Explicit Expressions for the Variance and Higher Moments of the Size of a Simultaneous Core Partition and its Limiting Distribution
By
Shalosh B. Ekhad and Doron Zeilberger
.pdf
.ps
.tex
Posted: Aug. 30, 2015.
This version: Sept. 1, 2015.
[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, as well as in arxiv.org.]
Dedicated to William Y.C. "Bill" Chen, the tireless apostle of enumerative and algebraic combinatorics in China (and beyond)
UPDATE ADDED March 31, 2020:
Chaim EvenZohar brilliantly proved the first challenge of our paper.
A donation of $100 to the OEIS Foundation, in his honor, has been made.
UPDATE ADDED Sept. 1, 2015:
See Marko Thiel and Nathan Williams very
interesting comments
where they point out
that the Second Challenge (except for the upper bound of "two pages") was already present in Paul Johnson's
article, and our Theorems 2 and 3 (that are now "official" theorems)
are given in their recent paper.
A donation of $100 to the
OEIS Foundation, in honor of Paul Johnson, Marko Thiel, and Nathan Williams, has been made.
Note that, in particular, all the theorems in our paper are now "real" theorems, even for ultraorthodox mathematicians
(like George Andrews and Christian Krattenthaler).
Jaclyn Anderson proved that if s and t are relatively prime positive integers, then there are only finitely many partitions
whose set of hooklengths (see wiki article)
is disjoint from the set {s,t}. In fact she proved that there are (s+t1)!/(s!t!) of them. Drew Armstrong
(see here) conjectured a beautiful expression for the average size, namely
(s1)(t1)(s+t+1)/24 ,
and this has been recently (rather painfully) proved by Paul Johnson
and reproved by Victor Y. Wang.
But the average is just the first question one can ask about a probability distribution.
In the present article, we state absolutely certain expressions (but "officially" still conjectures)
for the variance (showing in particular that
is is rather large, and there is no "concentration about the mean"), and the third through the sixth moments.
For the special case of (s,s+1)core partitions, we go all the way to the 9th moment.
[Of course, with bigger computers, and better coding, one would be able to go further, but enough is enough.]
We also pose two challenges, and will be glad to donate 100 dollars each, to the OEIS foundation in honor
of the first provers, regarding a "soft" and "global", yet rigorous, justification of our empirical approach, and
for proving an intriguing conjecture about the limiting distribution.
Maple Package

stCore,
To discover theorems about the moments of the random variable "Size" defined on the (finite) set
of socalled (s,t)core partitions, i.e. partitions whose set of hook lengths never contain s and never contain t
(where s and t are relatively prime positive integers)
Some Input and Output files for the Maple package stCore

If you want to see explicit formulas (as polynomials in s and t),
regarding the random variable "size" defined over the set of (s,t)cores, i.e. partitions
whose set of hook lengths is disjoint from {s,t}, where s and t are relatively prime positive integers, for

the mean (aka average, aka expectation, conjectured by Drew Armstrong
and proved by Paul Johnson, and reproved by Victor Y. Wang),

the variance (officially still "only" a conjecture, but absolutely certain)

the third through sixth moments (about the mean) (officially still "only" conjectures, but absolutely certain)

The limits of the standardized moments, from the third through the sixth
the input yields
the output

If you want to see explicit formulas (as polynomials in s),
regarding the random variable "size" defined over the set of (s,s+1)cores, i.e. partitions
whose set of hook lengths is disjoint from {s,s+1},

mean (aka average, aka expectation, conjectured by Drew Armstrong
and first proved (in this "most interesting" special case) by Richard Stanley and Fabrizio Zanello, and
later proved in general (see above) by Paul Johnson, and reproved by Victor Y. Wang,

variance (officially still "only" a conjecture, but absolutely certain)

the 3rd through 9th moments (about the mean) (officially still "only" conjectures, but absolutely certain)

The limits of the standardized moments, from the 3rd through the 9th
the input yields
the output

If you want to see the first 20 moments of the intriguing continuous probability distribution described in the paper
that we conjecture is the limiting distribution for the r.v. size defined on (s,t)core
the input yields
the output
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
Doron Zeilberger's Home Page