DOCSLIB.ORG

Weighted Partition Rank and Crank Moments. I. Andrews–Beck Type Congruences

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Weighted Partition Rank and Crank Moments. I. Andrews–Beck Type Congruences Weighted partition rank and crank moments. I. Andrews{Beck type congruences Shane Chern Abstract. Recently, George Beck conjectured and George Andrews proved a number of interesting congruences concerning Dyson's rank function and Andrews{Garvan's crank function. In this paper, we shall prove two identities involving the weighted rank and crank moments, from which we may deduce more congruences of Andrews{Beck type. Keywords. Partitions, Dyson's rank, Andrews{Garvan's crank, Andrews{Beck type con- gruences. 2010MSC. 11P83, 11P81, 05A19. 1. Introduction 1.1. Background. As usual, a partition of a positive integer n is a weakly de- creasing sequence of positive integers whose sum equals n. For example, 4 has five partitions: 4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1. If we denote by p(n) the number of partitions of n, then p(4) = 5. In the theory of partitions, one of the most well-known results is achieved by Ramanujan [1, Chapter 10]. He found that p(n) satisfies the following three con- gruences: p(5n + 4) ≡ 0 (mod 5); (1.1) p(7n + 5) ≡ 0 (mod 7) (1.2) and p(11n + 6) ≡ 0 (mod 11): (1.3) In his famous 1944 paper [7], Dyson defined the rank of a partition to be the largest part minus the number of parts. Dyson then conjectured that the rank function can provide a combinatorial interpretation of (1.1) and (1.2); this was later confirmed by Atkin and Swinnerton-Dyer [6]. Dyson also conjectured the existence of a crank function, which can combinatorially interpret all the three Ramanujan congruences. Such a function was not found until over four decades later by Andrews and Garvan [4]. Following [6], we define by N(m; k; n) the number of partitions of n with rank congruent to m modulo k. Atkin and Swinnerton-Dyer proved (1.1) and (1.2) by showing that for 0 ≤ i ≤ 4, 1 N(i; 5; 5n + 4) = p(5n + 4) 5 2 S. Chern and that for 0 ≤ i ≤ 6, 1 N(i; 7; 7n + 5) = p(7n + 5): 7 In a recent paper [3], Andrews mentioned that George Beck has made a number of new conjectures along this line. Instead of considering the N(m; k; n) function, Beck studied the total number of parts in the partitions of n with rank congruent to m modulo k, which is defined by NT (m; k; n). One of the results proved by Andrews reads as follows. Theorem. If i = 1 or 4, then for all n ≥ 0, NT (1; 5; 5n + i) + 2NT (2; 5; 5n + i) − 2NT (3; 5; 5n + i) − NT (4; 5; 5n + i) ≡ 0 (mod 5): (1.4) Andrews further remarked that (1.4) is trivial if one replaces the NT (m; k; n) function by the rank function N(m; k; n). This is simply due to the symmetry N(m; k; n) = N(k − m; k; n): However, the above phenomenon is generally false for the NT (m; k; n) function. Hence, the validness of (1.4) is to some extent exciting. In his proof of (1.4), Andrews did not apply the differentiation technique directly to the three-variable generating function X X x](λ)zrank(n)qn n≥0 λ`n where the inner summation runs through all partitions of n and ](λ) and rank(λ) respectively represent the number of parts in λ and the rank of λ. Instead, he transformed the generating function to a tractable representation so that he can use some results proved already in [6]. However, we observe that if we study a weighted rank moment, then we may obtain a connection with the second Atkin{Garvan rank moment [5] so that (1.4) follows as an immediate consequence. Likewise, a weighted crank moment will imply some new conjectures of Beck concerning the crank function. We will state our results in x1.3. 