MACMAHON S PARTITION ANALYSIS IX: -GON PARTITIONS GEORGE E. ANDREWS, PETER PAULE, AND AXEL RIESE Dedicated to George Szeeres on the occasion of his 90th birthday Abstract. MacMahon devoted a significant portion of Volume II of his famous boo Combinatory Analysis to the introduction of Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. In a series of papers we have shown that MacMahon s method turns into an extremely powerful tool when implemented in computer algebra. In this note we explain how the use of the pacage Omega developed by the authors has led to a generalization of a classical counting problem related to triangles with sides of integer length.. Introduction In his famous boo Combinatory Analysis [2, Vol. II, Sect. 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. We will use MacMahon s method and the Omega pacage for a study of a classical combinatorial problem related to triangles with sides of integer size. We start out by stating the well-nown base case which has been discussed at various places; see e.g. [, 9,, 0, 8], and [3, Ch. 4, Ex. 6]. Problem. Let t 3 n be the number of non-congruent triangles whose sides have integer length and whose perimeter is n. For instance, t 3 9 3, corresponding to 3 + 3 + 3, 2 + 3 + 4, + 4 + 4. Find n 3 t 3nq n. Obviously the corresponding generating function is T 3 q : n 3 t 3 n q n q a +a 2+a 3 where is the restricted summation over all positive integer triples a, a 2, a 3 satisfying a a 2 a 3 and a + a 2 > a 3. In order to see how Partition Analysis can be used to compute a closed form representation for n 3 t 3nq n, we need to recall the ey ingredient of MacMahon s method, the Omega operator. Date: February 22, 200. The first author was partially supported by NSF Grant DMS-9206993. The third author was supported by SFB Grant F305 of the Austrian FWF.
2 GEORGE E. ANDREWS, PETER PAULE, AND AXEL RIESE Definition. The operator is given by s s r A s,...,s r λ s λsr r : s 0 A s,...,s r, 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. 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 [2, Vol. II, pp. 02 06] of fundamental evaluations. Subsequently he extends this table by new rules whenever he is enforced to do so. Once found, most of these fundamental rules are easy to prove. This is illustrated by the following examples which are taen from MacMahon [2, Vol. II, Art. 354, p. 06]. s r0 Proposition. For integer s 0 and variables A, B being free of λ, λ s λa B λ λ s λa B λ i,j 0 A s A AB ; 2 AB Bs+ + AB s+ A B AB. 3 Proof. Rule 3 is a special case of the more general rule 3; see Lemma 2. Rule 2 is proved as follows. By geometric series expansion the left-hand side equals λ i j s A i B j λ A +j+s B j, j, 0 where the summation parameter i has then been replaced by + j + s. But now sets λ to, which completes the proof. Now we are ready for deriving the closed form expression for T 3 q with Partition Analysis. First, in order to get rid of the diophantine constraints, one rewrites the restricted sum expression in into what MacMahon has called the crude form of the generating function, T 3 q a, a 2,a 3 0 λ a2 a λ a3 a2 2 λ a+a2 a3 3 q a+a2+a3 qλ q λ q λ q, where the last line is by geometric series summation.
MACMAHON S PARTITION ANALYSIS IX: -GON PARTITIONS 3 Next by applying again rule 2 we eliminate successively, λ, and, T 3 q qλ q λ q q2 λ q 3 q 2 q 3 q q 3 q 2 q 3 q 4. 