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We remember that the Euclid’s formulas do not give all Pythagorean triples that involves a predetermined positive integer x, for example the triples (,,), (,,) and (,,). Moreover it can be laborious to find m and n such that x = m 2 − n 2 {\displaystyle x=m^{2}-n^{2}} while using ( 1 ) it is enough to find all the d ∈ C ( x ...
Heap's algorithm generates all possible permutations of n objects. It was first proposed by B. R. Heap in 1963. [ 1 ] The algorithm minimizes movement: it generates each permutation from the previous one by interchanging a single pair of elements; the other n−2 elements are not disturbed. In a 1977 review of permutation-generating algorithms ...
The convolution/sum of probability distributions arises in probability theory and statistics as the operation in terms of probability distributions that corresponds to the addition of independent random variables and, by extension, to forming linear combinations of random variables. The operation here is a special case of convolution in the ...
Combination. In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple ...
and the binomial coefficient is the coefficient of the x2 term. Arranging the numbers in successive rows for n = 0, 1, 2, ... gives a triangular array called Pascal's triangle, satisfying the recurrence relation. The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. In combinatorics the symbol is usually ...
Stars and bars (combinatorics) In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", [ 1 ] "balls and bars", [ 2 ] and "dots and dividers" [ 3 ]) is a graphical aid for deriving certain combinatorial theorems. It can be used to solve many simple counting problems, such as how many ways there are to put n ...
Decision tables are a concise visual representation for specifying which actions to perform depending on given conditions. Decision table is the term used for a Control table or State-transition table in the field of Business process modeling ; they are usually formatted as the transpose of the way they are formatted in Software engineering .
Combinatorial number system. 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.