1.2. Notation and terminology. Let P be the set of partitions. Again, the notation λ ` n means that λ is a partition of n. Below let λ always be a partition. Let jλj be the size of λ. We use ](λ) and !(λ) to denote the number of parts in λ and the number of ones in λ, respectively. Further, rank(λ) and crank(λ) denotes the rank and crank of λ. As defined previously, NT (m; k; n) denotes the total number of parts in the partitions of n with rank congruent to m modulo k. We also denote by N(m; n) the number of partitions of n whose rank is m. Then the second Atkin{Garvan rank moment N2(n) is defined by X1 2 N2(n) := m N(m; n): m=−∞ Further, M!(m; k; n) counts the total number of ones in the partitions of n with crank congruent to m modulo k. Weighted partition rank and crank moments 3 Finally, we adopt the standard q-series notation nY−1 m (A; q)n := (1 − Aq ) m=0 and Y1 m (A; q)1 := (1 − Aq ): m=0 1.3. Main results. Our first result deals with the following weighted rank mo- ment. Theorem 1.1. We have 2 X X qn Xn qm ](λ) rank(λ)qjλj = − : (1.5) (q; q)2 (1 − qm)2 λ2P n≥1 n m=1 It follows that X 1 ](λ) rank(λ) = − N (n): (1.6) 2 2 λ`n Remark. It is worth pointing out that the following generating function identity for N2(n) is used most frequently. X X − n n(3n+1)=2 n n 2 ( 1) q (1 + q ) N2(n)q = − : (q; q)1 (1 − qn)2 n≥0 n≥1 See Eq. (3.4) in [2]. Likewise, for the weighted crank moment, our result reads as follows. Theorem 1.2. We have X 1 X qn !(λ) crank(λ)qjλj = − : (1.7) (q; q)1 (1 − qn)2 λ2P n≥1 It follows that X !(λ) crank(λ) = −np(n): (1.8) λ`n Remark. Let M(m; n) denote number of partitions of n with crank m. Atkin and Garvan [5] defined the k-th crank moment Mk(n) by X1 k Mk(n) = m M(m; n): m=−∞ It can be shown by means of a relation due to Dyson [8] that M2(n) = 2np(n): It turns out that X 1 !(λ) crank(λ) = − M (n): (1.9) 2 2 λ`n This is analogous to (1.6). 4 S. Chern As consequences of Theorems 1.1 and 1.2, we have a number of Andrews{Beck type congruences, including (1.4). Corollary 1.3. If i = 1 or 4, then for all n ≥ 0, NT (1; 5; 5n + i) + 2NT (2; 5; 5n + i) − 2NT (3; 5; 5n + i) − NT (4; 5; 5n + i) ≡ 0 (mod 5): (1.10) Corollary 1.4. If i = 1 or 5, then for all n ≥ 0, NT (1; 7; 7n + i) + 2NT (2; 7; 7n + i) + 3NT (3; 7; 7n + i) − 3NT (4; 7; 7n + i) − 2NT (5; 7; 7n + i) − NT (6; 7; 7n + i) ≡ 0 (mod 7): (1.11) Corollary 1.5. For all n ≥ 0, M!(1; 5; 5n + 4) + 2M!(2; 5; 5n + 4) − 2M!(3; 5; 5n + 4) − M!(4; 5; 5n + 4) ≡ 0 (mod 5): (1.12) Corollary 1.6. For all n ≥ 0, M!(1; 7; 7n + 5) + 2M!(2; 7; 7n + 5) + 3M!(3; 7; 7n + 5) − 3M!(4; 7; 7n + 5) − 2M!(5; 7; 7n + 5) − M!(6; 7; 7n + 5) ≡ 0 (mod 7): (1.13) Corollary 1.7. For all n ≥ 0, M!(1; 11; 11n + 6) + 2M!(2; 11; 11n + 6) + 3M!(3; 11; 11n + 6) + 4M!(4; 11; 11n + 6) + 5M!(5; 11; 11n + 6) − 5M!(6; 11; 11n + 6) − 4M!