mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

s, not required to be distinct from each other. For instance,

480

is a Jordan–Pólya number because Every

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only ...

has a number of

symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

of an

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

of a tree. These numbers are named after

Camille Jordan Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...

and

George Pólya George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributi ...

, who both wrote about them in the context of symmetries of trees. These numbers grow more quickly than

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s but more slowly than

exponentials Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...

. As well as in the symmetries of trees, they arise as the numbers of transitive orientations of

comparability graph In graph theory and order theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partial ...

s and in the problem of finding factorials that can be represented as products of smaller factorials.

Sequence and growth rate

The

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

of Jordan–Pólya numbers begins: They form the smallest

multiplicatively closed set In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite ...

containing all of the factorials. The

n

th Jordan–Pólya number grows more quickly than any polynomial of

n

, but more slowly than any exponential function of

n

. More precisely, for every

\varepsilon>0

, and every sufficiently large

x

(depending on

\varepsilon

), the number

J(x)

of Jordan–Pólya numbers up to

x

obeys the inequalities

\exp\frac < J(x) <
\exp\frac.

Factorials that are products of smaller factorials

Every Jordan–Pólya number

n

, except 2, has the property that its factorial

n!

can be written as a product of smaller factorials. This can be done simply by expanding

n!=n\cdot(n-1)!

and then replacing

n

in this product by its representation as a product of factorials. It is

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

d, but unproven, that the only numbers

n

whose factorial

n!

equals a product of smaller factorials are the Jordan–Pólya numbers (except 2) and the two exceptional numbers 9 and 10, for which

9!=2!\cdot3!\cdot3!\cdot7!

and

10!=6!\cdot7!=3!\cdot5!\cdot7!

. The only other known representation of a factorial as a product of smaller factorials, not obtained by replacing

n

in the product expansion of

n!

, is

16!=2!\cdot5!\cdot14!

, but as

16

is itself a Jordan–Pólya number, it also has the representation

16!=2!^4\cdot 15!

References

{{DEFAULTSORT:Jordan-Polya number Integer sequences Factorial and binomial topics Algebraic graph theory Trees (graph theory)

Sequence and growth rate

Factorials that are products of smaller factorials

See also

References