4 This completes the generating function computation and Problem is solved. With our pacage Omega the whole computation can be done automatically and in one stroe. Note that setting-up the crude generating function is done also by the pacage: In[]: <<Omega2.m Out[] Axel Riese s Omega implementation version 2.33 loaded In[2]: OSum[q a+a2+a3, {a 2 a, a 3 a 2, a +a 2 > a 3, a }, λ] Out[2] Assuming a 2 0 Assuming a 3 0 In[3]: OR[%] Out[4] λ,, Eliminating... Eliminating... Eliminating λ... q λ q q λ q λ q 3 q 2 q 3 q 4 As already pointed out in [2], with Partition Analysis one is able to derive much more information. Namely, we can consider the full generating function S 3 x, x 2, x 3 : x a xa2 2 xa3 3, where denotes again the restricted summation over all positive integer triples a, a 2, a 3 satisfying a a 2 a 3 and a + a 2 > a 3. On this expression we can carry out essentially the same Partition Analysis steps as above for obtaining a closed form expression for it. For the crude form one gets S 3 x, x 2, x 3 a, a 2,a 3 0 λ a2 a λ a3 a2 2 λ a+a2 a3 3 x a xa2 2 xa3 3 x λ x λ x2 λ x3. available at http://www.risc.uni-linz.ac.at/research/combinat/risc/software/omega
4 GEORGE E. ANDREWS, PETER PAULE, AND AXEL RIESE Next by applying again rule 2, we eliminate successively, λ, and as above and obtain S 3 x, x 2, x 3 x λ x λ x 3 x2 x 3 λ x x 2 x 3 x 2 x 3 x x 2 x 3 x3 x x 2 x 3 x 2 x 3 x x 2 x 3 x x 2 x 2 5 3. This not only generalizes the generating function T 3 q, i.e. T 3 q S 3 q, q, q, but gives rise also to a complete, parameterized solution of the underlying diophantine set of equations a, a a 2, a 2 a 3, and a + a 2 > a 3. This can be seen by geometric series expansion of 5, namely S 3 x, x 2, x 3 x n2+n3+ x n+n2+n3+ 2 x n+n2+2n3+ 3. In other words, by choosing n,n 2,n 3 0 a n 2 + n 3 +, a 2 n + n 2 + n 3 +, and a 3 n + n 2 + 2n 3 +, and running through all non-negative integers n, n 2, n 3, one constructs in a oneto-one fashion all non-degenerate triangles with sides of integer size. In [3] we considered the following generalization of the triangle problem to -gons where 3. Definition 2. As the set of non-degenerate -gon partitions into positive parts we define τ : {a,..., a Z a a 2 a and a + + a > a }. As the set of non-degenerate -gon partitions of n into positive parts we define τ n : {a,..., a τ a + + a n}. The corresponding cardinality is denoted by t n : τ n. The term non-degenerate refers to the restriction to strict inequality, i.e. to a + + a > a. In the form of 4 we computed a rational expression for T 3 q n 3 t 3nq n. With the Omega pacage in hand, we are able to compute also the next cases in a purely mechanical manner. For instance, t 4 n q n q 4 + q + q 5 q 2 q 3 q 4 q 6, 6 and n 4 t 5 n q n q 5 q q q 2 q 4 q 5 q 6 q 8, 7 n 5 t 6 n q n q 6 q 4 + q 5 + q 7 q 8 q 3 q q 2 q 3 q 4 q 6 q 8 q 0. 8 n 6
MACMAHON S PARTITION ANALYSIS IX: -GON PARTITIONS 5 From these results we were able to derive a number of partition theoretical consequences. However, despite the fact that the particular instances of n t nq n can be computed so easily, we were not able to find a common underlying pattern. So we stated as an open problem: Problem 2. In view of the generating function representations 4, 6, 7, and 8: Is it possible to find a common pattern for all possible choices of? In Section 2 we provide an affirmative answer to this problem. More precisely, we give closed form expressions for T q as well as for the corresponding general version S x,..., x defined as follows: Definition 3. For integer 3, T q : n t n q n, and S x,..., x : x a xa. a,...,a τ Finally, Section 3 provides some concluding remars. 2. Generating Functions for -Gon Partitions In this section we prove the following main result for -gon partitions. Theorem. Let 3 and X i x i x for i. Then S x,..., x X X X 2 X X X 2 X X X 2 X X 3 X 2 X X 2. Since T q S q,..., q, Theorem implies the desired generating function representation. Corollary. For 3, T q q q q 2 q q2 2 q 9 q 2 q 4 q 2 2. 0 Remar. It is easily verified that 0 is bringing the representations 4, 6, 7, and 8 of the special cases 3, 4, 5, 6 under one umbrella. We shall prove Theorem with Partition Analysis. To this end we first need the crude form of S x,..., x. Proposition 2. For 3, S x,..., x x λ λ x2 λ λ x λ x3 λ x λ 2 λ λ x λ λ.