(7; 11; 11n + 6) − 3M!(8; 11; 11n + 6) − 2M!(9; 11; 11n + 6) − M!(10; 11; 11n + 6) ≡ 0 (mod 11): (1.14) At last, let spt(n) denote the total number of appearances of the smallest parts in all of the partitions of n. In [2], it was shown that 1 spt(n) = np(n) − N (n): 2 2 In view of (1.6) and (1.8), we immediately obtain the following interesting relation. Corollary 1.8. For all n ≥ 0, X X spt(n) = ](λ) rank(λ) − !(λ) crank(λ): (1.15) λ`n λ`n 2. The weighted rank moment We first study the weighted rank moment. It was shown in [3] that 2 X X X xnqn x](λ)zrank(λ)qn = : (2.1) (zq; q)n(xq=z; q)n n≥0 λ`n n≥0 Weighted partition rank and crank moments 5 Proof of Theorem 1.1. We first apply the operator [@=@x] to (2.1). X X x=1 ](λ)zrank(λ)qn ≥ ` n 0 λ n " # 2 X @ xnqn = ≥ @x (zq; q)n(xq=z; q)n n 0 " x=1 # 2 ( ) X xnqn @ xn = log ≥ (zq; q)n(xq=z; q)n @x (xq=z; q)n n 0 " x=1 !# 2 X qn @ Xn = n log x − log(1 − xqm=z) ≥ (zq; q)n(q=z; q)n @x n 0 " m=1# x=1 2 X qn n Xn qm = + m (zq; q)n(q=z; q)n x z − xq n≥0 m=1 x=1 ! 2 X qn Xn qm = n + m : (2.2) (zq; q)n(q=z; q)n z − q n≥1 m=1 We next make the following easy observation: for any n ≥ 1 where n can also be 1, [ ( )] " ( )# @ 1 Xn qm qm log = m + m 2 = 0: (2.3) @z (zq; q)n(q=z; q)n 1 − zq zq − z z=1 m=1 z=1 Applying the operator [@=@z] to (2.2) and using (2.3) yields X X z=1 ](λ) rank(λ)qn ≥ ` n 0 λ n " !# 2 X @ qn Xn qm = n + − m ≥ @z (zq; q)n(q=z; q)n z q n 1 " m=1 z=1# 2 ( ) X nqn @ 1 = log ≥ (zq; q)n(q=z; q)n @z (zq; q)n(q=z; q)n n 1 " z=1 # 2 ( ) X Xn qn qm @ 1 + m log m (zq; q)n(q=z; q)n z − q @z (zq; q)n(q=z; q)n(z − q ) n≥1 m=1 z=1 2 X qn Xn qm = − : (2.4) (q; q)2 (1 − qm)2 n≥1 n m=1 This is the first part of Theorem 1.1.
Load more

Recommended publications
  • Arxiv:2108.08639V1 [Math.CO]
    Generalizations of Dyson’s Rank on Overpartitions Alice X.H. Zhao College of Science Tianjin University of Technology, Tianjin 300384, P.R. China [email protected] Abstract. We introduce a statistic on overpartitions called the k-rank. When there are no overlined parts, this coincides with the k-rank of a partition introduced by Garvan. Moreover, it reduces to the D-rank of an overpartition when k = 2. The generating function for the k-rank of overpartitions is given. We also establish a relation between the generating function of self-3-conjugate overpartitions and the tenth order mock theta functions X(q) and χ(q). Keywords: Dyson’s rank, partitions, overpartitions, mock theta functions. AMS Classifications: 11P81, 05A17, 33D15 1 Introduction Dyson’s rank of a partition is defined to be the largest part minus the number of parts [8]. In 1944, Dyson conjectured that this partition statistic provided combinatorial interpre- tations of Ramanujan’s congruences p(5n + 4) ≡ 0 (mod 5) and p(7n + 5) ≡ 0 (mod 7), where p(n) is the number of partitions of n. Let N(m, n) denote the number of partitions of n with rank m. He also found the following generating function of N(m, n) [8, eq. arXiv:2108.08639v1 [math.CO] 19 Aug 2021 (22)]: ∞ 1 ∞ N(m, n)qn = (−1)n−1qn(3n−1)/2+|m|n(1 − qn). (1.1) (q; q) n=0 ∞ n=1 X X Here and in the sequel, we use the standard notation of q-series: ∞ (a; q) (a; q) := (1 − aqn), (a; q) := ∞ , ∞ n (aqn; q) n=0 ∞ Y j(z; q) := (z; q)∞(q/z; q)∞(q; q)∞.
