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. For instance, formula_1 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. 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 and in the problem of finding factorials that can be represented as products of smaller factorials. Sequence and growth rate. The sequence of Jordan–Pólya numbers begins: They form the smallest multiplicatively closed set containing all of the factorials. The formula_2th Jordan–Pólya number grows more quickly than any polynomial of formula_2, but more slowly than any exponential function of formula_2. More precisely, for every formula_5, and every sufficiently large formula_6 (depending on formula_7), the  number formula_8 of Jordan–Pólya numbers up to formula_6 obeys the inequalities formula_10 Factorials that are products of smaller factorials. Every Jordan–Pólya number formula_2, except 2, has the property that its factorial formula_12 can be written as a product of smaller factorials. This can be done simply by expanding formula_13 and then replacing formula_2 in this product by its representation as a product of factorials. It is conjectured, but unproven, that the only numbers formula_2 whose factorial formula_12 equals a product of smaller factorials are the Jordan–Pólya numbers (except 2) and the two exceptional numbers 9 and 10 , for which formula_17 and formula_18. The only other known representation of a factorial as a product of smaller factorials, not obtained by replacing formula_2 in the product expansion of formula_12, is formula_21, but as formula_22 is itself a Jordan–Pólya number, it also has the representation formula_23.