It also has received significant study for its purely combinatorial properties; notably, it is the motivating example of a differential poset. 0 -bMBÞ\E¢â ½Îö§FGŒ.ÈFœ¥«´À-ëiñCÍÈeY7e“]îOÕ~ã üñ³²ª²ú†qżf¢MÉ«7Ýy–‡òŠ7¤þÔ²YdÕm^g½óð¦Ä(ÿN1_¤e°žUù ò¥„ѳë}| ò€íOe°LY#”-ï_eË´Éæ7é Ý4ÃÿTW!ϋkÆfËl•SMZݵ;½„®ê!‹=úU“7yYÐUá2ÎÊâºZgÅ,«½"ÊÞ˂XdؽìÍîÓ[JÝ®¿mhG¨€2YÛn*v‰DÂ×®ÿ ¤`£À 1éi™Þ^Šd£kïC%wém[ف ‹ˆ¹UI‚ž3ÆÒSJЏùßN ©/Ü^õoÝs{÷…ÛÛ­ÛÛ?íö1ßÞ!ÐLp¾ÈX³eš¯j„C0/ƱQ!DFL⪕H~ÂÔïž,Ñyh’ÏÀ¨æy=[×u6G—¤5íÀWë The diagrams for the 5 partitions of the number 4 are listed below: An alternative visual representation of an integer partition is its Young diagram (often also called a Ferrers diagram). If A is a set of natural numbers, we let pA(n) denote the number of partitions 3 (q), (q) AND ˚(q) GEORGE E. ANDREWS, ATUL DIXIT, AND AE JA YEE Abstract. [1] As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of polyomino.[2]. In 1742, Leonhard Euler established the generating function of P(n). Partition Function q-Series Partition Function De nition Apartitionof a natural number n is a way of writing n as a sum of positive integers. because the integer ( ) A Ramanujan-type formula due to the Chudnovsky brothers used to break a world record for computing the most digits of pi: 1 π = 1 53360√640320 ∞ ∑ n=0(−1)n (6n)! Godfrey Harold Hardy said that Srinivasa Ramanujan was the first, and up to now the only, mathematician to discover any such properties of P(n). k − Ramanujan va néixer el 22 de desembre de 1887 a Erode, Tamil Nadu, Índia, on vivien els seus avis materns. In 1967, Atkin and J. N. O’Brien [4] discovered further congruences; for example, for all k 0, p 17303k+ 237 0 (mod 13): In mathematics, a partition is a way of writing a number as a sum of positive integers (called parts), such as and so there are five ways to partition the number 4. {\displaystyle n} Moreover, central to Ramanujan’s thoughts is the more general partition function p r(n) de ned by 1 (q;q)r 1 = X1 n=0 p r(n)qn; jqj<1; which is not discussed in [12]. No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence … N For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted by q(n). ³ºI/—y½ÈæbÄã±ïž¥°Ö³ªÂ¤§c¼L›:ÛÌ>åÙ-£OËZŒÈ\¶2Z¼®òëO ic)_Çú³ô&ã›×¤Khe4æ™[êN_dwìÐ~ÛO\PVóú§]¾œ:J:mnB'&²ï. In mathematics, a partition is a way of writing a number as a sum of positive integers (called parts), such as. (If order matters, the sum becomes a composition.) The Correct Formulas For The Number Of Partitions Of A Given Number As A Combination And As A Permutation That Srinivasa Ramanujan Had Missed This Discove by A.C.Wimal Lalith De … [15] “±`ãëeSódž±½g[âˆ9mvÝ.æó,õ s If A has k elements whose greatest common divisor is 1, then[19], One may also simultaneously limit the number and size of the parts. {\displaystyle n=0,1,2,\dots } Several generalizations of partitions have been studied, among which overpartitions, which are partitions where the last occurrence of a number can be overlined, overpartition pairs, and n-color partitions, which are related to a model of statistical … Originally published in 1927, this book presents the collected papers of the renowned Indian mathematician Srinivasa Ramanujan (1887–1920), with editorial contributions from G. H. Hardy (1877–1947). Chen e Kathy Q. Ji, em resposta ao questionamento de George E. Andrews, matemático estadunidense, sobre encontrar demonstrações combinatórias de duas identidades no Caderno Perdido de Ramanujan, nos mostra algumas formas ponderadas do Teorema de Euler sobre partições com partes … Mathematically, 1729 = 1 3 + 12 3 = 9 3 + 10 3 . #5 He discovered the three Ramanujan’s congruences. A Ramanujan-type formula due to the Chudnovsky brothers used to break a world record for computing the most digits of pi: Partition means p(4)=5. And in which 4 is expressed in 5 different ways. Of particular interest is the partition 2 + 2, which has itself as conjugate. {\displaystyle p(4)=5} Many integer partition theorems can be restated as an analytic identity, as a sum equal to a product. Ramanujan's work on partitions 83 VII. partition. The Hardy-Ramanujan number, which Ramanujan stated was the smallest number that can be expressed as the sum of cubed numbers in two different ways. En n nous donnons × … , 2 where ak is (−1)m if k = 3m2 − m for some integer m and is 0 otherwise. The partition 6 + 4 + 3 + 1 of the positive number 14 can be represented This statistic (which is unrelated to the one described above) appears in the study of Ramanujan congruences. = Theorem 3.12 suggests that there is a relationship between the right side of the First Rogers{Ramanujan Identity and … In 1913, a young, self-taught mathematical genius in India named Srinivasa Ramanujan is invited to England to work with G. H. Hardy, a Cambridge professor. the partition function nd their seed in some keen observations of Ramanujan. 