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\bf
A Constant Term Identity Featuring The Ubiquitous (And Mysterious)
Andrews-Mills-Robbins-Rumsey Numbers 1,2,7,42,429, ... \rm
{\it Doron Zeilberger*}
\footnote{*}{
Department of Mathematics, Temple University,
Philadelphia, PA19122. Supported in part by NSF grant DMS8800663.
}
{\bf Appeared in:} J. Comb. Theory Series A {\bf 66} (1994), 17-27.
{\it If $A=B$ then $B=A$}
(Axiom satisfied by the equality relation)
{\bf Abstract:} George Andrews's recent proof of the
Mills-Robbins-Rumsey conjectured formula for the number of
totally symmetric self-complementary plane partitions
is used to derive a new multi-variate constant
term identity, reminiscent of, but not implied by, Macdonald's
$BC_n$-
Dyson identity. The method of proof consists in translating to the
language of constant terms an expression of Doran for the
desired number in terms of sums of minors of a certain matrix.
The question of a direct proof of the identity, which would furnish
an alternative proof of the Mills-Robbins-Rumsey conjecture,
is raised, and a prize is offered for its solution.
{\bf 0. Prologue}
Sometimes,
it may occur to mathematician {\bf X}, in his attempt at proving
a conjectured equality $A=B$, to introduce
another quantity $C$, and to attempt to prove the two lemmas
$A=C$ and $C=B$. The original conjecture $A=B$ would then follow by
the transitivity of the $=$ relation.
Alas, it might happen that, after the successful completion
by {\bf X} of the first part of his program, but before finishing
the second part, the conjecture $A=B$ is proved by his rival {\bf Y}
by a completely different method. Should {\bf X} let thousands of hours,
fifty yellow pads, and ten ball-point pens go unrecorded in the archival
literature? Certainly not! All that {\bf X} has to do is
promote the equality $C=B$ from the status of lemma to that
of theorem and observe that its proof follows immediately
from his own lemma $A=C$, {\bf Y}'s theorem $A=B$, and the
symmetry and transitivity of the equality relation. To be on the safe
side, {\bf X} should argue also that in addition to the intrinsic interest
of $C=B$, the {\it method of proof} of the lemma $A=C$ is interesting,
and might lead to the proof of other conjectures.
The above scenario happened with the following specialization.
{\bf X}= Your faithful servant.
{\bf Y}= George Andrews.
A:= The number of totally symmetric self complementary plane partitions
(TSSCPP) whose 3D Ferres diagrams fit inside $[0,2n]^3$.
$$
B:= \,\prod_{i=0}^{n-1} {{(3i+1)!}\over {(n+i)!}} \, \, \,,
$$
the Andrews-Mills-Robbins-Rumsey sequence {1,2,7,42,429, ... }
[A3],[MRR1],[MRR2],[Ro1].
$$
C:= CT \, \prod_{1<= i