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

• Armin.txt, a Maple package to study the statistical distribution of the random variable "size" defined on (2n+1,2n+3)-core partitions with distinct parts.

• core.txt, that uses the Dyck paths approach to compute the Straub polynomials, but takes much longer. On the other hand it can handle arbitrary (s,t)-core partitions with distinct parts (when s and t are relatively prime), while Armin.txt only treats the case (s,s+2) with s odd.

# Sample Input and Output for Armin.txt

• If you want to see an experimentl math (easily rigorizable) proof of Armin Straub's conjecture 1.1 made here and first proved by Yan et. al. here

the input file generates the output file.

• If you want to see an experimentl math (new) proof of Armin Straub's conjecture 1.2 made here and first proved by Yan et. al. here

the input file generates the output file.

• If you want to see the first 21 Straub polynomials (the k-th Straub polynomial, S[k](q), is the generating functions for the set of (2n+1,2n+3)-core partitions with distinct parts according to size)

the input file generates the output file.

• If you want to see explicit expressions for the mean variance, and the third through 6th moments about the mean, as well as the limits of the scaled moments up to the 6th,

the input file generates the output file.

# Sample Input and Output for core.txt

• If you want to see the first eight Straub polynomials, using the Dyck paths approach,

the input file generates the output file.

[As you can see, they agree with those generated by the new method]

# Some Pictures of the Lattices Involved

• If you want to see what is left of the lattice P13,15 when the vertices labeled 1,.., 2i-3 are occupied, but 2i-1 is unoccupied (implying that 2,..., 2i-2 are unocuppied, and hence lots of vertices disapper), for i=1 to i=7, look   here   .

• If you want to see the lattices EO(6,a) for a=1,2,3,4,5, click   here   .

• If you want to see the lattices OE(6,a) for a=1,2,3,4,5,6 click   here   .

• If you want to see the lattice EO(7,8) and what is left of it, when the vertices labeled 1,.., 2i-3 are occupied (implying that 2,..., 2i-2 are unocuppied), but 2i-1 is unoccupied, and hence lots of vertices disappear, for i=1 to i=8, look   here   .

Articles of Doron Zeilberger