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# Proof of George Andrews’s and David Robbins’s *q*-TSPP conjecture

Edited by George E. Andrews, Pennsylvania State University, University Park, PA, and approved December 22, 2010 (received for review February 24, 2010)

## Abstract

The conjecture that the orbit-counting generating function for totally symmetric plane partitions can be written as an explicit product formula has been stated independently by George Andrews and David Robbins around 1983. We present a proof of this long-standing conjecture.

## 1. Proemium

In the historical conference *Combinatoire Énumerative* that took place at the end of May 1985 in Montréal, Richard Stanley raised some intriguing problems about the enumeration of plane partitions (see below), which he later expanded into a fascinating article (1). Most of these problems concerned the enumeration of “symmetry classes” of plane partitions that were discussed in more detail in another article of Stanley’s (2). All of the conjectures in the latter article have since been proved (see David Bressoud’s modern classic; ref. 3), except one, which until now has resisted the efforts of some of the greatest minds in enumerative combinatorics. It concerns the proof of an explicit formula for the *q*-enumeration of totally symmetric plane partitions, conjectured around 1983 independently by George Andrews and David Robbins (refs. 1 and 2, conj. 7; ref. 3, conj. 13; and already alluded to in ref. 4). In the present article, we finally turn this conjecture into a theorem.

A plane partition *π* is an array *π* = (*π*_{i,j})_{1≤i,j}, of nonnegative integers *π*_{i,j} with finite sum , which is weakly decreasing in rows and columns so that *π*_{i,j}≥*π*_{i+1,j} and *π*_{i,j}≥*π*_{i,j+1}. A plane partition *π* is identified with its 3D Ferrers diagram which is obtained by stacking *π*_{i,j} unit cubes on top of the location (*i*, *j*). The result is a left-, back-, and bottom-justified structure in which we can refer to the locations (*i*, *j*, *k*) of the individual unit cubes. If the diagram is invariant under the action of the symmetric group *S*_{3} on the coordinate axes, then *π* is called a totally symmetric plane partition (TSPP). In other words, *π* is called totally symmetric if whenever a location (*i*, *j*, *k*) in the diagram is occupied then all its up to five permutations {(*i*,*k*,*j*),(*j*,*i*,*k*),(*j*,*k*,*i*),(*k*,*i*,*j*),(*k*,*j*,*i*)} are occupied as well. Such a set of cubes, i.e., all cubes to which a certain cube can be moved via *S*_{3}, is called an orbit; the set of all orbits of *π* forms a partition of its diagram (see Fig. 1).

In 1995, Stembridge (5) proved Ian Macdonald’s conjecture that the number of totally symmetric plane partitions with largest part at most *n*, i.e., those whose 3D Ferrers diagram is contained in the cube [0,*n*]^{3}, is given by the elegant product formula Ten years after Stembridge’s completely human-generated proof, Andrews et al. (6) came up with a computer-assisted proof based on an ingenious matrix decomposition, but because no *q*-analog of their decomposition was found, their proof could not be extended to a proof for the *q*-case. A third proof of Stembridge’s theorem (7, 8), even more computerized, was recently found in the context of our investigations of the *q*-TSPP conjecture. We have now succeeded in completing all the required computations for an analogous proof of the *q*-TSPP conjecture, and can therefore announce:

Let *π*/*S*_{3} denote the set of orbits of a totally symmetric plane partition *π* under the action of the symmetric group *S*_{3}. Then the orbit-counting generating function (ref. 3, p. 200, and ref. 2, p. 106) is given by where *T*(*n*) denotes the set of totally symmetric plane partitions with largest part at most *n*.

Our proof is based on a result by Okada (9) who has shown that the theorem is implied by a certain—conjectured—determinant evaluation. These preliminaries are stated accurately in the next section, followed by a description of the holonomic ansatz (10), which we are going to pursue. This approach relies on a kind of oracle that tells us a description of a certificate function *c*_{n,j}; the odyssey how this function has been “guessed” is described in the following section. Once *c*_{n,j} is known, the determinant evaluation reduces further to proving the three identities **1**, **2**, and **3** stated below. Finally we come full circle by proving these identities. Technical details of the proof and our computations as well as the explicit certificates are provided electronically on our Web site http://www.risc.uni-linz.ac.at/people/ckoutsch/qtspp/. Further details which are suppressed here can be found in the detailed description of the proof for the case *q* = 1 which proceeds along the same lines (7, 8).

