Jordan–Pólya number

In mathematics, the Jordan–Pólya numbers are the numbers that can be obtained by multiplying together one or more factorials, not required to be distinct from each other.

Every tree has a number of symmetries that is a Jordan–Pólya number, and every Jordan–Pólya number arises in this way as the order of an automorphism group of a tree.

These numbers are named after Camille Jordan and George Pólya, who both wrote about them in the context of symmetries of trees.

[1][2] These numbers grow more quickly than polynomials but more slowly than exponentials.

As well as in the symmetries of trees, they arise as the numbers of transitive orientations of comparability graphs[3] and in the problem of finding factorials that can be represented as products of smaller factorials.

The sequence of Jordan–Pólya numbers begins:[4] They form the smallest multiplicatively closed set containing all of the factorials.

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

, but more slowly than any exponential function of

, and every sufficiently large

), the number

of Jordan–Pólya numbers up to

obeys the inequalities[5]

Every Jordan–Pólya number

, except 2, has the property that its factorial

can be written as a product of smaller factorials.

This can be done simply by expanding

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

It is conjectured, but unproven, that the only numbers

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

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

in the product expansion of

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