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, 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]

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.

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.

Sequence and growth rate

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The sequence of Jordan–Pólya numbers begins:[4]

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, ... (sequence A001013 in the OEIS)

They form the smallest multiplicatively closed set containing all of the 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

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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]

References

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  1. ^ Jordan, Camille (1869), "Sur les assemblages de lignes", Journal für die reine und angewandte Mathematik, 1869 (70): 185–190, doi:10.1515/crll.1869.70.185, S2CID 119829832
  2. ^ Pólya, George (1937), "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen", Acta Mathematica, 68: 145–254, doi:10.1007/BF02546665, S2CID 121878844
  3. ^ Golumbic, Martin Charles (1977), "Comparability graphs and a new matroid", Journal of Combinatorial Theory, Series B, 22 (1): 68–90, doi:10.1016/0095-8956(77)90049-1, MR 0439689
  4. ^ a b Sloane, N. J. A. (ed.), "Sequence A001013 (Jordan-Polya numbers: products of factorial numbers)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
  5. ^ De Koninck, Jean-Marie; Doyon, Nicolas; Razafindrasoanaivolala, A. Arthur Bonkli; Verreault, William (2020), "Bounds for the counting function of the Jordan-Pólya numbers", Archivum Mathematicum, 56 (3): 141–152, arXiv:2107.09114, doi:10.5817/am2020-3-141, MR 4156441, S2CID 226661345
  6. ^ Guy, Richard K. (2004), "B23: Equal products of factorials", Unsolved problems in number theory, Problem Books in Mathematics, vol. 1 (3rd ed.), New York: Springer-Verlag, p. 123, doi:10.1007/978-0-387-26677-0, ISBN 0-387-20860-7, MR 2076335