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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].

$$
  C:= CT \, \prod_{1<= i<j<=n} (1- x_i / x_j ) \prod_{i=1}^{n} (1+ {x_i}^{-1} )^{n-i} \{ \prod_{i=1}^{n} (1- x_i )^{-1} \prod_{1<= i<j<=n} (1- x_i  x_j )^{-1} \} \, \,, 
$$

where "CT" stands for the phrase "the constant term of", i.e., the
coefficient of $x_1^0... x_n^0$.

{\bf 1. Introduction}

George Andrews[A1] has recently proved the following
conjecture of Mills, Robbins, and Rumsey[MRR2] (see also [Ro1],[S])

{\bf Theorem A=B:} (Andrews[A1]) $A=B$.

In this paper we will prove

{\bf Theorem C=B}: $C=B$.

This identity closely resembles Macdonald's [Macd2] celebrated
$BC_n$-Dyson identity, but doesn't seem to be implied by it.
Its mere existence seems to indicate that Macdonald's identities
and conjectures are far from being the only explicitly evaluable
constant term expressions, and that there are still many more 
that are waiting to be
discovered. Even more exciting is the possibility that there exists
a common generalization of Macdonald's constant term identities
and the present one.
Section 7 presents yet another such identity.
We refer the readers to [Gu] and [Ga-Go] for an update on the status
of Macdonald's constant term conjectures.

In the constant term $C$, we view the rational function inside the
braces as a power series in positive powers of 
$x_1 , ... , x_n$. When this gets multiplied by the Laurent 
polynomial in front of the braces, one gets a well defined Laurent
series, with a well defined constant term. 

Theorem {\bf C=B\rm
 would follow from Andrews's theorem \bf
A=B}
and the following lemma:

{\bf Lemma A=C:} $A=C$.

I am offering 25 US dollars for a direct proof of theorem {\bf C=B},
that does not use Andrews's theorem {\bf A=B}. Such a proof, combined with
my lemma {\bf A=C} would, of course, give a new proof
of theorem {\bf A=B}.

The readers are welcome to look up the definition of TSSCPP in
[MRR2] or [S], if they wish, but
the present paper can be understood without it, provided one is willing
to believe Doran's[Do] result, to be recalled shortly, 
that the number of these
creatures, whatever they are, equals the sum of minors of a certain
matrix. 
In fact the larger message of this paper is in introducing a
new and potentially useful {\it method} for expressing sums of minors
of matrices whose entries are binomial coefficients in terms of
constant terms of rational functions, the Mills-Robbins-Rumsey
conjecture being the instructive example by which this method is
being illustrated. The present method can be viewed as a determinantal
extension of Egorychev's[E] method of "integral representation" (which
is tantamount to "constant term") for binomial coefficients sums.
MacMahon's celebrated master theorem[MacM] that expresses a 
certain determinant as a certain constant term also comes to mind,
but at first sight only appears to be a distant cousin.

{\bf 2. Doran's Sum of Minors expression for the Number of TSSCPP}

To prove lemma $A=C$, we need a result of Bill Doran[Do] (no relation).
Let $D$ be the sum of all the $n \times n$ 
minors of the following $n \times (2n-1)$ matrix:

$$
  X_{i,j} :=  (  { i-1 \atop j-i }  )  \, \, \,, 1 <= i <= n \, \, \,, \, \,1<=j<=2n-1 \, \, \,. 
$$

{\bf Theorem A=D}(Doran[Do]): $A=D$.

It remains to prove that $D=C$. I will prove a more general
result. Let $D'$ be the sum of all the 
$n \times n$ minors of the $n \times (2n+m-1)$ matrix 
$X$ given by

$$
  X_{i,j} :=  (  { m+i-1 \atop j-i }  )  \, \, \,, 1 <= i <= n \, \, \,, \, \,1<=j<=2n+m-1 \, \, \,. 
$$

and let $C'$ be given by

$$
  C':=CT \, \,\prod_{1<= i<j<=n} (1- x_i / x_j ) \prod_{i=1}^{n} (1+ {x_i}^{-1} )^{m+n-i} \{ \prod_{i=1}^{n} (1- x_i )^{-1} \prod_{1<= i<j<=n} (1- x_i  x_j )^{-1} \} \, \,. 
$$

I will prove

{\bf Lemma D'=C'}: $D'=C'$.

The proof of this lemma will be given in the next section. Assuming
it for the moment, we have

{\bf Corollary D=C}: $D=C$.

{\bf Proof:\rm
 Take $m=0$ in Lemma \bf
D'=C'}.

{\bf Completion of the proof of Lemma A=C}: Combine
Doran's[Do] Lemma $A=D$ and Corollary $D=C$.