    [Show full text]
  • BG-Ranks and 2-Cores
    BG-ranks and 2-cores William Y. C. Chen Kathy Q. Ji Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China <[email protected]> <[email protected]> Herbert S. Wilf Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA <[email protected]> Submitted: May 5, 2006; Accepted: Oct 29, 2006; Published: Nov 6, 2006 Mathematics Subject Classification: 05A17 Abstract We find the number of partitions of n whose BG-rank is j, in terms of pp(n), the number of pairs of partitions whose total number of cells is n, giving both bijective and generating function proofs. Next we find congruences mod 5 for pp(n), and then we use these to give a new proof of a refined system of congruences for p(n) that was found by Berkovich and Garvan. 1 Introduction If π is a partition of n we define the BG-rank β(π), of π as follows. First draw the Ferrers diagram of π. Then fill the cells with alternating 1's, chessboard style, beginning with a +1 in the (1; 1) position. The sum of these entries is β(π), the BG-rank of π. For example, the BG-rank of the partition 13 = 4 + 3 + 3 + 1 + 1 + 1 is −1. This partition statistic has been encountered by several authors ([1, 2, 3, 9, 10]), but its systematic study was initiated in [1]. Here we wish to study the function pj(n) = j fπ : jπj = n and β(π) = j gj : We will find a fairly explicit formula for it (see (2) below), and a bijective proof for this formula.
    [Show full text]
  • Freeman Dyson, the Mathematician∗
    GENERAL ARTICLE Freeman Dyson, the Mathematician∗ B Sury A mathematician of the highest rank with ideas ever-brimming with a full tank. He designed special lenses to ‘see’ Ramanujan’s congruences partitioned through the yet-invisible crank! The one person who comes closest to the legacy of Hermann B Sury is with the Indian Weyl was Freeman Dyson, who contributed enormously to Statistical Institute in Bengaluru. He is the National both Physics and Mathematics. His books and talks are tes- Co-ordinator for the taments to his prolificity in writing widely on the world at Mathematics Olympiad large and on science and mathematics in particular. To name Programme in India, and also a few of his books, he has (talked and) written on ‘Bombs and the President of the Indian Mathematical Society. Poetry’, ‘Imagined Worlds’, ‘Origins of Life’ and ‘Birds and Frogs’. Remarkably, Dyson contributed handsomely to what is termed ‘pure mathematics’. One would expect a physicist- mathematician to interest himself mainly in problems of an ‘applied’ nature. Here, we take a necessarily brief peek into some of his ‘purely mathematical’ work. The suggested read- ing at the end can be referred to for more details. 1. Statistical Mechanics – Dyson’s Conjecture In an enormously influential paper, ‘Statistical theory of energy Keywords levels of complex systems’, I. J. Math. Phys., 3:140–156, 1962, Congruences for partitions, rank and crank, random matrix the- Dyson proposed a new type of ensemble transforming the whole ory, Riemann zeta function, Dyson subject into purely group-theoretic language. In the paper, Dyson transform, quasi-crystals.
    [Show full text]
  • What Is the Crank of a Partition? Daniel Glasscock, July 2014
    What is the crank of a partition? Daniel Glasscock, July 2014 These notes complement a talk given for the What is ... ? seminar at the Ohio State University. Introduction The crank of an integer partition is an integer statistic which yields a combinatorial explanation of Ramanujan's three famous congruences for the partition function. The existence of such a statistic was conjectured by Freeman Dyson in 1944 and realized 44 years later by George Andrews and Frank Gravan. Since then, groundbreaking work by Ken Ono, Karl Mahlburg, and others tells us that not only are there infinitely many congruences for the partition function, but infinitely many of them are explained by the crank. In this note, we will introduce the partition function, define the crank of a partition, and show how the crank underlies congruences of the partition function. Along the way, we will recount the fascinating history behind the crank and its older brother, the rank. This exposition ends with the recent work of Karl Mahlburg on crank function congruences. Andrews and Ono [3] have a good quick survey from which to begin. The partition function The partition function p(n) counts the number of distinct partitions of a positive integer n, where a partition of n is a way of writing n as a sum of positive integers. For example, p(4) = 5 since 4 may be written as the sum of positive integers in 5 essentially different ways: 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1. The numbers comprising a partition are called parts of the partition, thus 2 + 1 + 1 has three parts, two of which are odd, and is not composed of distinct parts (since the 1 repeats).
