MacMahon s Partition Analysis VIII: Plane Partition Diamonds

MacMahon s Partition Analysis VIII: Plane Partition Diamonds George E. Andrews * Department of Mathematics The Pennsylvania State University University Park, PA 6802, USA E-mail: andrews@math.psu.edu Peter Paule Research Institute for Symbolic Computation Johannes Kepler University Linz A 4040 Linz, Austria E-mail: Peter.Paule@risc.uni-linz.ac.at and Axel Riese Research Institute for Symbolic Computation Johannes Kepler University Linz A 4040 Linz, Austria E-mail: Axel.Riese@risc.uni-linz.ac.at DEDICATED TO DOMINIQUE FOATA ON THE OCCASION OF HIS 65TH BIRTHDAY In his famous book Combinatory Analysis MacMahon introduced Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. However, MacMahon failed in his attempt to use his method for a satisfactory treatment of plane partitions. It is the object of this article to show that nevertheless Partition Analysis is of significant value when treating non-standard types of plane partitions. To this end plane partition diamonds are introduced. Applying Partition Analysis a simple closed form for the full generating function is derived. In the discovering process the Omega package developed by the authors has played a fundamental rôle. * Partially supported by NSF Grant DMS-9206993 Supported by SFB Grant F305 of the Austrian FWF.

2 G.E. ANDREWS, P. PAULE, AND A. RIESE. INTRODUCTION In his famous book Combinatory Analysis [6, Vol. II, Section VIII, pp. 9 70] MacMahon introduced Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. In Chapter II of Section IX he starts out to consider plane partitions as a natural application domain for his method. MacMahon begins by discussing the most simple case [6, Vol. II, p. 83], namely where non-negative integers a i are placed at the corner of a square such that the following order relations are satisfied: a a 2, a a 3, a 2 a 4 and a 3 a 4. ( By using Partition Analysis he derives that D := x a xa2 2 xa3 3 xa4 4 = x 2 x 2 x (2 3 ( x x x 2 x x 3 x x 2 x 3 x x 2 x 3 x 4, where the sum is taken over all non-negative integers a i satisfying (. Furthermore, he observes that if x = x 2 = x 3 = x 4 = q, the resulting generating function is ( q q 2 2 ( q 3. In order to see how Partition Analysis works on (2 we need to recall the key ingredient of MacMahon s method, the Omega operator Ω. Definition.. The operator Ω is given by Ω s = s r= A s,...,s r λ s λsr r := s =0 A s,...,s r, s r=0 where the domain of the A s,...,s r is the field of rational functions over C in several complex variables and the λ i are restricted to a neighborhood of the circle λ i =. In addition, the A s,...,s r are required to be such that any of the series involved is absolute convergent within the domain of the definition of A s,...,s r. We emphasize that it is essential to treat everything analytically rather than formally because the method relies on unique Laurent series representations of rational functions.

PLANE PARTITION DIAMONDS 3 Another fundamental aspect of Partition Analysis is the use of elimination rules which describe the action of the Omega operator on certain base cases. MacMahon begins the discussion of his method by presenting a catalog [6, Vol. II, pp. 02 03] of twelve fundamental evaluations. Subsequently he extends this table by new rules whenever he is forced to do so. Once found, most of these fundamental rules are easy to prove. This is illustrated by the following examples which are taken from MacMahon s list. Proposition.. For integer s, Ω Ω ( λa ( B λ s = ( λa λb ( C λ = ( A A s B ; (3 ABC ( A B AC BC. (4 We prove (3; the proof of (4 is analogous and is left to the reader. Proof (of (3. By geometric series expansion the left hand side equals Ω i,j 0 λ i sj A i B j = Ω j,k 0 λ k A sj+k B j, where the summation parameter i has been replaced by sj +k. But now Ω sets λ to which completes the proof. Now we are ready for deriving the closed form expression for D with Partition Analysis. Proof (of (2. First, in order to get rid of the diophantine constraints, one rewrites the sum expression in (2 into what MacMahon called the crude form of the generating function, D = Ω = Ω a,a 2,a 3,a 4 0 λ a a2 λ a a3 2 λ a2 a4 3 λ a3 a4 4 x a xa2 2 xa3 3 xa4 4 ( λ λ 2 x λ 3 λ x 2 λ 4 λ 2 x 3 x 4. λ 3λ 4 Next by rule (3 we eliminate successively λ, λ 3, and λ 4, D = Ω ( λ2 x λ2 λ 3 x x 2 λ 4 λ 2 x 3 x 4 λ 3λ 4 = Ω ( λ2 x λ2 x x 2 λ 4 λ 2 x 3 λ 2x x 2x 4 λ 4 (5