{\bf 3. Proof of Lemma D'=C':} 

We will have a fairly long string of equalities. Whenever the equality
requires explanation we will label the equal sign with an integer,
and give the explanation in section 4, under the heading of that
integer. Some readers may prefer to consult section 4 simultaneously.

$$
  D'  \,:= \, \sum_{1 <= j_1 < j_2 < ... < j_n <= 2n+m-1}  det  (  { m+i-1 \atop {j_r -i} }  )_{1 <= i <= n \,, \,1 <= r <= n}  \,= \, 
$$

$$
  \sum_{0 <= j_1 < j_2 < ... < j_n <= 2n+m-2}  det  (  { m+i-1 \atop {j_r -i+1} }  )_{1 <= i <= n \,, \,1 <= r <= n} 
$$

$$
   \,=^{(1) \,} \sum_{0 <= j_1 < j_2 < ... < j_n <= 2n+m-2}  \, \,\sum_{ \pi  \epsilon  S_n} sgn ( \pi ) \prod_{r=1}^{n}   (  { m+ \pi (r) -1 \atop {j_r - \pi (r) +1} }  )  
$$

$$
   \,=^{(2)}  \, \sum_{0 <= j_1 < j_2 < ... < j_n <= 2n+m-2}  \, \,\sum_{ \pi  \epsilon  S_n} sgn ( \pi ) CT  [ \prod_{r=1}^{n}  {{(1+ x_r )^{m+ \pi (r) -1}} 
\over { {x_r}^{ j_r - \pi (r) + 1} }}  ] 
$$

$$
  = \, \sum_{0 <= j_1 < j_2 < ... < j_n <= 2n+m-2} CT  [ {{ \prod_{r=1}^{n}  {(1+ x_r )^m } \over { {x_r}^{ j_r } }}  \sum_{ \pi  \epsilon  S_n} sgn ( \pi ) \prod_{r=1}^{n} [(1+ x_r ) x_r ]^{ \pi (r) -1 } }  ] 
$$

$$
   \,=^{(3)} CT  [ { \prod_{r=1}^{n} {(1+ x_r )^m} \prod_{1 <= i < j <= n } [ (1+ x_i ) x_i - (1+ x_j ) x_j ]  \, \,\sum_{0 <= j_1 < j_2 < ... < j_n <= 2n+m-2} x_1^{- j_1 } ... x_n^{- j_n} }  ] 
$$

$$
   \,=^{(4)} CT  [ { \prod_{r=1}^{n} {(1+ x_r )^m} \prod_{1 <= i < j <= n } [ (1+ x_i ) x_i - (1+ x_j ) x_j ] \sum_{0 <= j_1 < j_2 < ... < j_n < \infty } x_1^{- j_1 } ... x_n^{- j_n} }  ] 
$$

$$
   \,=^{(5)} CT  [ { \prod_{r=1}^{n} {(1+ x_r^{-1} )^m} \prod_{1 <= i < j <= n } [ (1+ x_i^{-1} ) x_i^{-1} - (1+ x_j^{-1} ) x_j^{-1} ] \sum_{0 <= j_1 <= j_2 <= ... <= j_n < \infty } x_1^{j_1 } ... x_n^{j_n} }  ] 
$$

$$
   \,=^{(6) \,} CT  [  \prod_{r=1}^{n} {(1+ x_r^{-1} )^m} \prod_{1 <= i < j <= n } [ (1+ x_i^{-1} ) x_i^{-1} - (1+ x_j^{-1} ) x_j^{-1} ] \cdot 
$$

$$
  (1- x_n )^{-1} (1- x_n x_{n-1} )^{-1} ... (1- x_n x_{n-1} ... x_1 )^{-1} ] 
$$

$$
   \,=^{(7)} {{1}\over {n!}} CT  [  \prod_{r=1}^{n} {(1+ x_r^{-1} )^m} \prod_{1 <= i < j <= n } [ (1+ x_i^{-1} ) x_i^{-1} - (1+ x_j^{-1} ) x_j^{-1} ] \cdot 
$$

$$
  \sum_{ \pi  \epsilon  S_n} sgn ( \pi ) \pi [ (1- x_n )^{-1} (1- x_n x_{n-1} )^{-1} ... (1- x_n x_{n-1} ... x_1 )^{-1} ] ] 
$$

$$
   \,=^{(8)} {{1}\over {n!}} CT  [  \prod_{r=1}^{n} {(1+ x_r^{-1} )^m} \prod_{1 <= i < j <= n } [ (1+ x_i^{-1} ) x_i^{-1} - (1+ x_j^{-1} ) x_j^{-1} ] \cdot 
$$

$$
{  { \prod_{ 1 <= i < j <= n} ( x_j - x_i ) } \over { \prod_{i=1}^{n} (1- x_i ) \prod_{ 1 <= i < j <= n} (1- x_i x_j )}} ] 
$$