Our proof is noteworthy not only for its obvious significance in enumerative combinatorics, where it settles a long-standing conjecture, attempted by many people. It is also noteworthy for computational reasons, because the computations we performed went far beyond what has been thought to be possible with currently known algebraic algorithms, software packages, and computer hardware.

## 2. The Telemachiad

In order to prove the *q*-TSPP conjecture, we exploit an elegant reduction by Okada (9) to the problem of evaluating a certain determinant. This determinant is also listed as Conjecture 46 in Krattenthaler’s essay (11) on the art of determinant evaluation.

Let, as usual, *δ*_{i,j} be the Kronecker delta function and let, also as usual, denote the *q*-binomial coefficient. Define the discrete function *a*_{i,j} for *i*, *j*≥1 by Okada’s crucial insight (9) is that Theorem 1 holds if So for proving the *q*-TSPP conjecture, it is sufficient to prove this conjectured determinant evaluation. For this purpose, we apply a computational approach originally proposed in ref. 10, which is applicable to identities of the form where *a*_{i,j} and *b*_{n} (with *b*_{n} ≠ 0 for all *n*≥1) are given explicitly (as it is the case here).

The approach rests on the following induction argument on *n*. For *n* = 1, the identity is trivial. Suppose the identity holds for *n* - 1; then the linear system has a unique solution (*c*_{n,1},…,*c*_{n,n}). The component *c*_{n,j} of this solution is precisely the (*n*, *j*) cofactor divided by the (*n*, *n*) cofactor of the *n* × *n* determinant. The division is meaningful because the (*n*, *n*) cofactor is just the (*n* - 1) × (*n* - 1) determinant, which by induction hypothesis is equal to *b*_{n-1}, which by general assumption is nonzero. Because the *n* × *n* determinant can be expressed in terms of the matrix entries *a*_{n,j} and the normalized cofactors *c*_{n,j} via the induction step is completed by showing that this sum evaluates to *b*_{n}.

The difficulty is that this last summation involves the function *c*_{n,j} (of the discrete variables *n* and *j*) for which we do not have an explicit expression for general *n* and *j*. In order to achieve our goal, we guess a suitable description of a function *c*_{n,j} and then prove that it satisfies the three identities [1][2][3]Once these identities are proved, then, by the argument given before, the *c*_{n,j} must be precisely the normalized (*n*, *j*) cofactors of the *n* × *n* determinant and the determinant evaluation follows as a consequence. So in a sense, the function *c*_{n,j} plays the role of a certificate for the determinant identity.

## 3. The Odyssey

In our setting, the certificate function *c*_{n,j} will be described implicitly by a system of linear recurrence equations in *n* and *j* with coefficients depending polynomially on *q*, *q*^{j}, and *q*^{n}. Such recurrence equations can be phrased as the elements of some noncommutative operator algebras such as where the symbols *S*_{x} represent the shifts *x*↦*x* + 1. If a function is annihilated by certain operators (namely, it satisfies certain recurrence equations), then it is also annihilated by all the elements in the (left) ideal generated by those operators. We speak of an annihilating ideal and represent such ideals by (left) Gröbner bases, so that, for instance, ideal membership can be decided effectively.

The annihilating ideals we use for representing functions are such that they uniquely determine the function up to some finitely many initial values which we can list explicitly. Technically, this requirement means that the ideals have dimension zero and that some particular polynomial coefficients appearing in the recurrence system must not vanish simultaneously for infinitely many points (*n*, *j*). These recurrence systems are similar but somewhat simpler than *q*-holonomic systems (12), which satisfy some additional requirements that are not needed for our proof. We have checked that all the ideals arising in our proof are indeed of the desired form, but in the interest of clarity we suppress a more detailed description of these checks here. For a complete analysis including all technical details, we refer to the supplementary material on our Web site. Also in the interest of clarity, we will from now on identify recurrence equations with their corresponding operators. The details are to a large extent analogous to the special case *q* = 1, which as mentioned in the introduction has been written up in detail in refs. 7 and 8.