4 G.E. ANDREWS, P. PAULE, AND A. RIESE = Ω ( λ2 x λ2 x x 2 x 3. λ 2 x x 2 x 3 x 4 Finally, applying rule (4 eliminates λ 2 and completes the proof of (2. After considering some further questions about plane partitions on a square, MacMahon writes down the crude form for the general case, i.e. the Ω expression for the full generating function for plane partitions of m rows, l columns and each part not exceeding n; see [6, Vol. II, p. 86] and also []. But a few lines later MacMahon writes: Our knowledge of the Ω operation is not sufficient to enable us to establish the final form of result. This will be accomplished by the aid of new ideas which will be brought forward in the following chapters. Despite MacMahon s negative statement, in this article our object is to show that Partition Analysis nevertheless is an extremely valuable tool in studying plane partitions of non-standard type. In Section 2 we will consider plane partition diamonds for which Partition Analysis enables to derive an elegant expression for the full generating function. In the concluding Section 3 we present two further types of possible plane partition generalizations. 2. PLANE PARTITION DIAMONDS It will be convenient to introduce alternative descriptions for relations. For instance, an alternative description of the inequalities ( is Figure below. It is understood that an arrow pointing from a i to a j is a 2 a a 4 a 3 FIG.. The inequalities ( interpreted as a i a j. In the spirit of MacMahon s Partition Analysis we introduce another equivalent description of the inequalities (, namely λ a a2 λ a a3 2 λ a2 a4 3 λ a3 a4 4. (6

PLANE PARTITION DIAMONDS 5 Each λ variable stands for an arrow, i.e. if its exponent is a i a j, it is interpreted as a i a j. Now we are in the position to describe plane partition diamonds. Instead of gluing such squares together as in the case of standard plane partitions, we consider the configurations shown in Figure 2. Note that a 2 a 5 a 8 a 3n a... a 3n+ a 4 a 7 a 0 a 3n 2 a 3 a 6 a 9 a 3n FIG. 2. Diamond of length n the arrows can be also interpreted as a flow, so a is considered as the source and a 3n+ as the sink. For n we call such a configuration a diamond of length n. A more formal description can be given as follows. Definition 2.. For n, Λ n := λ a3n 2 a3n 4n 3 λ a3n 2 a3n 4n 2 λ a3n a3n+ 4n λ a3n a3n+ 4n. Obviously, Λ is the product in (6 and represents one-to-one the four inequalities (. In view of Figure 2 this leads to the following definition. Definition 2.2. is called diamond of length n. For n the product D n := Λ Λ 2 Λ n Obviously, Λ Λ 2 Λ n gives a precise description of the 4n inequalities described by Figure 2. Definition 2.3. For n we define D n := D n (x,..., x 3n+ := x a xa3n+ 3n+, where the sum ranges over all non-negative integers a,..., a 3n+ which satisfy the inequalities encoded by D n.

6 G.E. ANDREWS, P. PAULE, AND A. RIESE Thus D n is the full generating function for all diamonds of fixed length n. We are also interested in diamonds with sink ρ, ρ N. Definition 2.4. D (ρ n For n, ρ 0 we define := D n (ρ (x,..., x 3n+ := x a xa3n+, where the sum ranges over all non-negative integers a,..., a 3n+ which satisfy the inequalities encoded by D n and where a 3n+ ρ. Ω 2.. The Crude Generating Function In this subsection we will define the crude forms of D n and D n (ρ, i.e. the expressions for D n and D n (ρ. To this end we need a few definitions. and Definition 2.5. For k, n, f k := ( λ 4k λ 4k 3 x 3k λ 4k λ 4k 2 x 3k λ 4k+ λ 4k+2, λ 4k λ 4k x 3k+ g n := λ4n+λ 4n+2 λ 4n λ 4n x 3n+, x3n+ λ 4n λ 4n h := λ λ 2 x. Note that if x 3n+ is replaced by λ 4n+ λ 4n+2 x 3n+ then Proposition 2.. For n, f n g n turns into f n. (7 D n = Ω h f f n g n. (8 Proof. We proceed by induction on n. The case n = corresponds to the most simple plane partition case (5. Suppose (8 is true for n, then D n+ = Ω Λ Λ n+ x a xa3n+4 a i 0