$$
   \,=^{(3)} {{1}\over {n!}} CT [ \prod_{r=1}^{n} {(1+ x_r^{-1} )^m}  \prod_{ 1 <= i < j <= n} ( x_j - x_i )  \prod_{i=1}^{n} (1- x_i )^{-1} \prod_{ 1 <= i < j <= n} (1- x_i x_j )^{-1} \cdot  
$$

$$
  \sum_{ \pi  \epsilon  S_n } sgn ( \pi )  \prod_{r=1}^{n} [ (1+ x_r^{-1} ) x_r^{-1} ]^{ \pi (r) -1}  \,] 
$$

$$
   \,=^{(9)} CT [ \prod_{r=1}^{n} {(1+ x_r^{-1} )^m}  \prod_{ 1 <= i < j <= n} ( x_j - x_i )  \prod_{i=1}^{n} (1- x_i )^{-1} \prod_{ 1 <= i < j <= n} (1- x_i x_j )^{-1} \cdot  
$$

$$
  \prod_{r=1}^{n} [ (1+ x_r^{-1} ) x_r^{-1} ]^{r-1}  \,] 
$$

$$
   \,= \, CT [ \prod_{r=1}^{n} {(1+ x_r^{-1} )^{m+r-1} }  \prod_{ 1 <= i < j <= n} ( 1-  x_i / x_j )  \prod_{i=1}^{n} (1- x_i )^{-1} \prod_{ 1 <= i < j <= n} (1- x_i x_j )^{-1}   \,] \quad, QED \,. 
$$

{\bf 4. Explanations Of The Above Equalities}

{\bf 1.} The definition of the determinant.

{\bf 2.} The binomial theorem and the fact that 
$CT[ f(x) g(y)]= CT [f(x)] CT [g(y)] \,$.

{\bf 3.} The Vandermonde determinant identity:

$$
 \sum_{ \pi  \epsilon  S_n } sgn( \pi ) \prod_{r=1}^{n}  y_r^{\pi (r) -1}  \,= \, \prod_{ 1 <= i < j <= n} ( y_i - y_j ) \, \, \,. 
\eqno(V)
$$        

See [Ge] and [A2], sec. 4.4 for Gessel's beautiful combinatorial
proof.

{\bf 4.} The added terms are of higher degree then the polynomial in front
of the sigma, and so add $0$ to the constant term.

{\bf 5.} We make the transformation $ x_i \rightarrow x_i^{-1}$,
$i= 1, ... , n $, which obviously does not change the constant
term. We also changed the summation range from
{$ 0 < j_1 < ... < j_n < \infty $} to 
{$ 0 <= j_1 <= ... <= j_n < \infty $}, which doesn't change
anything, since the extra monomials all have at least two of
their exponents equal, and hence their contribution to the constant term
is zero, thanks to the anti-symmetry of the kernel.
\eject
{\bf 6.} Here we used

$$
  \sum_{0 <= j_1 <= j_2 <= ... <= j_n < \infty } x_1^{j_1 } ... x_n^{j_n}   \,= \, (1- x_n )^{-1} (1- x_n x_{n-1} )^{-1} ... (1- x_n x_{n-1} ... x_1 )^{-1}  \, \,. 
$$

For $n=1$ this is just the sum of an infinite geometric series.
Assuming this is true for $n-1$ variables,
$x_2 , ... , x_n$, we have

$$
  \sum_{0 <= j_1 <= j_2 <= ... <= j_n < \infty } {x_1}^{ j_1} ... {x_n}^{ j_n} \,= \, 
$$

$$
  [ \sum_{ j_1 =0}^{\infty}  {( x_1 ... x_n )}^{ j_1 } ] \cdot [ \sum_{ 0<= j_2 - j_1 <= ... <= j_n - j_1 < \infty } x_2^{j_2 - j_1 } ... x_n^{j_n - j_1 } ] 
$$

$$
   \,= \, (1- x_1 ... x_n )^{-1}  (1- x_2 ... x_n )^{-1}  ... (1- x_n )^{-1}  \, \, \,. 
$$

{\bf 7.} Here we "averaged" over all the images under the symmetric group,
noting that the constant term is not affected. More explicitly, 
for any Laurent series $f( x_1 , ... , x_n )$ let
the symmetric group act naturally by:

$$
  \pi (f) ( x_1 , ... , x_n ) = f ( x_{ \pi (1)} , ... , x_{ \pi (n)} ) \, \,, 
$$

and let

$$
  f^\# = {{1}\over {n!}} \sum_{ \pi  \epsilon  S_n } \pi (f) \, \,, 
$$

then $CT(f) = CT ( f^\# )$. We also used the obvious fact that 
if $f^\&$ denotes the "anti-symmetrizer":

$$
  f^\& = {{1}\over {n!}} \sum_{ \pi  \epsilon  S_n} sgn ( \pi ) \pi (f) \, \,, 
$$

