Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-15T23:08:37.971Z Has data issue: false hasContentIssue false

14 - Asymptotic formulas for partitions with bounded multiplicity

Published online by Cambridge University Press:  18 December 2014

Pierre Liardet
Affiliation:
Aix Marseille University, Marseille
Alain Thomas
Affiliation:
Aix Marseille University, Marseille
Gerhard Larcher
Affiliation:
Johannes Kepler Universität Linz
Friedrich Pillichshammer
Affiliation:
Johannes Kepler Universität Linz
Arne Winterhof
Affiliation:
Austrian Academy of Sciences, Linz
Chaoping Xing
Affiliation:
Nanyang Technological University, Singapore
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] F. C., Auluck and C. B., Haselgrove, On Ingham's Tauberian theorem for partitions. Proc. Cambridge Philos. Soc. 48, 566–570, 1952.Google Scholar
[2] W. De Azevedo, Pribitkin, Revisiting Rademacher's formula for the partition function p(n). Ramanujan J. 4, 455–467, 2000.Google Scholar
[3] N. A., Brigham, A general asymptotic formula for partitions. Proc. Am. Math. Soc. 1, 182–191, 1950.Google Scholar
[4] N. G. de, Bruijn, On Mahler's partition problem. Indag. Math. 10, 210–220, 1948.Google Scholar
[5] J.-M., Dumont, N., Sidorov and A., Thomas, Number of representations related to linear recurrent basis. Acta Arith. 88(4), 371–396, 1999.Google Scholar
[6] M., Dutta, On new partition of numbers. Rend. Sem. Mat. Univ. Padova 25, 138–143, 1956.Google Scholar
[7] P., Erdős, On an elementary proof of some asymptotic formulae in the theory of partitions. Ann. Math. (2) 43, 437–450, 1942.Google Scholar
[8] P., Erdős and B., Richmond, Concerning periodicity in the asymptotic behavior of partition functions. J. Aust. Math. Soc. Ser. A 21, 447–456, 1976.Google Scholar
[9] P., Erdős and B., Richmond, Partitions into summands of the form [mα]. Proceedings of the Seventh Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1977), pp. 371–377. Utilitas Math., Winnipeg, Man., 1978.
[10] D.-J., Feng, P., Liardet and A., Thomas, Partition functions in numeration systems with bounded multiplicity. Unif. Distrib. Theory 9(1), 43–77, 2014.Google Scholar
[11] P., Flajolet and R., Sedgewick, Analytic Combinatorics. Cambridge University Press, Cambridge, 2009.
[12] I. S., Gradshteyn and I. M., Ryzhik, Table of Integrals, Series, and Products, seventh edition. Academic Press, San Diego, CA, 2007.
[13] P., Hagis, Jr., Partitions with a restriction on the multiplicity of the summands. Trans. Am. Math. Soc. 155, 375–384, 1971.Google Scholar
[14] G. H., Hardy and S., Ramanujan, Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. Ser. 2 17, 75–115, 1918.Google Scholar
[15] L.-K., Hua, On the number of partitions of a number into unequal parts. Trans. Am. Math. Soc. 51, 939–961, 1942.Google Scholar
[16] A. E., Ingham, A Tauberian theorem for partition. Ann. Math. 42(5), 1075–1090, 1941.Google Scholar
[17] K., Mahler, On a special functional equation. J. London Math. Soc. 15, 115–123, 1940.Google Scholar
[18] G., Menardus, Asymptotische Aussagen über Partitionen. Math. Z. 59, 388–398, 1954.Google Scholar
[19] D. J., Newman, The evaluation of the constant in the formula for the number of partitions of n. Am. J. Math. 73, 599–601, 1951.Google Scholar
[20] W. B., Pennington, On Mahler's partition problem. Ann. Math. 57(3), 531–546, 1953.Google Scholar
[21] V., YuProtasov, Asymptotic behavior of the partition function. Sb. Math. 191, 381–414, 2000.Google Scholar
[22] H., Rademacher, On the expansion of the partition function in a series, Ann. Math. 44, 416–422, 1943.Google Scholar
[23] B., Reznick, Some binary partition functions. Analytic Number Theory (AllertonPark, IL, 1989). Progress in Mathematics, volume 85, pp. 451–47. Birkhäuser, Boston, MA, 1990.
[24] L. B., Richmond, Asymptotic relations for partitions. J. Number Theory 7, 389–405, 1975.Google Scholar
[25] B., Richmond, Mahler's partition problem. Ars Combin. 2, 169–189, 1976.Google Scholar
[26] K. F., Roth and G., Szekeres, Some asymptotic formulae in the theory of partitions. Q. J. Math. Oxford 5(2), 241–259, 1954.Google Scholar
[27] W., Schwartz, Einige Anwendungen Tauberscher Sätze in der Zahlentheorie. C: Mahler's Partitionsproblem. J. Reine Angew. Math. 228, 182–188, 1967.Google Scholar
[28] H. N. V., Temperly, Statistical mechanics and the partition of numbers I. The transition of liquid helium. Proc. R. Soc. London 109, 361–375, 1949.Google Scholar
[29] E. M., Wright, Asymptotic partition formulae, III. Partitions into kth powers. Acta Math. 63, 143–191, 1934.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×