Explicit (Polynomial!) Expressions for the Expectation, Variance and Higher Moments of the Size of of a (2n+1,2n+3)-core partition with Distinct parts

By Anthony Zaleski and Doron Zeilberger

.pdf            latex  

published, in J. Difference Equations and Applications Volume 23 (2017), 1241-1254.

First Written: Nov. 16, 2016

This version: Dec. 6, 2016

Abstract: Armin Straub's beautiful article concludes with two inriguing conjectures about the number, and maximal size, of (2n+1,2n+3)-core partitions with distinct parts. This was proved by an ingenious, but complicated, argument by Sherry H.F. Yan, Guizhi Qin, Zemin Jin, Robin D.P. Zhou. In the present article, we first comment that these conjectures can be proved faster by "experimental mathematics" methods, that are easily rigorizable, and then go on to state explicit expressions for the mean, variance, and the third-through the 7th moments (about the mean) of the random varialble "size" defined on (2n+1,2n+3)-core partitions with distinct parts. In particular we show that this random varaible is not asympotically normal, and the limit of the coefficient of variation is (14010)1/2/150 = 0.789092305..., the scaled-limit of the 3rd-moment (skewness) is (396793/390815488)(467*7680)1/2= 1.92278748.., the that the scaled-limit of the 4th-moment (kurtosis) is 145309380/16792853= 8.6530490.... We are offering to donate 100 dollars to the OEIS foundation in honor of the first to identify the limiting distribution.

Added Nov. 30, 2016: Read Armin Straub's insightful remarks  

Maple packages

Sample Input and Output for Armin.txt

Sample Input and Output for core.txt

Some Pictures of the Lattices Involved

Articles of Doron Zeilberger

Doron Zeilberger's Home Page

Anthony Zaleski's Home Page