then if $g$ is anti-symmetric, then

$$
  (gf)^\# = g ( f^\& ) \, \, \,. 
$$

{\bf 8.} Here we used the identity:

$$
 \sum_{ \pi  \epsilon  S_n}  \,sgn ( \pi ) \, \pi [ (1- x_n )^{-1} (1- x_n x_{n-1} )^{-1} ... (1- x_n x_{n-1} ... x_1 )^{-1}  ] \,= \, 
\eqno(L)
$$        

$$
{  { \prod_{ 1 <= i < j <= n } ( x_j - x_i ) } \over { \prod_{i=1}^{n} (1- x_i ) \prod_{ 1 <= i < j <= n } (1- x_i x_j ) }} \, \, \, \,. 
$$

This is easily seen to be equivalent to Littlewood's identity that
sums all the Schur functions (e.g. [Macd1],  ex I.5.4, p. 45). Here
I give another proof which, incidentally, also proves Littlewood's
identity.
Let's call the left side of (L) $ f( x_1 , ... , x_n )$,
and its right side $g( x_1 , ... , x_n )$.
Separating the sum over $S_n$ in the definition of $f$ into
the $n$ subsets of $S_n$, according to the values of $\pi (n )$,
we have (chopping $\pi (n) = i$ amounts to losing $n-i$ inversions)

$$
  f( x_1 , ... , x_n ) = (1- x_1 x_2 ... x_n )^{-1} \sum_{i=1}^{n}  (-1)^{(n-i)} f( x_1 , x_2 , ... , \hat{ x_i } , ... x_n ) \, \, \, \,. 
$$

Since $f( x_1 ) = g ( x_1 )$ is obviously true, $f=g$
is true for $n=1$. (L) would follow by induction if we can prove that

$$
  g( x_1 , ... , x_n ) = (1- x_1 x_2 ... x_n )^{-1} \sum_{i=1}^{n}  (-1)^{(n-i)} g( x_1 , x_2 , ... , \hat{ x_i } , ... x_n ) \, \,, 
$$

which is equivalent to

$$
  (1- x_1 ... x_n )  \,= \, \sum_{i=1}^{n}  \,(-1)^{n-i}  \, g ( x_1 , ... , 
\hat{x_i} , ... , x_n ) / g ( x_1 , ... , x_n ) \, \, \,. 
$$

But it is readily seen that
$$
  (-1)^{n-i} {{g ( x_1 , ... , \hat {x_i} , ... , x_n )} \over { g ( x_1 , ... , x_n )} }\,= \,  (1- x_i )  \prod_{1 <= j <= n , j \neq  i}  (1- x_i x_j )/( x_i - x_j ) \, \, \,. x_n
$$

It thus remains to prove that

$$
  (1- x_1 ... x_n ) =  \sum_{i=1}^{n} (1- x_i )  \prod_{1 <= j <= n , j \neq  i}  ( x_i - x_j )/(1- x_i x_j ) \, \, \,. 
$$

This is a rational function identity resembling those of Good[Go],
Gustafson and Milne[Gu-Mi], Gross and Richards [Gr-Ri],
and Milne [Mi]. It is easily proved by Lagrange-interpolating
the degree-$n$ polynomial (in $z$)
$f(z):= (1- z x_1 ) ... ( 1- z x_n )$ at the $n+1$ points
{$1, x_1 , ... , x_n $}, then substituting 
$z=0$ and finally making trivial adjustments.

{\bf 9.} This is (7) in reverse.

\bf
5. Robbins's Conjectured Expression For D', And a More General
(and hence easier) Constant Term Conjecture\rm

Dave Robbins[Ro2], in a private communication, made the following
conjecture regarding the sum of minors D`. Let
$B'(m,n)$ be defined by $B'(0,n)=B(n)$, and

$$
  {{B'(m+1,n)}\over {B'(m,n)}} \,= \, 2 \prod_{j=1}^{n-1} {{(2m+j+2)(3m+2n+2+j)}\over {(m+1+j)(3m+2+2j)}} \, \, \,, 
$$

then 

{\bf Conjecture D'=B'} (Robbins[Ro2]): $D'=B'$.

Conjecture {\bf D'=B'} would, of course, follow from the following
constant term conjecture

{\bf Conjecture C'=B'}: $C'=B'$.

{\bf C'=B'\rm
, being more general than \bf
C=B}, should be easier to
prove. Its proof will, in particular, solve the $25$ dollars problem
stated above. 

