mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 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 (n-1) \times (n-2) \t ...

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, including only woody plants with secondary growth, plants that are ...

has a number of

symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order 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 (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian 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 ...

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

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

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 *Exp ...

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

transitive orientation Transitivity or transitive may refer to: Grammar * Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects * Transitive verb, a verb which takes an object * Transitive case, a grammatical case to mark a ...

s of

comparability graph In graph 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, partially orderable graph ...

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 calle ...

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 Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

\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 (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

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)