    [Show full text]
  • Ramanujan, His Lost Notebook, Its Importance
    RAMANUJAN, HIS LOST NOTEBOOK, ITS IMPORTANCE BRUCE C. BERNDT 1. Brief Biography of Ramanujan Throughout history most famous mathematicians were educated at renowned centers of learning and were taught by inspiring teachers, if not by distinguished research mathematicians. The one exception to this rule is Srinivasa Ramanujan, born on December 22, 1887 in Erode in the southern Indian State of Tamil Nadu. He lived most of his life in Kumbakonam, located to the east of Erode and about 250 kilometers south, southwest of Madras, or Chennai as it is presently named. At an early age, he won prizes for his mathematical ability, not mathematics books as one might surmise, but books of English poetry, reflecting British colonial rule at that time. At the age of about fifteen, he borrowed a copy of G. S. Carr's Synopsis of Pure and Applied Mathematics, which was the most influential book in Ramanujan's development. Carr was a tutor and compiled this compendium of approximately 4000{5000 results (with very few proofs) to facilitate his tutoring at Cambridge and London. One or two years later, Ramanujan entered the Government College of Kumbakonam, often called \the Cambridge of South India," because of its high academic standards. By this time, Ramanujan was consumed by mathematics and would not seriously study any other subject. Consequently he failed his examinations at the end of the first year and lost his scholarship. Because his family was poor, Ramanujan was forced to terminate his formal education. At about the time Ramanujan entered college, he began to record his mathematical discoveries in notebooks.
    [Show full text]
  • Dyson's Ranks and Maass Forms
    ANNALS OF MATHEMATICS Dyson’s ranks and Maass forms By Kathrin Bringmann and Ken Ono SECOND SERIES, VOL. 171, NO. 1 January, 2010 anmaah Annals of Mathematics, 171 (2010), 419–449 Dyson’s ranks and Maass forms By KATHRIN BRINGMANN and KEN ONO For Jean-Pierre Serre in celebration of his 80th birthday. Abstract Motivated by work of Ramanujan, Freeman Dyson defined the rank of an inte- ger partition to be its largest part minus its number of parts. If N.m; n/ denotes the number of partitions of n with rank m, then it turns out that n2 X1 X1 X1 q R.w q/ 1 N.m; n/wmqn 1 : I WD C D C Qn .1 .w w 1/qj q2j / n 1 m n 1 j 1 D D1 D D C C We show that if 1 is a root of unity, then R. q/ is essentially the holomorphic ¤ I part of a weight 1=2 weak Maass form on a subgroup of SL2.Z/. For integers 0 Ä r < t, we use this result to determine the modularity of the generating function for N.r; t n/, the number of partitions of n whose rank is congruent to r .mod t/. We I extend the modularity above to construct an infinite family of vector valued weight 1=2 forms for the full modular group SL2.Z/, a result which is of independent interest. 1. Introduction and statement of results The mock theta-functions give us tantalizing hints of a grand synthe- sis still to be discovered.
    [Show full text]
  • Arxiv:1908.08660V1 [Math.NT] 23 Aug 2019 Ffrhrpriincnrecs Npriua,Te Defin Were They Cranks Particular, and in Ranks of Congruences
    AN INEQUALITY BETWEEN FINITE ANALOGUES OF RANK AND CRANK MOMENTS PRAMOD EYYUNNI, BIBEKANANDA MAJI AND GARIMA SOOD Dedicated to Professor Bruce C. Berndt on the occasion of his 80th birthday Abstract. The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite ana- logues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews’ famous identity for smallest parts function. In the same paper, they also con- jectured an inequality between finite analogues of rank and crank moments, analogous to Garvan’s conjecture. In the present paper, we give a proof of this conjecture. 1. Introduction Let p(n) denote the number of unrestricted partitions of a positive integer n. To give a combinatorial explanation of the famous congruences of Ramanujan for the partition function p(n), namely, for m ≥ 0, p(5m + 4) ≡ 0 (mod 5), p(7m + 5) ≡ 0 (mod 7), Dyson [16] defined the rank of a partition as the largest part minus the number of parts. He also conjectured that there must be another statistic, which he named ‘crank’, that would explain Ramanujan’s third congruence, namely, arXiv:1908.08660v1 [math.NT] 23 Aug 2019 p(11m + 6) ≡ 0 (mod 11). After a decade, Atkin and Swinnerton-Dyer [9] confirmed Dyson’s observations for the first two congruences for p(n). Also, in 1988, ‘crank’ was discovered by Andrews and Garvan [7]. An interesting thing to note is that by using the partition statistic ‘crank’, Andrews and Garvan were able to explain not only the third congruence but also the first two.