{\bf 6. Sums of Minors of Generalized Binomial Coefficients Matrices}

Our approach for expressing the sums of minors of the matrix
$X$ took advantage of the fact that its entries were representable
as constant terms as follows:

$$
  X_{i,j} = CT {{(1+x)^{m+i-1}} \over {x^{j-i}} }= CT  [ 
{{ (1+x)^m {((1+x)x)^{i-1}} \over {x^{j-1} }}            }     ] \, \, \,. 
$$

Scanning the proof of {\bf D'=C'} given in section 3, we see that
it is still valid when $(1+x)^m$ is replaced by a general
polynomial $f(x)$ and $(1+x)x$ is replaced by a general polynomial
$g(x)$.  We thus have

{\bf General Theorem}: Let $f(x)$ and $g(x)$ be polynomials and consider
the $n \times ( \,deg(f)+(n-1) deg(g)+1)$ matrix whose entries are given
by 

$$
  X_{i,j \,:=} CT[ {{f(x) g(x)^{i-1}} \over { x^{j-1}}}] \, \, \,. 
$$

The sum of all its $n \times n$ minors equals

$$
  CT \,\prod_{1<= i<j<=n} ( x_j - x_i ) \prod_{i=1}^{n} f ( {x_i}^{-1} )  g ( {x_i}^{-1} )^{n-i} \{ \prod_{i=1}^{n} (1- x_i )^{-1} \prod_{1<= i<j<=n} (1- x_i  x_j )^{-1} \} \, \,= 
$$

$$
  {{1}\over {n!}} CT \,\prod_{1<= i<j<=n} ( x_j - x_i ) \prod_{i=1}^{n} f ( {x_i}^{-1} )  \prod_{1<= i<j<=n} ( g( { x_j }^{-1} )  \,- \,  g( { x_i }^{-1} )) \{ \prod_{i=1}^{n} (1- x_i )^{-1} \prod_{1<= i<j<=n} (1- x_i  x_j )^{-1} \} \, \,. 
$$

{\bf Proof\rm
: Do a \it
global "replace"} in section 3, replacing
$(1+x\_)^m$ by $f(x)$ and $x\_(1+x\_)$ by $g(x)$.
(Here we used the MATHEMATICA convention of transformation
rules where "x\_" means "anything, to be called x".) The second equality
follows from explanation 4 of section 4.
\medskip
{\bf 7. Another Constant Term Identity}

Taking $f(x)= (1+x)^m$, $g(x)=(1+x)$, the matrix
$X$ becomes

$$
 X_{i,j} :=  (  { m+i-1 \atop j-1 }  )  \, \, \,, 1 <= i <= n \, \, \,, \, \,1<=j<=n+m \, \, \,. 
\eqno(*)
$$         

The sum of its $n \times n$ minors can be computed explicitly,
since each minor is nothing but a Schur function evaluated at
$x_1 = x_2 = ... = x_n =1$, by the Jacobi-Trudi
formula (e.g. [Macd1], p.25, (3.5)), and these can be summed
($q=1$ in [Macd1], p.52, (4), see also [De1][De2] for a superb exposition
about summing Schur functions) to yield

$$
  \prod_{j=0}^{n-1} \prod_{i=1}^{m} {{2i+j}\over {i+j}} \, \, \,, 
$$

as observed empirically by Dave Robbins[Ro2].
Combining the general theorem with
the above, we get the following elegant multi-variate constant term 
identity.

{\bf IDENTITY:}

$$
  {{1}\over {n!}} CT \,\prod_{1<= i \neq  j<=n} ( \,1- {{x_i}\over{ x_j}}  \,) \prod_{i=1}^{n}  ( 1+ {x_i}^{-1} )^m \{ \prod_{i=1}^{n} (1- x_i )^{-1} \prod_{1<= i<j<=n} (1- x_i  x_j )^{-1} \} \, \,= \, \prod_{j=0}^{n-1} \prod_{i=1}^{m} {{2i+j}\over {i+j}} \, \, \,. 
$$

I am offering $5$ US dollars for a direct proof of this identity.

{\bf REFERENCES}

[A1] Andrews, George E., \it
Plane Partitions V: the T.S.S.C.P.P
conjecture\rm
, preprint.

[A2] Andrews, George E., \it
q-SERIES: THEIR DEVELOPMENT AND APPLICATIONS
IN ANALYSIS, NUMBER THEORY, COMBINATORICS, PHYSICS, AND COMPUTER 
ALGEBRA\rm
, CBMS {\bf 66}, AMS, Providence, 1986.

[A3] Andrews, George E., \it
Plane Partitions III: the weak
Macdonald conjecture\rm
, Invent. Math. {\bf 53} (1979), 193-225.

[De1] De\*'sarme\*'nien, Jacques, \it
La de\*'monstration des
identite\*'s de Gordon et MacMahon et de deux identite\*'s nouvelles\rm
,
Se\*'m. Lotharingien de Combinatoire, {\bf 15} (1986), ed. by A.
Kerber, Publications IRMA, Strasbourg.

