Using GENERATINGFUNCTIONOLOGY to Enumerate Distinct-Multiplicity Partitions

Doron Zeilberger

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(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and

Written: Jan. 18, 2012

Last Update of this page (but not article): Sept. 11, 2012 [Thanks to Daniel Kane and Robert Rhoades]

Previous Updates: Feb. 1, 2012 [thanks to Daniel Kane], Feb. 4, 2012 (page and article) [Thanks to Vaclav Kotesovec], Feb. 8, 2012 [Thanks to Mark Ward]

In Fond Memory of

Herbert Saul WILF (June 13, 1931- Jan. 7, 2012),

Who asked so many good questions,

and knew so well what is an ANSWER

ל ע ל ו י   נ ש מ ת ו   ש ל

ה ר ב ר ט   ש א ו ל   ו י ל ף

( כ" ח   ס י ו ן   ה' ת ר צ"א - י"ב   ט ב ת   ה' ת ש ע"ב )

ז כ ר   ג א ו ן   ל ב ר כ ה

ש ש א ל   ש א ל ו ת   כ ל   כ ך   ט ו ב ו ת

ו י ד ע   ה י ט ב   מ ה   ז א ת   ת ש ו ב ה

Added Feb. 1, 2012: Daniel Kane seems to have found the true asymptotics of log f(n), see the end of the current version of the article. As soon as Daniel would write it up, I will put a link to it here. Meanwhile, he kindly allowed me to post his Email message .

Added Feb. 4, 2012: According to Vaclav Kotesovec's numerics, Daniel Kane's proposed asymptotics does not seem to quite fit, as stated (but the n1/3 part may still be right). See Vaclav Kotesovec's graph .

Added Feb. 8, 2012: Mark Ward, in collaboration with Jim Fill and Svante Janson also found (private communication, see Mark Ward's Email message) the same asymptotics as Daniel Kane (so it must be correct, apparently 508 terms do not suffice to see what is going on exactly) and are working on refined asymptotics. I hope that they would join forces, and as soon as the article is posted in the arxiv (and/or their websites), I will link to it here.

Added Sept. 11, 2012: Daniel Kane and Robert Rhoades found a more refined asymptitics!

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