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Proof of a Conjecture of Amitai Regev about Three-Rowed Young Tableaux (and much more!)
By Shalosh B. Ekhad and Doron Zeilberger

.pdf
.ps
.tex

Written: Dec. 8, 2006.

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

I grew up in Kiryat Motzkin, so I particularly like the Motzkin numbers.
(The town is named after Leo Motzkin, the numbers after his son Theodore Motzkin).
Amitai Regev, back in 1981, proved that the number of three-rowed Young-tableaux
witb n cells equals the Motzkin number M(n), and very recently, he conjectured
that the number of three-rowed Young tableaux of n cells where the "3" entry
occupies the (1,2)-cell is M(n-1)-M(n-3). In this paper,
Shalosh and I prove this conjecture, and make lots of similar
conjectures and prove them at the same time.

Important: This article is accompanied by Maple
package
AMITAI
that automatically conjectures and then automatically proves
closed-form expressions for the number of ballot paths of n steps
starting at an *arbitrary* point in three dimensions.
It also makes conjectures for such expressions in four dimensions
but only proves them semi-rigorously.

## Sample Input and Output
- If you want a verbose proof of the original conjecture, the
input file
will produce the
output file
- If you want the statement and its verbose proof for the number of walks starting at [4,2,0] the
input file
will produce the
output file
- If you want the statement and its verbose proof for the number of walks starting at [6,4,0] the
input file
will produce the
output file
- If you want closed-formed expressions, phrased in terms of the Motzkin
sequence M(n), that are completely rigorously proved for
n-step walks that start at points [b1,b2,0] with b1 ranging from 0 to 10 and
b2 from 0 to b1, the
input file
will produce the
output file
- If you want closed-formed expressions, completely rigorously proved,
phrased in terms the Motzkin sequence for
the number of Young tableaux with n cells and at most 3 rows, and such
that the [i,j] entry has m in it, for m between 1 and 15 and
all possible [i,j] (of course i<=3 and i*j<=m) the
input file
will produce the
output file
- If you want closed-formed expressions, completely rigorously proved for
n-step walks that start at points [b1,b2,b3,0] with b1 ranging from 0 to 5 and
b2 from 0 to b1, and b3 from 0 to b2, the
input file
will produce the
output file

Added Jan. 15, 2007: This article was submitted to Journal of Integer Sequences,
and accepted subject to minor revisions, and to reformatting it in LaTeX.
Since we didn't agree with the minor revisions, and do not have
time to reformat it in LaTeX, we decided to publish it in our
Personal Journal.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

Doron Zeilberger's Home Page