[De2] De\*'sarme\*'nien, Jacques, \it
Une ge\*'ne\*'ralisation des
formules de Gordon et de MacMahon\rm
, C.R. Acad. Sci. Paris, 
Ser. I {\bf 309} (1989), 269-272.

[Do] Doran, William F., \it
A connection between alternating sign
matrices and totally symmetric, self-complementary plane partitions\rm
,
preprint.

[E] Egorychev, G. P., \it
INTEGRAL REPRESENTATION AND THE COMPUTATION
OF COMBINATORIAL SUMS\rm
, American Mathematical Society Translations
{\bf 59}, AMS, Providence, 1984.

[Ga-Go] Garvan, Frank G., and Gonnet, Gaston, \it
Macdonald' constant
term conjectures for exceptional root systems\rm
, Bulletin (new
series) of the Amer. Math. Soc, to appear.

[Ge] Gessel, Ira, {\it Tournaments and Vandermonde's determinant},
J. Graph Theory {\bf 3}(1979), 305-307.

[Go] Good, Irving J., {\it Short proof of a conjecture of Dyson},
J. Math. Phys. {\bf 11}(1970), 1884.

[Gr-Ri] Gross, Kenneth I., and Richards, Donald St. P., \it
Constant
term identities and hypergeometric functions on spaces of hermitian
matrices\rm
, preprint.

[Gu] Gustafson, Robert A., \it
A generalization of Selberg's beta
integral\rm
, Bulletin (new series) of the Amer. Math. Soc.
{\bf 22}(1990), 97-105.

[Gu-Mi] Gustafson, Robert A., and Milne, Stephen C., \it
Schur
functions, Good's identity, and hypergeometric series well poised in
SU(n)\rm
, Adv. in Math. {\bf 57}(1985), 209-225.

[MRR1] Mills, William H., Robbins, David P., and Rumsey, Howard,
{\it Proof of the Macdonald conjecture}, Invent. Mat {\bf 66}(1982),
73-87.

[MRR2] Mills, William H., Robbins, David P., and Rumsey, Howard,
{\it Self complementary totally symmetric plane partitions}, J.
Combin. Theory Ser. A {\bf 42}(1986), 277-292.

[MacM] MacMahon, Percy A., {\it Combinatory Analysis}, Vol. 1,
Cambridge University Press, London, 1916. (Reprinted: Chelsea, New
York, 1960).

[Macd1] Macdonald, Ian G., {\it SYMMETRIC FUNCTIONS AND HALL POLYNOMIALS},
Oxford University Press, Oxford, 1979.

[Macd2] Macdonald, Ian G., {\it Some conjectures for root systems},
SIAM J. Math. Anal. {\bf 13}(1982), 988-1007.

[Mi] Milne, Stephen C., \it
A q-analog of the Gauss Summation Theorem
for hypergeometric series in U(n)\rm
, Adv. Math. {\bf 72}(1988) 59--131. 

[Ro1] Robbins, David P., {\it The story of 1, 2, 7, 42, 429, 7436, ...},
to appear in The Mathematical Intelligencer.

[Ro2] Robbins, David P., {\it private communication}.

[S] Stanley, Richard, \it
A baker's dozen of conjectures concerning
plane partitions\rm
, in: COMBINATOIRE ENUMERATIVE, ed. by G. Labelle
and P. Leroux, Lecture Notes in Mathematics {\bf 1234}, Springer,
Berlin, 1986.

Feb, 1991. Revised: Mar, 1991.

\bf
EPILOGUE: Some Philosophical (and Technical) Ramblings On
Andrews's Proof, In Particular, And Of The Whole Genre, In General\rm

{\bf Note: This epilogue to
``A Constant Term Identity .. 1,2,7,42,429, ''
 was NOT included in the original version, published in
JCT(A) 66 (1994), 17-27,
to comply with the request of the editor, George Andrews.}

{\bf 1. Comments On Andrews's Presentation Of His Proof}


George Andrews's great achievement is obscured by the apparent
complexity of his proof, that seems very brute-force-ish and
uninspiring, and that contains a sequence of seven hair-raising lemmas.
I claim that Andrews's proof can be made much more streamlined and
palatable
by clearly separating the inspiration part from the perspiration part,
especially since the latter will be soon completely doable by
computers, and is already so to a large extent. Furthermore, there
is no point in actually presenting the computer-generated proofs
of the claimed identities, since they are readily reproducible.
All one has to do is {\it report} that they were successfully
proved by the author's computer.

Andrews's method consists in a slight (but crucial) change
in a determinant expression of Stembridge[Ste] for the
desired number. Calling the (modified) Stembridge matrix $M(n)$,
Andrews pulls out of the hat a lower triangular matrix, $R(n)$
with certain closed form coefficients, such that
$V(n):=M(n) \cdot R(n)$ is lower triangular, with
certain explicitly given diagonal terms. 
Since both $V(n)$, and $R(n)$ are lower
triangular, their determinants are easy to compute (namely
the product of the diagonal entries), and it follows that
$det(M(n))=det(V(n))/det(R(n))$.