A priori, there is no reason why the normalized cofactors *c*_{n,j} should admit a recursive description of the kind we are aiming at, but there is also no reason why they should not. It turns out that they do, and this situation is fortunate because for functions described in this way, techniques are known by which the required identities **1**, **2**, and **3** can be proven algorithmically (12–15). In order to find a recursive description for *c*_{n,j}, we first computed explicitly the normalized cofactors for a few hundred specific indices *n* and *j* by directly solving the linear system quoted in the previous section. Using an algorithm reminiscent of polynomial interpolation, we then constructed a set of recurrences compatible with the values of *c*_{n,j} at the (finitely many) indices we computed.

Polynomial interpolation applied to a finite sample *u*_{1},…,*u*_{k} of an infinite sequence *u*_{n} will always deliver some polynomial *p* of degree at most *k* - 1 which matches the given data. If it turns out that this polynomial matches some further sequence terms *u*_{k+1},*u*_{k+2},…, then it is tempting to conjecture that *u*_{n} = *p*(*n*) for all *n*. The more specific points *n* are found to match, the higher is the evidence in favor of this conjecture.

Very much analogously, it is possible to extract recurrence equations from some finite number of values *c*_{n,j}. The equations become trustworthy if they also hold for points (*n*, *j*) which were not used for their construction. In this way, we have discovered a system of potential recurrence equations for the *c*_{n,j}, which, despite being respectable in size (about 30 Mbytes), appears to be a rather plausible candidate for a recursive description of the normalized cofactors *c*_{n,j}. The system is available for download on our Web site. It consists of three equations involving shifts of order up to four (with respect to both *n* and *j*), with polynomial coefficients of degrees up to 52 (with respect to *q*^{n}) and 24 (with respect to *q*^{j}). Some further statistics on the set of operators can be found in a preliminary version of this article (16).

## 4. The Nostos

Now we switch our point of view. We discard the definition that *c*_{n,j} be the normalized (*n*, *j*) cofactor and redefine the discrete function *c*_{n,j} as the unique solution of the guessed recurrence system whose (finitely many) initial values agree with the normalized (*n*, *j*) cofactor. If we succeed in proving that *c*_{n,j} defined in this way satisfies the identities **1**, **2**, and **3**, then we are done. For this purpose, we provide operators which belong to the annihilating ideal of *c*_{n,j} or related ideals and have certain features which imply the desired identities. So in a sense, these operators play the rôle of certificates for the identities under consideration.

Because of their astronomical size (up to 7 Gbytes; equivalent to more than one million printed pages; corresponding to about 2.5 tons of paper), these certificates are not included explicitly in this article but provided only electronically on our Web site. Also because of their size, it was not possible to construct them by simply applying the standard algorithms from refs. 12–15. A detailed explanation of how exactly we found the certificates is beyond the scope of this article and will be given in a separate publication. But this lack of explanation does not at all affect the soundness of our proof, because the correctness of the certificates can be checked independently by simply performing ideal membership tests. Our certificates are so big that even this “simple” calculation is not quite trivial, but a reader with a sufficient amount of patience and programming expertise will be able to do it.

We proceed by explaining the properties of the certificates provided on our Web site and why they imply [**1**], [**2**], and [**3**].

### A Certificate for [1].

To certify that *c*_{n,n} = 1 for all *n*≥1, we provide a recurrence in the annihilating ideal of *c*_{n,j} which is of the special form By virtue of the substitution *j*↦*n*, it translates into a recurrence for the diagonal sequence *c*_{n,n}. It is a relatively cheap computation to check that this recurrence contains the operator *S*_{n} - 1 annihilating the constant sequence 1 as a (right) factor. This fact implies that *c*_{n,n} and the constant sequence 1 both satisfy the same seventh-order recurrence. Therefore, after checking *c*_{1,1} = *c*_{2,2} = ⋯ = *c*_{7,7} = 1 and observing that *p*_{7}(*q*,*q*^{n},*q*^{n}) ≠ 0 for all , it can be concluded that *c*_{n,n} = 1 for all *n*.

