to graph a system of inequalities. A system of inequalities is a set of inequalities which are collectively satisfied by a certain range of values for the variables. To graph a system of inequalities, each inequality making up the system is graphed individually with the side of
From playlist Solve a System of Inequalities by Graphing
Learn how to graph a system of linear inequalities of two vertical boundary lines
👉 Learn how to graph a system of inequalities. A system of inequalities is a set of inequalities which are collectively satisfied by a certain range of values for the variables. To graph a system of inequalities, each inequality making up the system is graphed individually with the side of
From playlist Solve a System of Inequalities by Graphing
How to graph a system of linear inequalities in slope intercept form
👉 Learn how to graph a system of inequalities. A system of inequalities is a set of inequalities which are collectively satisfied by a certain range of values for the variables. To graph a system of inequalities, each inequality making up the system is graphed individually with the side of
From playlist Solve a System of Inequalities by Graphing
Solve a system of inequalities with vertical and horizontal lines
👉 Learn how to graph a system of inequalities. A system of inequalities is a set of inequalities which are collectively satisfied by a certain range of values for the variables. To graph a system of inequalities, each inequality making up the system is graphed individually with the side of
From playlist Solve a System of Inequalities by Graphing
Marian Mrozek (8/30/21): Combinatorial vs. Classical Dynamics: Recurrence
The study of combinatorial dynamical systems goes back to the seminal 1998 papers by Robin Forman. The main motivation to study combinatorial dynamics comes from data science. Combinatorial dynamics also provides very concise models of dynamical phenomena. Moreover, some topological invari
From playlist Beyond TDA - Persistent functions and its applications in data sciences, 2021
Graphing a system of linear inequalities with a feasible solution
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From playlist Solve a System of Inequalities by Graphing
Aaron Sidford: Introduction to interior point methods for discrete optimization, lecture I
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Konstantin Mischaikow (8/28/21): Solving systems of ODEs via combinatorial homological algebra
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From playlist Beyond TDA - Persistent functions and its applications in data sciences, 2021
Matrix Ansatz and Algebraic Bethe Ansatz for the Exclusion Process by Kirone Mallick
PROGRAM URL : http://www.icts.res.in/program/NESP2015 DATES : Monday 26 Oct, 2015 - Friday 20 Nov, 2015 VENUE : Ramanujan Lecture Hall, ICTS Bangalore DESCRIPTION : This program will be organized as an advanced discussion workshop on some topical issues in nonequilibrium statstical phys
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Masayuki Ohzeki: "Quantum annealing and machine learning - new directions of quantum"
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From playlist Solve a System of inequalities by Graphing | Standard Form
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Stochastic rewriting systems evolving over graph-like structures are a versatile modeling paradigm that covers in particular biochemical reaction systems. In fact, to date rewriting-based frameworks such as the Kappa platform [1] are amongst the very few known approaches to faithfully enco
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👉 Learn how to graph a system of inequalities. A system of inequalities is a set of inequalities which are collectively satisfied by a certain range of values for the variables. To graph a system of inequalities, each inequality making up the system is graphed individually with the side of
From playlist Solve a System of inequalities by Graphing | Standard Form