In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map is a bijection. In case the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. Over a finite field, every function, so in particular every permutation of the elements of that field, can be written as a polynomial function.
In the case of finite rings Z/nZ, such polynomials have also been studied and applied in the interleaver component of error detection and correction algorithms.[1][2]
^Takeshita, Oscar (2006). "Permutation Polynomial Interleavers: An Algebraic-Geometric Perspective". IEEE Transactions on Information Theory. 53: 2116–2132. arXiv:cs/0601048. doi:10.1109/TIT.2007.896870.
^Takeshita, Oscar (2005). "A New Construction for LDPC Codes using Permutation Polynomials over Integer Rings". arXiv:cs/0506091.
and 26 Related for: Permutation polynomial information
In mathematics, a permutationpolynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map x ↦ g...
symmetric polynomial if for any permutation σ of the subscripts 1, 2, ..., n one has P(Xσ(1), Xσ(2), ..., Xσ(n)) = P(X1, X2, ..., Xn). Symmetric polynomials arise...
of permutations occurred around 1770, when Joseph Louis Lagrange, in the study of polynomial equations, observed that properties of the permutations of...
Newton polynomial Orthogonal polynomials Orthogonal polynomials on the unit circle Permutationpolynomial Racah polynomials Rogers polynomials Rogers–Szegő...
of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group...
mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G...
chess, the impetus for studying rook polynomials is their connection with counting permutations (or partial permutations) with restricted positions. A board...
Chebyshev polynomials. One of the main reasons for interest in them is that for fixed α, they give many examples of permutationpolynomials; polynomials acting...
the permutations of X (i.e. the bijective functions from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If...
entries 0.: 26 An n × n permutation matrix can represent a permutation of n elements. Pre-multiplying an n-row matrix M by a permutation matrix P, forming PM...
in Wiktionary, the free dictionary. QPP may refer to: Quadratic permutationpolynomial Quebec Pension Plan (QPP) Queensland People's Party Queerplatonic...
within a distance of S in the output). a contention-free quadratic permutationpolynomial (QPP). An example of use is in the 3GPP Long Term Evolution mobile...
X_{i}} by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the polynomial – in fact, it is...
cryptography, a pseudorandom permutation (PRP) is a function that cannot be distinguished from a random permutation (that is, a permutation selected at random with...
automorphisms of the symmetric group Sn on the polynomial ring in n indeterminates, where a permutation acts on a polynomial by simultaneously substituting each...
the cycles of some permutation of the Galois group of P. Another example: P being as above, a resolvent R for a group G is a polynomial whose coefficients...
in polynomial time for permutation graphs by using a longest decreasing subsequence algorithm. likewise, an increasing subsequence in a permutation corresponds...
cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can...
of permutations of the numbers 1 to n {\textstyle n} in which exactly k {\textstyle k} elements are greater than the previous element (permutations with...
Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high...
there are n ! {\displaystyle n!} ( n {\displaystyle n} factorial) such permutation operations, the order (number of elements) of the symmetric group S n...
elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed...
and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation...
Separable permutation, a permutation that can be obtained by direct sums and skew sums of the trivial permutation Separable polynomial, a polynomial whose...
the base change matrix P used). Minimal polynomial Frobenius normal form Jordan normal form, up to a permutation of the Jordan blocks Index of nilpotence...
Global Information