Search results
Results From The WOW.Com Content Network
An n-tuple can be formally defined as the image of a function that has the set of the n first natural numbers as its domain. Tuples may be also defined from ordered pairs by a recurrence starting from ordered pairs; indeed, an n-tuple can be identified with the ordered pair of its (n − 1) first elements and its n th element.
An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1. In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing. More generally, an unordered n-tuple is a set of the form {a 1, a 2,... a n}. [5] [6] [7]
The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, always equals the unordered pair {b, a}. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional ...
"Ordered" means that the elements of the data type have some kind of explicit order to them, where an element can be considered "before" or "after" another element. This order is usually determined by the order in which the elements are added to the structure, but the elements can be rearranged in some contexts, such as sorting a list. For a ...
The ordered sequential types are lists (dynamic arrays), tuples, and strings. All sequences are indexed positionally (0 through length - 1) and all but strings can contain any type of object, including multiple types in the same sequence. Both strings and tuples are immutable, making them perfect candidates for dictionary keys (see below).
As there are only a finite number of possible n-tuples with these constraints, the sequence of n-tuples must sooner or later repeat itself. Every Ducci sequence therefore eventually becomes periodic. If n is a power of 2 every Ducci sequence eventually reaches the n-tuple (0,0,...,0) in a finite number of steps. [1] [3] [4]
That is, an ordered -tuple is in the same orbit as any other that is a re-ordered version of it. A path in the n {\displaystyle n} -fold symmetric product is the abstract way of discussing n {\displaystyle n} points of X {\displaystyle X} , considered as an unordered n {\displaystyle n} -tuple, independently tracing out n {\displaystyle n} strings.
By taking conjugates, the number p k (n) of partitions of n into exactly k parts is equal to the number of partitions of n in which the largest part has size k. The function p k (n) satisfies the recurrence p k (n) = p k (n − k) + p k−1 (n − 1) with initial values p 0 (0) = 1 and p k (n) = 0 if n ≤ 0 or k ≤ 0 and n and k are not both ...