A note on partitions into distinct parts and odd parts

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1 c,, 5 () Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. A note on partitions into distinct parts and odd parts DONGSU KIM * AND AE JA YEE Department of Mathematics Korea Advanced Institute of Science and Technology Taejon, Republic of Korea dskim@math.kaist.ac.kr Editor: Abstract. Bousquet-Mélou and Eriksson showed that the number of partitions of n into distinct parts whose alternating sum is k is equal to the number of partitions of n into k odd parts, which is a refinement of a well-known result by Euler. We give a different graphical interpretation of the bijection by Sylvester on partitions into distinct parts and partitions into odd parts, and show that the bijection implies the above statement. Keywords: Partitions. Introduction A finite nonincreasing sequence of positive integers, λ λ λ 2 λ l, is called an integer partition of n, where n l i λ i. We follow standard notations in []. The length of a partition λ is the number of its parts, denoted l(λ). Let λ, called the size of λ, denote the number which λ is a partition of. Though each part in a partition is positive, we sometimes allow a zero as a part. Let P d denote the set of all partitions whose parts are all distinct, and let P o denote the set of all partitions whose parts are all odd. If λ P d is a partition of odd length, then we can add a zero as the last part, which changes the length of λ into even. So we assume that each partition in P d has an even length. It is well known, originally due to Euler, that the generating function of P d is equal to that of P o, which proves that for any nonnegative integer n, the number of partitions of n into distinct parts is equal to the number of partitions of n into odd parts. In terms of generating functions, it is written as follows: λ P d q λ ( q; q) (q; q 2 ) λ P o q λ. () There are well known proofs of this identity. One of them is by Sylvester, which can be found on pages in [2] and pages 3 in [5]. It actually proves more general result that the number of partitions of n into odd parts (repetitions allowed) with exactly k distinct parts appearing is equal to the number of partitions of n * Partially supported by KOSEF : and SAMSUNG

2 2 into distinct parts such that exactly k sequences of consecutive integers occur in each partition ([], Theorem 2.2). Recently there has been another proof by Bousquet-Mélou and Eriksson, [3, ], establishing the identity q λ x λ a q λ x l(λ), (2) λ P d λ P o l(λ) where λ a denotes the alternating sum of λ, i.e. i ( )i λ i. They obtain more general results on the lecture hall partitions and show that the above identity is a limit case. But they do not give a combinatorial bijection establishing the identity (2). Identity (2) is stronger than identity (). It says that the number of partitions of n into distinct parts whose alternating sum is k is equal to the number of partitions of n into k odd parts. In this paper, we give a different description for the bijection by Sylvester and bring out a hidden property of the bijection, i.e. it is a simple, bijective proof of identity (2). Our description uses a special way of drawing a diagram associated to a partition in P d. Our diagram is different from that used by Sylvester. The existence of different diagrams shows the richness of the graphical methods in the theory of partitions. It turns out that the bijection establishes combinatorially (xq; q 2 ) i0 q 2i2 i x i, (3) which is a special case of the Cauchy formula (aq; q) i0 q i2 a i (q; q) i (aq; q) i () under the substitutions q q 2 and a x/q. The Cauchy formula itself has a combinatorial interpretation in terms of partitions given by Frobenius notation. But our result doesn t follow from it. 2. A new description of Sylvester s bijection We first describe a mapping φ from P d into P o, establishing the identity (2). This in fact coincides with the inverse of the bijection by Sylvester described in [5], Theorem 2.2. But the diagrams involved look totally different so that a separate description is worthwhile. By allowing the last part to be a zero, we may assume that any λ P d has even length. Let λ λ λ 2... λ 2k be a partition in P d. We need to obtain the unique partition φ(λ) in P o with λ a parts. We stack λ boxes in l(λ) rows as follows (Figure ). We first arrange λ boxes horizontally to form the first row. The second row consisting of λ 2 boxes is put on the top of the first row, aligning the leftmost box with the leftmost box in the row below. The third row consisting

