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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 ...
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 ...
A k-combination of a set S is a k-element subset of S: the elements of a combination are not ordered. Ordering the k-combinations of S in all possible ways produces the k-permutations of S. The number of k-combinations of an n-set, C(n,k), is therefore related to the number of k-permutations of n by: (,) = (,) (,) = _! =!
Moreover it can be laborious to find m and n such that = while using it is enough to find all the () to obtain all Pythagorean triples. In particular if we need to find all primitive Pythagorean triples that involve a predetermined positive integer x now we can use only the d ∈ C ( x ) {\displaystyle d\in C(x)} that satisfy the conditions ( 2 ).
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 ...
The change-making problem addresses the question of finding the minimum number of coins (of certain denominations) that add up to a given amount of money. It is a special case of the integer knapsack problem, and has applications wider than just currency. It is also the most common variation of the coin change problem, a general case of ...
A truth table is a structured representation that presents all possible combinations of truth values for the input variables of a Boolean function and their corresponding output values. A function f from A to F is a special relation, a subset of A×F, which simply means that f can be listed as a list of input-output pairs. Clearly, for the ...
For the computer science data structure, see Multiset (abstract data type). In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, [ 1 ] allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the ...