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Combinatorial Modelling
Combinatorial modelling is the process which lets us identify a suitable mathematical model to reformulate a problem. These combinatorial models will provide, through the combinatorics theory, the operations needed to solve the problem. Implicit combinatorial models Simple combinatorial problems are the ones that can be solved by applying just one combinatorial operation (variations, permutations, combinations, …). These problems can be classified into three different models, called implicit combinatorial models. Selection A selection problem requires to choose a sample of ''k'' elements out of a set of ''n'' elements. It is needed to know if the order in which the objects are selected matters and whether an object can be selected more than once or not. This table shows the operations that the model provides to get the number of different samples for each of the selections: Examples ''1.- At a party there are 50 people. Everybody shakes everybody’s hand once. How often ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Numbers Of The Second Kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \left\. Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. Stirling numbers of the second kind are one of two kinds of Stirling numbers, the other kind being called Stirling numbers of the first kind (or Stirling cycle numbers). Mutually inverse (finite or infinite) triangular matrices can be formed from the Stirling numbers of each kind according to the parameters ''n'', ''k''. Definition The Stirling numbers of the second kind, written S(n,k) or \lbrace\textstyle\rbrace or with other notations, count the number of ways to partition a set of n labelled objects into k nonempty unlabelled subsets. Equivalently, they count the number of different equivalence relations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lah Number
In mathematics, the Lah numbers, discovered by Ivo Lah in 1954, are coefficients expressing rising factorials in terms of falling factorials. They are also the coefficients of the nth derivatives of e^. Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of ''n'' elements can be partitioned into ''k'' nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers. Unsigned Lah numbers : : L(n,k) = \frac. Signed Lah numbers : : L'(n,k) = (-1)^n \frac. ''L''(''n'', 1) is always ''n''!; in the interpretation above, the only partition of into 1 set can have its set ordered in 6 ways: :, , , , or ''L''(3, 2) corresponds to the 6 partitions with two ordered parts: :, , , , or ''L''(''n'', ''n'') is always 1 since, e.g., partitioning into 3 non-empty subsets results in subsets of length 1. : Adapting the Karamata– Knuth notation for Stirling numbers, it has been proposed to use the following alterna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twelvefold Way
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number. The idea of the classification is credited to Gian-Carlo Rota, and the name was suggested by Joel Spencer. Overview Let and be finite sets. Let n=, N, and x=, X, be the cardinality of the sets. Thus is an -set, and is an -set. The general problem we consider is the enumeration of equivalence classes of functions f: N \to X. The functions are subject to one of the three following restrictions: # No condition: each in may be sent by to any in , and each may occur multiple times. # is injective: each value f(a) for in must be distinct from every other, and so each in may occur at most once in the image of . # is surjective: for each in there must be at least one in such that f(a) = b, thus each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