PLANE PARTITION DIAMONDS 7 = Ω Λ Λ n x a xa3n (λ 4n+λ 4n+2 x 3n+ a3n+ a i 0 ( λ4n+3 ( a3n+2 λ a3n+3 4n+4 x 3n+2 x 3n+3 λ 4n+ λ 4n+2 ( x a3n+4 3n+4 λ 4n+3 λ 4n+4 = Ω h f f n f n+ g n+, where the last line is by the induction hypothesis and by (7. Proposition 2.2. For n and ρ 0, D (ρ n = Ω h f f n g n ( x3n+ λ 4n λ 4n ρ. (9 Proof. The induction proof with respect to n is entirely analogous to the proof of Proposition 2. and is left to the reader. 2.2. The Diamond Generating Function In this subsection we will prove our main theorem. Theorem 2.. For n, D n (x,..., x 3n+ = where X k = x x 2 x k, k. ( X X 2 ( X 3n+ X X 3 X 4X 6 X3 x 2 X6 x 5 X 3n 2X 3n, X3n x 3n Before we turn to the proof of Theorem 2., we introduce two corollaries. Corollary 2.. For n, D n (q,..., q = ( + q2 + q 5 + q 8 ( + q 3n ( q q 2 q 3 ( q 3n+. Proof. The proof is immediate from Theorem 2.. Corollary 2.2. For n and ρ 0, D (ρ n (x,..., x 3n+ = X ρ 3n+ D n(x,..., x 3n+, (0

8 G.E. ANDREWS, P. PAULE, AND A. RIESE where X k = x x 2 x k as in Theorem 2.. Proof. We fix n and prove the statement by induction on ρ. For ρ = 0 the assertion is trivial. Suppose (0 is true for ρ. Obviously, D (ρ+ n = D (ρ n x ρ 3n+ xρ 3n+ D(ρ n, where x ρ k F (x,..., x m stands for the coefficient of x ρ k in F (x,..., x m. From (9, x ρ 3n+ D(ρ But by Theorem 2. we have n = x ρ 3n+ Xρ 3n+ D n = Xρ 3n+ x 0 3n+ D n. x ρ 3n+ x 0 3n+ D n = ( X 3n+ D n x 0 3n+ X 3n+ = ( X 3n+ D n. Collecting these facts and applying (9 we obtain D (ρ+ n = X ρ 3n+ D n X ρ 3n+ ( X 3n+D n = X ρ+ 3n+ D n, which completes the proof. Basically we are ready for the proof of Theorem 2.. However, it will be convenient to introduce an elementary lemma. Definition 2.6. Let k, and y,..., y k, z 0 be distinct elements from a suitable field. Define p(y; z := k ( i= y i z where y = (y,..., y k. For j k define p j (y; z := k ( i= i j y i z,. Lemma 2.. Let k and y = (y,..., y k, then p(y; z = k j= p j (y; y j. yj z

PLANE PARTITION DIAMONDS 9 Proof. The assertion is immediate by partial fraction decomposition. Proof (of Theorem 2.. We proceed by induction on n. For n = the statement corresponds to the simplest case of classical plane partitions, namely (2. Suppose the theorem holds for n. By Proposition 2., D n+ = Ω h f f n+ g n+ = Ω h f f n ( λ 4n λ 4n 3 x 3n ( λ 4n+λ 4n+2 λ 4n λ 4n x 3n+ λ 4n+3 λ 4n+ x 3n+2 ( λ 4n+4 λ 4n+2 x 3n+3 x 3n+4. λ 4n+3λ 4n+4 λ 4n λ 4n 2 x 3n By applying case n = of Theorem 2. to the last four factors we obtain, with D n+ = Ω h f f n x2 3n+ x3n+2x3n+3 λ 2 4n λ2 4n p(y; λ 4n λ 4n ( λ 4n λ 4n 3 x 3n λ 4n λ 4n 2 x 3n y = (y, y 2, y 3, y 4, y 5 = (x 3n+, x 3n+ x 3n+2, x 3n+ x 3n+3, x 3n+ x 3n+2 x 3n+3, x 3n+ x 3n+2 x 3n+3 x 3n+4. ( Now applying Lemma 2. yields D n+ = p j (y; y j D n (x,..., x 3n, y j j= x 2 3n+x 3n+2 x 3n+3 j= p j (y; y j y 2 j D (2 n (x,..., x 3n, y j. By Corollary 2.2 with ρ = 2 (implied by the induction hypothesis, D n+ = p j (y; y j D n (x,..., x 3n, y j j=

