A tree that (as an abstract
graph) has 480 symmetries (automorphisms). There are 2 ways of permuting the two children of the upper left vertex, 2 ways of permuting the two children of the upper middle vertex, and 5! = 120 ways of permuting the five children of the upper right vertex, for 2 · 2 · 120 = 480 symmetries altogether.
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. For instance, is a Jordan–Pólya number because . 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 th Jordan–Pólya number grows more quickly than any polynomial of , but more slowly than any exponential function of . More precisely, for every , and every sufficiently large (depending on ), the number of Jordan–Pólya numbers up to obeys the
inequalities[5]
Factorials that are products of smaller factorials
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 and then replacing in this product by its representation as a product of factorials. It is
conjectured, but
unproven, that the only numbers whose factorial equals a product of smaller factorials are the Jordan–Pólya numbers (except 2) and the two exceptional numbers 9 and 10, for which and . The only other known representation of a factorial as a product of smaller factorials, not obtained by replacing in the product expansion of , is , but as is itself a Jordan–Pólya number, it also has the representation .[4][6]