In mathematics, the Jordan鈥揚贸lya 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鈥揚贸lya number because . Every tree has a number of symmetries that is a Jordan鈥揚贸lya number, and every Jordan鈥揚贸lya number arises in this way as the order of an automorphism group of a tree. These numbers are named after Camille Jordan and George P贸lya, 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鈥揚贸lya numbers begins: [4]
They form the smallest multiplicatively closed set containing all of the factorials.
The th Jordan鈥揚贸lya 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鈥揚贸lya numbers up to obeys the inequalities [5]
Every Jordan鈥揚贸lya 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鈥揚贸lya 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鈥揚贸lya number, it also has the representation . [4] [6]
In mathematics, the Jordan鈥揚贸lya 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鈥揚贸lya number because . Every tree has a number of symmetries that is a Jordan鈥揚贸lya number, and every Jordan鈥揚贸lya number arises in this way as the order of an automorphism group of a tree. These numbers are named after Camille Jordan and George P贸lya, 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鈥揚贸lya numbers begins: [4]
They form the smallest multiplicatively closed set containing all of the factorials.
The th Jordan鈥揚贸lya 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鈥揚贸lya numbers up to obeys the inequalities [5]
Every Jordan鈥揚贸lya 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鈥揚贸lya 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鈥揚贸lya number, it also has the representation . [4] [6]