3 3 of λ 3 boxes is put on the top of the second row, aligning the rightmost box with the rightmost box in the row below. We repeat this process, alternating between leftmost and rightmost. Then we get a diagram of height 2k, the top row of which may contain no box. It happens if λ 2k 0. We put a vertical bar through the whole diagram, called the separator, dividing the first row into two pieces, λ a boxes to the right and λ λ a boxes to the left. Then the columns to the right of the separator may be regarded as a partition into odd parts. Let π π π 2... π λ a be the partition formed with the heights of the columns to the right of the separator. It is clear that π 2k and π λ a and any odd integer between and 2k appears in π as a part at least once. We now prepare another partition µ µ µ 2... µ 2k, where µ i is the number of boxes to the left of the separator in row i. Then µ 2i µ 2i > µ 2i+. For i, 2,..., k, add 2µ 2i ( µ 2i + µ 2i ) to the rightmost 2i in π. Let σ be the partition obtained by sorting the resulting sequence. It is clear that each part in σ is odd and l(σ) λ a. Define φ(λ) σ. An example is given in Figure Figure. λ λ a. φ(λ) Theorem The mapping φ from P d into P o is a bijection such that λ φ(λ) and λ a l(φ(λ)), establishing the identity (2). Moreover, the inverse of φ is the Sylvester bijection in [5], Section 29. Proof: The above mapping φ(λ) can be defined in terms of formulas as well. Let ρ ρ ρ 2... ρ 2k be the partition defined by ρ 2i λ 2i λ 2i + λ 2i+ + λ 2k λ 2k, for i, 2,..., k and ρ 2i ρ 2i+ for i, 2,..., k. Note that ρ 2k > 0. For each i, ρ i is the length of row i to the right of the separator. Let µ µ µ 2... µ 2k be the partition defined by µ 2i λ 2i ρ 2i and µ 2i µ 2i for i, 2,..., k. Note that µ 2k 0. For each i, µ i is the length of row i to the left of the separator. Let π π π 2... π ρ be the conjugate of ρ. Then π 2k, π ρ and each odd integer between and 2k occurs in π at least once. For each 2i, add 2µ 2i 2(λ 2i ρ 2i ) to the rightmost 2i in π. Let σ be

4 the partition obtained by sorting the resulting sequence. Then φ(λ) σ. It is clear that λ σ and λ a l(σ). The inverse of φ is defined as follows. Let σ σ σ 2... σ l be a partition into odd parts. Let k be the largest integer such that σ k 2k. Sort the integers, 3,..., 2k, σ k+, σ k+2,..., σ l to get a new partition π π π 2... π l. Then π 2k and π l. Let ρ ρ ρ 2... ρ π be the conjugate of π. Since each part in π is odd, ρ 2i ρ 2i+, for i, 2,..., k ; and since 2i, for i, 2,..., k, appears in π, we obtain ρ 2i > ρ 2i, for i, 2,..., k. Let λ λ λ 2... λ 2k be the partition such that λ 2i ρ 2i + 2 (σ 2i 2i+) and λ 2i ρ 2i + 2 (σ 2i 2i+), for i, 2,..., k. Then λ 2i > λ 2i for i, 2,..., k, since ρ 2i > ρ 2i ; λ 2i > λ 2i+ for i, 2,..., k, since σ 2i (2i ) > σ 2i+ (2i + ). Then φ (σ) λ. If we examine the description of the bijection by Sylvester in [5], Section 29, we find that it coincides with the description of the inverse of φ. The diagrams which show up in the description of φ (in Figure ) can be counted directly as well. The diagram to the left of the separator is a partition λ with 2i parts, for some i, where each part appears exactly twice and 0 is allowed as a part. The diagram to the right of the separator is a partition µ with 2i 2l(λ) parts, where each part but the largest part appears exactly twice and the largest appears exactly once. A standard technique in partition theory [] gives q i2 i q i 2x i i0 as the generating function of these pairs (λ, µ), where x traces the number of columns to the right of the separator, i.e. µ, and the index i denotes the half of the height of the diagram, i.e. 2l(λ). It agrees with the righthand side of (3). So identity (3) follows from Theorem 2., since diagrams described in Figure correspond to partitions into distinct parts. 3. Remarks We may apply the same method, aligning alternatively at the left and at the right, to the partitions in which parts are different by at least 2. We obtain q λ x λ a q i2 2i x 2i (q 2 ; q 2 ) i (xq; q 2 + qx ) i λ i0 i0 q i2 +i x 2i, where the sum is over the set of partitions λ in which parts are different by at least 2. For the partitions in which parts are different by at least 2 and the smallest part is at least 2, we obtain q λ x λ a q i2 +2i x 2i (q 2 ; q 2 ) i (xq; q 2 + q 2 x 2 ) i λ i0 i0 q i2 +6i x 2i +.

5 5 But we don t know how to directly evaluate the righthand sides of these generating functions. Acknowledgments We thank George Andrews for bringing the subject of this paper to our attention. We also thank an anonymous referee who has pointed out that our bijections are identical to those by Sylvester. References. G. E. Andrews. The theory of partitions. Addison-Wesley, Reading, MA, 976. Encyclopedia of Mathematics and Its Applications, Vol G. E. Andrews. J. J. Sylvester, Johns Hopkins and Partitions. In A Century of Mathematics in America, Part I, pages 2 0. American Mathematical Society, M. Bousquet-Mélou and K. Eriksson. Lecture hall partitions. The Ramanujan Journal, :0, M. Bousquet-Mélou and K. Eriksson. Lecture hall partitions II. The Ramanujan Journal, :65 85, Percy A. MacMahon. Combinatory Analysis, Volume II. Chelsea Publishing Company, New York, N.Y., 98. Originally published in two volumes at Cambridge, 98. Published at New York as two volumes in one, 98.