Andrews's breakthrough, in addition to modifying the Stembridge
matrix, was in the actual {\it finding} of the miracle matrix
$R(n)$, which was experimental mathematics \it
par
excellence\rm
proof that $M(n) \cdot R(n)$ is indeed lower-triangular, and the
computation of its diagonal entries
are purely routine, 
at least in principle, thanks to the holonomic theory in [Z1].
Furthermore, since the above task is equivalent to
proving a certain double-sum hypergeometric identity, the proof
of the above matrix product will also become purely routine,
{\it in practice}, once Herb Wilf and I will be successful in our
current efforts at generalizing our WZ method[W-Z1][W-Z2] and my "fast"
[Z2] algorithm to multiple sums. Under this optimistic hypothesis,
a modern rendition of Andrews's proof will go as follows.
Exhibit the matrix $R(n)$, and state

{\bf Andrews's Theorem:} 
$V(n):=M(n) \cdot R(n)$ is a lower-triangular matrix,
and its diagonal terms are 

$$
  v(i,i)= -  (  { 3i+1  \atop i }  )^2 /  (  { 2i  \atop i }  )^{2 \, \, \,}. 
$$

{\bf Proof(2000 version):} Routine. QED

Consequently, all of section 3, and a large part of section 4 of
[A1]   will seem in a few years a quaint waste of paper.
But even, at this very moment, Andrews's paper can be shortened
considerably, if one adopts the above convention that there is no
point in reproducing purely routine calculations done by
readily available software. To do the WZ method, all one needs
is MACSYMA or MAPLE, on which Gosper's algorithm is built-in,
and my more general algorithm[Z2][Z3] for evaluating single-sum
hypergeometric series, implemented via MAPLE programs, are free
for the asking. Consequently, whenever one makes use of the WZ
method[W-Z1], or of my more
general "fast" algorithm[Z2][Z3],
it's enough to {\it report} that the program was indeed run,
and the outcome was positive. All those who really want
to "see" the proof (i.e. the certificate) are welcome to 
run the program themselves.

More specifically, lemmas 2-6 of [A1] are currently purely routine.
In fact,
for one of these lemmas, lemma 2, Andrews {\it did} make use of the WZ
method.
Andrews writes (my capitals):

"The underlying hypergeometric series ... is not one of the
well-known summable series. Consequently, we MUST rely
on the WZ-method([W-Z1])."


It turns out that lemmas 3-6, being single hypergeometric sums,
are also doable by either [W-Z1] or [Z2], so one
MAY do them that way, without bothering to derive them from
known identities. Consequently, even today, it would have been legitimate
to only state lemmas 2-6, and proclaim their proofs as routine.

Similar remarks apply to Andrews's previous proofs in his continuing
saga on plane partitions (see [A1] for references).

\bf
2.A Modern Rendition Of The Robbins-Mills-Rumsey Proofs of
Andrews's and Macdonald's Plane Partition Conjectures\rm

Like Andrews's proofs, the Mills-Robbins-Rumsey[MRR2] ingenious proofs
of the generating function for cyclically symmetric plane partitions,
conjectured by Macdonald([Macd1], p. 53), and of a conjecture of Andrews
(see [MRR2]) can be shrunk to few lines, if one isolates the purely
routine identities that are being used, and refrains from presenting
their readily producible proofs.

In [MRR], they present a method for evaluating determinants of the form
$d(m):=det( a_{i,j} )_{1<= i <= m , 1 <= j <= m}$,
for a certain explicitly given double-indexed sequence
$a_{i,j}$ ($1<= i < \infty , 1<= j < \infty$). The way they do it is
to exhibit a doubly indexed sequence $C_j^{(m)}$, 
$1<= j <= m < \infty$, such that, for
every fixed $m$, the vector $C_j^{(m)}$, $j=1, ... , m$,
is proportional to the cofactors of the last row in
$( a_{m,j} )_{ 1 <= j <= m}$. To show that
this is indeed true, they prove that 

$$
 \sum_{j=1}^{m} a_{i,j} C_j^{(m)} = 0 \, \,. 
\eqno(i)
$$          

Then they evaluate
$$
 \sum_{j=1}^{m} a_{m,j} C_j^{(m)} = e(m) \,, say 
\eqno(ii)
$$         

that in their case turns out to be closed form, and finally
they prove

$$
 C_m^{(m)} = f(m)  \,(another \,explicit \,expression) \, \,. 
\eqno(iii)
$$          

Since the actual cofactor of $a_{m,m}$ in 
$( a_{i,j} )_{1<= i <= m , 1 <= j <= m}$ is
$det ( a_{i,j} )_{1<= i <= m-1 , 1 <= j <= m-1} = d(m-1)$,
if follows that