    [Show full text]
  • Congruences Modulo Powers of 5 for the Rank Parity Function
    CONGRUENCES MODULO POWERS OF 5 FOR THE RANK PARITY FUNCTION DANDAN CHEN, RONG CHEN, AND FRANK GARVAN Abstract. It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for rank parity function is f(q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function. 1. Introduction Let p(n) be the number of unrestricted partitions of n. Ramanujan discovered and later proved that (1.1) p(5n + 4) ≡ 0 (mod 5); (1.2) p(7n + 5) ≡ 0 (mod 7); (1.3) p(11n + 6) ≡ 0 (mod 11): In 1944 Dyson [16] defined the rank of a partition as the largest part minus the number of parts and conjectured that residue of the rank mod 5 (resp. mod 7) divides the partitions of 5n + 4 (resp. 7n + 5) into 5 (resp. 7) equal classes thus giving combinatorial explanations of Ramanujan's partition congruences mod 5 and 7. Dyson's rank conjectures were proved by Atkin and Swinnerton-Dyer [7]. Dyson also conjectured the existence of another statistic he called the crank which would likewise explain Ramanujan's partition congruence mod 11. The crank was found by Andrews and the third author [4] who defined the crank as the largest part, if the partition has no ones, and otherwise as the difference between the number of parts larger than the number of ones and the number of ones.
    [Show full text]
  • Srinivasa Ramanujan: Going Strong at 125, Part II
    Srinivasa Ramanujan: Going Strong at 125, Part II Krishnaswami Alladi, Editor Ken Ono Modular Forms, Partitions, and Mock Theta Functions Ramanujan’s work on modular forms, partitions, and mock theta functions is rich with tantalizing examples of deeper mathematical structures. In- deed, his tau-function, his partition congruences, and his mock theta functions are prototypes that have helped to shape the modern mathematical landscape. Ramanujan was fascinated by the coefficients of the function 1 1 X n Y n 24 (1) ∆(z) = τ(n)q := q (1 − q ) n=1 n=1 = q − 24q2 + 252q3 − 1472q4 + · · · ; where q := e2πiz and Im(z) > 0. This function is a Ramanujan passport photo. weight 12 modular form. In other words, ∆(z) is a function on the upper half of the complex plane The 125th anniversary of the birth of Srinivasa such that Ramanujan was on December 22, 2012. To mark az + b = (cz + d)12 (z) the occasion, the Notices is publishing a feature ∆ cz + d ∆ article of which this is the second and final install- a b ment. This installment contains pieces by Ken Ono, for every matrix c d 2 SL2(Z). He conjectured Kannan Soundararajan, Robert Vaughan, and Ole (see p. 153 of [12]) that Warnaar on various aspects of Ramanujan’s work. τ(nm) = τ(n)τ(m), The first installment appeared in the December for every pair of coprime positive integers n and 2012 issue, and contained an introductory piece m, and that by Krishnaswami Alladi plus pieces by George An- drews, Bruce Berndt, and Jonathan Borwein on τ(p)τ(ps ) = τ(ps+1) + p11τ(ps−1), Ramanujan’s work.
