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), 12411254.
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 thirdthrough
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 scaledlimit of the 3rdmoment (skewness) is
(396793/390815488)(467*7680)^{1/2}= 1.92278748.., the that the scaledlimit of the 4thmoment (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 kth 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

P_{9,10}

P_{13,15}

If you want to see what is left of the lattice P_{13,15} when the vertices labeled 1,.., 2i3 are
occupied, but 2i1 is unoccupied (implying that 2,..., 2i2 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,.., 2i3 are
occupied (implying that 2,..., 2i2 are unocuppied), but 2i1 is unoccupied, and hence lots of vertices disappear,
for i=1 to i=8, look here .
Articles of Doron Zeilberger
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