0 G.E. ANDREWS, P. PAULE, AND A. RIESE Observing that we obtain (x x 3n 2 x 2 3n+x 3n+2 x 3n+3 p j (y; y j D n (x,..., x 3n, y j j= = ( X 3n+ X 3n+3 p j (y; y j D n (x,..., x 3n, y j. j= D n (x,..., x 3n, y = X 3n+ X 3n y D n(x,..., x 3n, x 3n+ D n+ = ( X 3n+ X 3n+3 X 3n+ j= But it is routine (computer algebra computation that j= p j (y; y j X 3n y j p j (y; y j X 3n y j D n. = ( X 3n+ X 3n+2 X 3n+3 X 3n+4 ( X3n+3 x 3n+2 for y = (y,..., y 5 as in (. Hence, D n+ = ( X 3n+2 X 3n+3 X 3n+4 X 3n+ X 3n+3 X3n+3 x 3n+2 D n and the proof of Theorem 2. is completed. 3. CONCLUSION As shown in a series of articles [2, 3, 4, 5] Partition Analysis is ideally suited for being supplemented by computer algebra methods. In these papers the Mathematica package Omega which had been developed by the authors, was used as an essential tool. The package is freely available from the Web via http://www.risc.uni-linz.ac.at/research/combinat/risc/software/ Omega. The Omega package played a crucial rôle also in discovering Theorem 2. above. For instance, the simplest case (2 is treated automatically by the package as follows.

PLANE PARTITION DIAMONDS After loading the file Omega2.m by In[]:= <<Omega2.m Out[]= Axel Riese s Omega implementation version 2.30 loaded one computes the crude form of the generating function D as follows: In[2]:= OSum[x a x a2 2 x a3 3 x a4 4, {a a 2, a a 3, a 2 a 4, a 3 a 4 }, λ] Assuming a 0 Assuming a 2 0 Assuming a 3 0 Assuming a 4 0 Out[2]= Ω λ,λ 2,λ 3,λ 4 ( x λ λ 2 ( ( x2 λ3 λ x 4 ( λ 3 λ 4 x 3 λ 4 λ 2 Finally, elimination can be done in one stroke and within a second: In[3]:= OR[%] Eliminating λ 4... Eliminating λ 3... Eliminating λ 2... Eliminating λ... Out[3]= x 2 x 2 x 3 ( x ( x x 2 ( x x 3 ( x x 2 x 3 ( x x 2 x 3 x 4 Already this elementary application indicates the usefulness of the Omega package for a further, more detailed study of possible new plane partition generalizations. We conclude by mentioning two further examples which have been found experimentally. The first example is a variation of the diamond theme. Instead of arranging diamonds in a row, we arrange them in hook shape as shown in Figure 3. Note that the ordering imposed is such that we now have 2 sources, a and a 0, and sink, namely a 5. The corresponding full

2 G.E. ANDREWS, P. PAULE, AND A. RIESE a 6 a 8 a 5 a 0 a 7 a 4 a 9 a 2 a 3 a FIG. 3. Diamond hook generating function turns out to be x a xa0 ( x 2 = x 2 x 3 x 8 x 9 x 2 0 ( x x 0 x x 2 x x 3 x 8 x 0 ( x 9 x 0 x x 2 x 3 x 8 x 9 x 0 x x 2 x 3 x 4 ( x x 2 x 3 x 4 x 7 x 8 x 9 x 0 x x 2 x 3 x 4 x 6 x 7 x 8 x 9 x 0 ( x x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 0. If we set all x i to q the product turns into + q 2 ( q 2 ( q 2 3 ( q 3 2 ( q 8 q 9 q 0. Other types of generalized plane partitions can be obtained by a variation of the order taken in the classical case. Consider, for instance, the reversed hook from Figure 4. Here we have again 2 sources, a and a 8, and sink, namely a 5. Despite the fact that the full generating function does not factor, the case x i = q, i 8, again is nice: ( q 2 ( q 2 2 ( q 3 q 4 2 ( q 5. We conclude by the remark that without the Omega package such observations can hardly be made.

PLANE PARTITION DIAMONDS 3 a 5 a 4 a 2 a a 3 6 a a 7 a 8 FIG. 4. Reversed standard hook REFERENCES. G.E. Andrews, MacMahon s partition analysis II: Fundamental theorems, Ann. Comb. 4 (2000. 2. G.E. Andrews, P. Paule and A. Riese, MacMahon s partition analysis III: The Omega package, SFB Report 99-24, J. Kepler University, Linz, 999, (to appear. 3. G.E. Andrews, P. Paule and A. Riese, MacMahon s partition analysis VI: A new reduction algorithm, (to appear. 4. G.E. Andrews, P. Paule and A. Riese, MacMahon s partition analysis VII: Constrained compositions, (to appear. 5. G.E. Andrews, P. Paule, A. Riese and V. Strehl, MacMahon s partition analysis V: Bijections, recursions, and magic squares, SFB Report 00-8, J. Kepler University, Linz, 2000, (to appear. 6. P.A. MacMahon, Combinatory Analysis, 2 vols., Cambridge University Press, Cambridge, 95 96 (Reprinted: Chelsea, New York, 960.