$$
  {{d(m)}\over {d(m-1)}}  \,= \, {{e(m)}\over {f(m)}} \, \, \,. 
$$

In general, whenever, $a_{i,j}$ and $C_j^{(m)}$ are holonomic,
all the above claims are routinely verifiable([Z1]) albeit only in
principle (today).
If $a_{i,j}$ is closed form, and so is $C_j^{(m)}$, then
(i)-(iii) are routinely provable also in practice. However, in
the specific determinants evaluated in [MRR1], $C_i^{(m)}$
turns out to be given by a single sum, so (i) and (ii) are
double sums, and a priori it seems that one has to wait a few
years before they become routinely verifiable. However, it turns
out, that in the two cases considered  in [MRR1], both inner sums
of (i) and (ii) are explicitly summable (and hence can be found,
from scratch, by [Z2][Z3]), and hence the whole thing is
routinely verifiable, even today. 

All the above also applies to q-analogs, and q-binomial coefficients,
using the theory of q-holonomic functions, and q-creative
telescoping[Z4]. Let's illustrate the above remarks by giving
new renditions of the $q=1$ case of Andrews's conjecture, and
of the generating function for cyclically symmetric plane
partitions.

{\bf Theorem([MRR1])}: Let

$$
  a_{i,j} := \delta_{i,j} +  (  {i+j \atop j-1}  ) \, \,, 
$$

with the above notation,

$$
  {{d(m)}\over {d(m-1)}} = {{ (  {3m+1 \atop m}  )} \over { (  {2m \atop m}  )}} \, \,. 
$$

{\bf Proof(1990 version of the proof in [MRR1])}: Introduce

$$
  C_j^{(m)}  \,:= \,\sum_{i=1}^{m} (-1)^{i+j}  (  {m+1 \atop i-j}  )  (  {m+i \atop i}  )  (  {2m-i \atop m-i}  ) \, \, \,, 
$$

$$
  e(m):={ (  {3m+1 \atop m}  )} \, \, \,, f(m):={ (  {2m \atop m}  )} \, \,. 
$$

(i),(ii),(iii) are routinely provable, and indeed were proved,
by my computer, Shalosh B. Ekhad. 
The theorem follows by the above general remarks. QED

Next, let's give a modern rendition of the determinant evaluation that
implies the generating function for cyclically symmetric plane partitions
([MRR1]). Here one has q- (alias basic) 
hypergeometric identities, but these too,
are routinely verifiable[Z4] (for single sums). Once again the double
sums that occur have explicitly evaluable inner sums, so everything
below is indeed routine, even toady.

Recall that the q-analogs of the binomial coefficients, the so-called
Gaussian polynomials are denoted by 

$$
   [  {m \atop n}  ], 
$$

and defined by

$$
   [  {m \atop n}  ]:=  {{(1- q^{m-n+1} ) ... (1- q^m )} \over {(1- q ) ... (1- q^n )}} \, \,. 
$$

{\bf Theorem([MRR1])}: Let $\omega$ be a primitive cubic root of
unity, and let

$$
  a_{i,j} := \delta_{i,j} +  \omega q^{i-1}  [  {i+j-2 \atop j-1}  ] \, \,, 
$$

with the above notation for $d(m)$

$$
  {{d(m)}\over {d(m-1)}} = {{(1- \omega^2 q^{m-1} )  [  {3m-2 \atop m-1}  ]} 
\over {(1- \omega q^{m-1} )  [  {2m-2 \atop m-1}  ]}} \, \,. 
$$

{\bf Proof(1990 version of the proof in [MRR1])}: Introduce

$$
  V( \alpha , \beta )_j  \,:= \, q^{j(m-1) + \beta )}  [  {m-2+j \atop {j- \alpha } }  ]   [  {2m-2-j \atop {m- \beta - j } }  ]  \, \,, 
$$

$$
  L_j  \,:= \, V(0,2)_j + \omega V(0,0)_j + \omega^2 V(2,0) \, \,, 
$$

and, finally,
$$
  C_j^{(m)}  \,:= \,\omega^{-1} \sum_{i=1}^{m} q^{j(1-m)}   [  {-m+i-j \atop i-j}  ] L_i  \, \,, \, \, 
$$

$$
  e(m):= {{(1+ \omega ) (1- \omega^2 q^{m-1} ) (1- q^{m-1} )}     \over {(1- \omega q^{m-1} ) (1- q^m )  [  {3m-2 \atop m-1}  ] }}\, \, \,, 
$$

$$
  f(m):={ [  {2m-2 \atop m-1}  ]} \, \,. 
$$

(i),(ii),(iii) are routinely provable[Z4], and indeed were proved,
by my computer, Shalosh B. Ekhad. 
The theorem follows by the above general remarks. QED
\end