Similarly, we showed that *c*_{n,0} = 0 for all *n*≥1 and that *c*_{n,j} = 0 for all *j* > *n*. This knowledge greatly simplifies the following proofs of the summation identities.

### Certificates for [2].

In order to prove the first summation identity, we translate [**2**] into the equivalent formulation [4]taking into account that *c*_{n,0} = 0. We provide certificates for **4**.

To this end, let be the summand of the sum on the left-hand side. A recursive description for can be computed directly from the defining equations of *c*_{n,j} and the fact that the rest of the summand is a *q*-hypergeometric factor. In the corresponding operator ideal *I*^{′}, we were able to find two different recurrence equations for which are of the special form and respectively, where the *p*_{μ,ν} and *r*_{μ,ν} are certain rational functions in and *u*_{n,i,j} and *v*_{n,i,j} are -linear combinations of certain shifts of which are determined by the Gröbner basis of *I*^{′} as described in ref. 15. Next observe that for *j* ≤ 0 (because of the *q*-binomial coefficient) and also for *j* > *n* (because *c*_{n,j} = 0 for *j* > *n*). When summing the two recurrence equations for *j* from -∞ to +∞, the right-hand side telescopes to zero and the left-hand side turns into a recurrence for the sum in [**4**]. This method is known as creative telescoping. Finally we obtain two annihilating operators *P*_{1} and *P*_{2} for the left-hand side of [**4**].

For the right-hand side of [**4**], we can again construct an ideal of recurrences from the defining equations of *c*_{n,j}. It turns out that this ideal contains *P*_{1} and *P*_{2}, so that both sides of [**4**] are annihilated by these two operators. Additionally, *P*_{1} and *P*_{2} have been constructed such that they require only finitely many initial values to produce a uniquely determined bivariate sequence (and this finite number of needed values is known explicitly). The proof was completed by checking that the two sides of [**4**] agree at these finitely many points (*n*, *j*).

### The Certificate for [3].

To certify the final identity, we rewrite [**3**] equivalently into [5]where (*a*; *q*)_{n}≔(1 - *a*)(1 - *qa*)⋯(1 - *q*^{n-1}*a*) denotes the *q*-Pochhammer symbol. As before, we use creative telescoping to provide a certified operator *P* which annihilates the sum on the left-hand side. In the present case, a single operator is sufficient because the sum depends only on a single variable *n* (there is no *i* there). Using the operator *P* and the defining equations for *c*_{n,j}, we then construct a recurrence for the entire left-hand side, which turns out to have order 12. This recurrence is a left multiple of the second-order operator which can be seen to annihilate the right-hand side, and its leading coefficient does not vanish for any index *n*. Therefore the proof of Theorem 1 is completed by checking [**5**] for *n* = 1,2,…,12. *Quod erat demonstrandum*.

## 5. Epilogue

Paul Erdős famously believed that every short and elegant mathematical statement has a short and elegant “proof from The Book,” and if humans tried hard enough, they would eventually find it. Kurt Gödel, on the other hand, meta-proved that there exist many short and elegant statements whose shortest-possible proof is very long. It is very possible that the *q*-TSPP theorem *does* have a yet-to-be-found *proof from The Book*, but it is just as possible that it does not, and although we are sure that the present proof is not the shortest possible (there were a lot of random choices in designing the proof), it may well be the case that the shortest-possible proof is still very long, and would still require heavy-duty computer calculations.

## Acknowledgments

C.K. was supported by grants National Science Foundation—Division of Mathematical Sciences 0070567 and Fonds zur wissenschaftlichen Förderung (FWF) P20162-N18. M.K. was supported by FWF Y464-N18. D.Z. was supported in part by the US National Science Foundation.

## Footnotes

^{2}To whom correspondence should be addressed. E-mail: mkauers{at}risc.jku.at.Author contributions: C.K., M.K., and D.Z. performed research and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

## References

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- Stanley R

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- Andrews GE

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- Koutschan C

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- Koutschan C

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- Foata D,
- Han G-N

- Krattenthaler C

- ↵
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- Takayama N

- ↵
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*q*-TSPP Conjecture (modulo a finite amount of routine calculations) ArXiv:0808.0571.

*q*-TSPP conjecture

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