    [Show full text]
  • BG-Ranks and 2-Cores
    BG-ranks and 2-cores William Y. C. Chen Kathy Q. Ji Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China <[email protected]> <[email protected]> Herbert S. Wilf Department of Mathematics University of Pennsylvania Philadelphia, PA 19104, USA <[email protected]> Submitted: May 5, 2006; Accepted: Oct 29, 2006; Published: Nov 6, 2006 Mathematics Subject Classification: 05A17 Abstract We find the number of partitions of n whose BG-rank is j, in terms of pp(n), the number of pairs of partitions whose total number of cells is n, giving both bijective and generating function proofs. Next we find congruences mod 5 for pp(n), and then we use these to give a new proof of a refined system of congruences for p(n) that was found by Berkovich and Garvan. 1 Introduction If π is a partition of n we define the BG-rank β(π), of π as follows. First draw the Ferrers diagram of π. Then fill the cells with alternating 1's, chessboard style, beginning with a +1 in the (1; 1) position. The sum of these entries is β(π), the BG-rank of π. For example, the BG-rank of the partition 13 = 4 + 3 + 3 + 1 + 1 + 1 is −1. This partition statistic has been encountered by several authors ([1, 2, 3, 9, 10]), but its systematic study was initiated in [1]. Here we wish to study the function pj(n) = j fπ : jπj = n and β(π) = j gj : We will find a fairly explicit formula for it (see (2) below), and a bijective proof for this formula.
    [Show full text]
  • Partition Bijections, a Survey
    PARTITION BIJECTIONS, A SURVEY IGOR PAK Abstract. We present an extensive survey of bijective proofs of classical par- titions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and general- izations are added in the form of exercises. Contents 1. Introduction 3 2. Basic results 8 2.1. Partitions and Young diagrams 8 2.2. Generating functions 10 2.3. Basic geometry of Young diagrams 11 2.4. Number of distinct parts 14 2.5. Dyson’s rank 14 2.6. q-binomial theorem 17 2.7. Heine transformation 17 3. Euler’s Theorem 20 3.1. Partitions into distinct parts vs. partitions into odd parts 20 3.2. Glaisher’s bijection 20 3.3. Franklin’s Extension 20 3.4. Sylvester’s bijection 21 3.5. Iterated Dyson’s map 22 4. Partition Theorems of Lebesgue, G¨ollnitzand Schur 23 4.1. Lebesgue identity 23 4.2. First proof of Lebesgue identity 23 4.3. G¨ollnitzTheorem 25 4.4. Second proof of Lebesgue identity 26 4.5. Schur’s Partition Theorem 27 5. Euler’s Pentagonal Theorem 29 5.1. The identity 29 5.2. Franklin’s involution 30 5.3. Sylvester’s identity 32 5.4. Bijective proof of Euler’s recurrence relation 33 5.5. Gauss identity 33 6. Jacobi’s triple product identity 35 6.1. Variations on the theme 35 6.2. Direct bijection 35 6.3. Involutive proof 37 Date: September 18, 2002. 1 2 IGOR PAK 7. Rogers-Ramanujan identities 38 7.1.
    [Show full text]
  • A Formula for the Partition Function That" Counts"
    A formula for the partition function that “counts” Yuriy Choliy and Andrew V. Sills Abstract. We derive a combinatorial multisum expression for the num- ber D(n, k) of partitions of n with Durfee square of order k. An immedi- ate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation D˜ (n, k) to the quasipolynomial ˜ representation of D(n, k). Numerically, the sum P1 k √n D(n, k) ap- pears to be extremely close to the initial term of the Hardy–R≤ ≤ amanujan– Rademacher convergent series for p(n). Mathematics Subject Classification (2010). Primary 05A17 Secondary 11P81. Keywords. integer partitions, partition function, Durfee square. 1. Introduction arXiv:1811.09327v1 [math.CO] 22 Nov 2018 1.1. Preliminaries: Definitions and Notation A partition λ of an integer n is a nonincreasing finite sequence of positive integers (λ1, λ2, λ3,...,λℓ) that sum to n. Each λj is called a part of λ. The number of parts ℓ = ℓ(λ) is the length of λ. The number of times mj = mj (λ) the positive integer j appears as a part in λ is the multi- plicity of j in λ. Sometimes it is convenient to notate the partition λ as m1 m2 m3 λ = 1 2 3 , with the convention that the superscript mj may be h ···i mj omitted if mj = 1 for that j, and j may be omitted if mj = 0 for that j. Thus, e.g., (6, 4, 3, 3, 1, 1, 1, 1) = 14324 6 .
    [Show full text]