In combinatorial mathematics a 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 be simply read off from the coefficients and exponents. This compact way of storing information in an algebraic form is frequently used in combinatorial enumeration.
Each permutation π of a finite set of objects partitions that set into cycles; the cycle index monomial of π is a monomial in variables a1, a2, … that describes the cycle type of this partition: the exponent of ai is the number of cycles of π of size i. The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also sometimes used in place of cycle index.
Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group's action. This is the main ingredient in the Pólya enumeration theorem. Performing formal algebraic and differential operations on these polynomials and then interpreting the results combinatorially lies at the core of species theory.
In combinatorial mathematics a cycleindex is a polynomial in several variables which is structured in such a way that information about how a group of...
cyclic permutations takes a set A to the set of all cycles on A. Removing a single element from a cycle reduces it to a list: C' = L. We can integrate the...
number of cycle types of a set with n elements equals the value of the partition function p ( n ) {\displaystyle p(n)} . Polya's cycleindex polynomial...
to measure the business cycle: consumer confidence index, retail trade index, unemployment and industry/service production index. Stock and Watson claim...
of the variation in national output over the duration of the business cycle. Index of industrial production (India) "Industrial Production and Capacity...
The solar cycle, also known as the solar magnetic activity cycle, sunspot cycle, or Schwabe cycle, is a nearly periodic 11-year change in the Sun's activity...
The estrous cycle (from Latin oestrus 'frenzy', originally from Ancient Greek οἶστρος (oîstros) 'gadfly') is a set of recurring physiological changes...
recurring cycles. See also Index of wave articles, Time, and Pattern. Astronomy – Axial precession – CNO cycle – Eclipse cycle – Eclipse – Full moon cycle – Galactic...
in various phases of cell cycle, and the immunohistochemical evaluation of cell cycle-associated proteins. Mitotic indexing is the oldest method of assessing...
The Corruption Perceptions Index (CPI) is an index that ranks countries "by their perceived levels of public sector corruption, as determined by expert...
Cell Cycle is a biweekly peer-reviewed scientific journal covering all aspects of cell biology. It was established in 2002 with Mikhail V. Blagosklonny...
The Fenian Cycle (/ˈfiːniən/), Fianna Cycle or Finn Cycle (Irish: an Fhiannaíocht) is a body of early Irish literature focusing on the exploits of the...
before the Great Depression than it has been since the end of World War II." Cycles in the country's agricultural production, industrial production, consumption...
In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any...
Index is an index prepared by the World Economic Forum. The inaugural index from 2020 ranked 82 countries and has not been updated since. The Index measures...
The Hainish Cycle consists of a number of science fiction novels and stories by Ursula K. Le Guin. It is set in a future history in which civilizations...
information are needed to calculate a Pearl Index for a particular study: the total number of months or cycles of exposure by women in the study the number...
credit cycle indices. At this point, to determine a TTC PD, one follows three steps: Converting the PIT PD to PIT DD Subtracting the credit cycleindex from...
internet penetration, retail sales, price index, and changes in credit conditions. The leading business cycle dating committee in the United States of...
The carbon cycle is that part of the biogeochemical cycle by which carbon is exchanged among the biosphere, pedosphere, geosphere, hydrosphere, and atmosphere...
Solar cycle 25 is the current solar cycle, the 25th since 1755, when extensive recording of solar sunspot activity began. It began in December 2019 with...
media related to Dihedral groups. Coordinate rotations and reflections Cycleindex of the dihedral group Dicyclic group Dihedral group of order 6 Dihedral...
Global Information