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Pentagonal number theorem
The pentagonal number theorem is an identity of Leonhard Euler stating that ∏(1 − x^m) equals a series with generalized pentagonal-number exponents, underlying his recurrence for the partition function.
Pentagram
A pentagram, also called a pentacle or pentalpha, is a five-pointed star polygon {5/2} whose points each measure 36 degrees and which is filled with golden-ratio proportions.
Pentomino
Pentomino: a polyomino of five edge-connected squares; 12 free, 18 one-sided, 63 fixed forms named F–Z by Solomon Golomb (1953). The 12 pieces tile 6×10 rectangles in 2,339 ways and inspired games like Blokus and Tetris.
Percentage
A percentage, from the Latin per centum, expresses a number as a fraction of 100; 100% equals the whole, and percentages let proportions from different totals be compared.
Percentile
A percentile, also known as a centile, is a value below which a given percentage of ordered data falls; coined by Francis Galton in 1885, percentiles are widely used in child growth charts and educational testing.
Percentile rank
A percentile rank is the percentage of scores in a comparison group that fall at or below a given score; a rank of 77 means 77% scored lower.
Percolation
Percolation is a mathematical model of random media, introduced by Broadbent and Hammersley in 1957, centered on a critical threshold where isolated clusters become system-spanning.
Percolation theory
Percolation theory is the mathematical study of connectivity in random media, founded by Broadbent and Hammersley in 1957, where an infinite connected cluster first appears at a critical probability.
Perfect field
A perfect field is a field in which every irreducible polynomial is separable, meaning it has no repeated roots in any field extension; Steinitz introduced the concept in 1910.
Perfect graph
A perfect graph is a graph in which, for every induced subgraph, the chromatic number equals the size of its largest clique, a property introduced by Claude Berge.
Perfect number
A perfect number is a positive integer that equals the sum of its proper divisors, as in 6 = 1 + 2 + 3 and 28.
Perfect numbers
A perfect number is a positive integer equal to the sum of its proper divisors, such as 6 and 28; every even one arises from a Mersenne prime.
Perfect power
A perfect power is a positive integer that can be written as m^k, with m > 1 and k at least 2, such as 4, 8, 9, and 27.
Perimeter
Perimeter is the total length of the boundary of a closed geometric figure; for a polygon it is the sum of side lengths, and for a circle it is the circumference.
Perimeter of an ellipse
The perimeter of an ellipse is the total length of its boundary, given exactly by 4a times the complete elliptic integral and usually computed with Ramanujan's 1914 approximation formulas.
Period mapping
A period mapping assigns to each point in the base of a family of algebraic varieties the cohomology of the fiber, viewed as a point in a moduli space of Hodge structures.
Period-doubling bifurcation
A period-doubling bifurcation, also called a flip bifurcation, is a local bifurcation where a periodic orbit loses stability and a new orbit of twice the period appears.
Periodic function
A periodic function, also called a periodic signal or periodic wave, repeats its values at regular intervals, like sine and cosine, which repeat every 2π radians.
Permanent (mathematics)
The permanent of a square matrix sums the product of entries over all permutations, like a determinant but with every term positive; computing it is #P-complete.
PERMANOVA
PERMANOVA (permutational multivariate analysis of variance) is a non-parametric method testing whether two or more groups differ in centroid location, introduced by Marti J. Anderson in 2001.
Permutation
A permutation is a rearrangement of the elements of a set, or a one-to-one mapping of a set onto itself; there are n! of n elements.
Permutation group
A permutation group is a group whose elements are permutations of a set, with composition as the operation; by Cayley's theorem, every group is isomorphic to one.
Permutation matrix
A permutation matrix is a square matrix with exactly one 1 in each row and each column, representing permutations of n objects; multiplying by it reorders rows or columns.
Permutation model
A permutation model, also called a Fraenkel–Mostowski model, is a model of Zermelo–Fraenkel set theory with atoms built by symmetric groups of permutations, devised to prove the Axiom of Choice independent.
Permutation notation
Permutation notation is the collection of mathematical symbols for denoting and counting permutations, including the factorial n!, the count nPr, and two-line and one-line notations.
Permutation polynomial
A permutation polynomial is a polynomial over a finite field, usually, that induces a bijection on the field's elements, with applications in cryptography, coding theory, and combinatorial design.
Permutation representation
A permutation representation is a way of realizing a group's elements as permutations of a set, equivalently a group action of the group on that set.
Permutation test
A permutation test is a nonparametric method of hypothesis testing in which the observed test statistic is compared against the distribution obtained by rearranging, or permuting, the data.
Permutohedron
The permutohedron is a convex polytope whose vertices are all permutations of a vector with distinct coordinates, a higher-dimensional generalization of the hexagon central to geometric combinatorics.
Perpendicular
Perpendicular describes two lines, segments, rays, vectors, or planes that meet at a right angle of 90 degrees; in vector terms, perpendicular vectors have a dot product of zero.