6 GEORGE E. ANDREWS, PETER PAULE, AND AXEL RIESE Proof. For fixed integer 3, S x,..., x a a 2,...,a 0 x a xa λa2 a λ a a λ a+ +a a by the definition of. The rest follows by geometric series summation. The next step is the successive elimination of λ,,..., λ from the crude form. For this it is convenient to introduce a lemma. Lemma. Let 3 and let y,..., y be free of λ,..., λ. Then y λ y λ y2 λ y λ 2 λ y λ y y y y y y y. Proof. We proceed by induction on. For 3, y λ y λ y2 λ y3 y λ y λ y3 y 2 y 3 λ by 2 with s 0 y y 2 y 3. y 2 y 3 y 3 y y 2 y 3 by 2 with s For the induction step we apply again rule 2 with s 0, y λ y λ y2 λ y λ 2 λ y λ λ y+ λ y + y λ y λ y2 λ y λ 2 λ y y + λ y + y y + y y + y y y + y y + ; for the last line we used the induction hypothesis. Now we are in the position to state the crude form of S x,..., x. Proposition 3. Let 3 and X i x i x for i. Then S x,..., x X X Proof. By Proposition 2, S x,..., x λ 3 X λ X 2 λ X 3 X λ 2 y λ λ. 2 y λ y2 λ y λ 2 λ y λ, where y x λ,..., y x λ and y x /λ. By Lemma this is equal to x x λ 3 x λ x x x 2 x x λ x x λ 2
MACMAHON S PARTITION ANALYSIS IX: -GON PARTITIONS 7 which is the right-hand side of 2. In order to complete the proof of Theorem we need another elementary lemma; namely, the special case m,, and j i i of our reduction algorithm described in [4]. However, for the sae of better readability we state and prove it explicitly. Lemma 2. Let, a 0, and let y, y,..., y be free of λ. Then λ a y λ y λ y 2 y λ y y y y a+ y y y 2 y 2 y y y. Remar. Formula 3 of Proposition is the special case. Proof. The left hand-side of 3 equals s,...,s 0 r 0 s,...,s 0 y s ys yr λ s+2 s2+ + s +a r y s ys s + + s +a and the lemma follows by applying m r0 yr y m+ / y. Finally we come to the proof of Theorem. Proof of Theorem. By Proposition 3, S x,..., x X X X X r0 λ 3 X λ X 2 λ X 3 X λ 2 X X 2 X 2 X X 2 X 2 X X 3 X 2 X X 2 y r X where the last equality is by Lemma 2 with a 3 and y X, y X 2, y 2 X 3,..., y 2 X. This completes the proof of Theorem. 3. Conclusion As shown in a series of articles [3, 4, 5, 6, 7], Partition Analysis is ideally suited for being supplemented by computer algebra methods. In these papers the Mathematica pacage Omega which had been developed by the authors, was used as an essential tool. The Omega pacage played a crucial role also in discovering Theorem above. However, it is important to note that the computations 4, 6, 7, and 8 for T q with 3, 4, 5, 6 have not led us to Theorem. Rather than this, the main point in the study of -gon partitions was the careful Omega investigation of the full generating function S x,..., x ; only in this generality the underlying pattern was finally revealed. 3,
8 GEORGE E. ANDREWS, PETER PAULE, AND AXEL RIESE Another remar concerns the constructive use of Theorem. As a matter of fact, formula 9 can be used to construct -gon partitions in the same way as explained in the introduction with the special case 5. In [2] refinements of the base case 3 of Theorem and Corollary have been considered. We expect that experiments with the Omega pacage will lead to more general results in this direction. References [] G.E. Andrews, A note on partitions and triangles with integer sides, Amer. Math. Monthly 86 979, 477 478. [2], MacMahon s partition analysis II: Fundamental theorems, Ann. Comb. 4 2000. [3] G.E. Andrews, P. Paule and A. Riese, MacMahon s partition analysis III: The Omega pacage, SFB Report 99-24, J. Kepler University, Linz, 999. to appear [4], MacMahon s partition analysis VI: A new reduction algorithm, SFB Report 0-4, J. Kepler University, Linz, 200. to appear [5], MacMahon s partition analysis VII: Constrained compositions, SFB Report 0-5, J. Kepler University, Linz, 200. to appear [6], MacMahon s partition analysis VIII: Plane Partition Diamonds, SFB Report 0-6, J. Kepler University, Linz, 200. to appear [7] 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 [8] R. Honsberger, Mathematical Gems III, Math. Assoc. of America, Washington, 985. [9] J.H. Jordan, R. Walsh and R.J. Wisner, Triangles with integer sides, Notices Amer. Math. Soc. 24 977, A-450. [0] J.H. Jordan, R. Walsh, and R.J. Wisner, Triangles with integer sides, Amer. Math. Monthly 86 979, 686 689. [] C.I. Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New Yor, 968. [2] P.A. MacMahon, Combinatory Analysis, 2 vols., Cambridge University Press, Cambridge, 95 96. Reprinted: Chelsea, New Yor, 960 [3] R.P. Stanley, Enumerative Combinatorics Volume, Wadsworth, Monterey, California, 986. Department of Mathematics, The Pennsylvania State University, University Par, PA 6802, USA E-mail address: andrews@math.psu.edu Research Institute for Symbolic Computation, Johannes Kepler University Linz, A 4040 Linz, Austria E-mail address: Peter.Paule@risc.uni-linz.ac.at Research Institute for Symbolic Computation, Johannes Kepler University Linz, A 4040 Linz, Austria E-mail address: Axel.Riese@risc.uni-linz.ac.at