1 ; p [8][9] This result was proved by Leonhard Euler in 1748[10] and later was generalized as Glaisher's theorem. Notation. For positive i we let m i:= \multiplicity" of size i parts. {\displaystyle \lambda _{k}-k} 2558 KRISHNASWAMI ALLADI AND ALEXANDER BERKOVICH a weighted partition identity connecting partitions into distinct parts and Rogers-Ramanujan partitions (see x4).The proof of Theorem 1 is given in x5andx6, with x5 describing the necessary prerequisites, namely, the method of weighted words of Alladi-Andrews-Gordon [5], and x6 giving the details of the proof. + The left The Hardy-Ramanujan Asymptotic Partition FormulaFor n a positive integer, let p(n) denote the number of unordered partitions of n, that is, unordered sequences of positive integers which sum to n; then the value of p(n) is given asymptotically by p(n) ∼ 1 4n √ 3 eτ √ n/6. If we count the partitions of 8 with distinct parts, we also obtain 6: This is a general property. + INTRODUCTION But overnight Srinivasa Ramanujan created it. Regarding the contribution of Ramanujan to the theory of partitions… Some more problems of the analytic theory of numbers 58 V. A lattice-point problem 67 VI. . International Journal of Mathematics and Mathematical Sciences (1987) Volume: 10, page 625-640; ISSN: 0161-1712; Access Full Article top Access to full text Full (PDF) How to cite top It grows as an exponential function of the square root of its argument. Ramanujan and the Partition Function By Sir Timothy Gowers, FRS, Fellow, Rouse Ball Professor of Mathematics Ramanujan is now known as perhaps the purest mathematical genius there has ever been, and the body of work he left behind has had a deep influence on mathematics that continues to this day. 3.2 Conjugate partitions 16 3.3 An upper bound on p(n)19 3.4 Bressoud’s beautiful bijection 23 3.5 Euler’s pentagonal number theorem 24 4 The Rogers-Ramanujan identities 29 4.1 A fundamental type of partition identity 29 4.2 Discovering the first Rogers-Ramanujan identity 31 4.3 Alder’s conjecture 33 4.4 Schur’s theorem 35 1 He was "discovered" by G. H. Hardy and J. E. Littlewood, two world-class mathematicians at Cambridge, and enjoyed an extremely fruitful period of … 1 {\displaystyle 4} PARTITIONS ASSOCIATED WITH THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ! Related to the Partition Theory of Numbers, Ramanujan also came up with three remarkable congruences for the partition function p(n).They are p(5n+4) = 0(mod 5); p(7n+4) = 0(mod 7); p(11n+6) = 0(mod 11).For example, the first congruence means that if an integer is 4 more than a multiple of 5, then number of its partitions … This one involves Ramanujan's pi formula. + manujan’s partition congruences for the modulus 5 and 7. Primary 11P83; Secondary 05A17. λ In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. by the following diagram: The 14 circles are lined up in 4 rows, each having the size of a part of the partition. 4 Continuing the biography and a look at another of Ramanujan's formulas. Two sums that differ only in the order of their summands are considered the same partition. 2 [3] The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. ( The rank of a partition is the largest number k such that the partition contains at least k parts of size at least k. For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. La seva mare, Komalatammal o Komal Ammal (Ammal en tamil és equivalent a senyora en català o madam en anglès), era una mestressa de casa i també una cantant en un temple de … In England Ramanujan made further advances, especially in the partition of numbers (the number of ways that a positive integer can be expressed as the sum of positive integers; e.g., 4 can be expressed as 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1). , C The number of partitions of n into parts none of which exceed r is the coefficient pr(n) ... ∗G. Abstract. a;b(n) denote the number of partitions of ninto elements of S a;b. [6] In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. 5 1 When r > 1 and s > 1 are relatively prime integers, let pr;s(n) denote the number of partitions of n into parts containing no multiples of r or s. We say that such a partition of an integer n is (r,s)-regular. The formula has been used in statistical physics and is also used (first by Niels Bohr) to calculate quantum partition functions of atomic nuclei. A centennial tribute. Child stated that the different types of partitions … Puis nous d emontrons deux nouvelles g en eralisa-tions d’identit es de partitions d’Andrews aux surpartitions. [14], The asymptotic growth rate for p(n) is given by, where Asymptotic theory of partitions 113 IX. I. = {\displaystyle 1+1+2} In 1981, S. Barnard and J.M. Hardy, G.H. Such partitions are said to be conjugate of one another. 2 Based on one of the results of Andrews, Dixit, and Yee, mod 2 congruences are obtained. {\displaystyle \lambda _{k}} Let p(N, M; n) denote the number of partitions of n with at most M parts, each of size at most N. Equivalently, these are the partitions whose Young diagram fits inside an M × N rectangle. Godfrey Harold Hardy said that Srinivasa Ramanujan was the first, and up to now the only, mathematician to discover any such properties of P(n). The partition function The notation λ ⊢ n means that λ is a partition of n. Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. Ramanujan’s proof of p(5n+ 4) 0(mod5) here is considerably briefer than it is in [12]. Both have several possible conventions; here, we use English notation, with diagrams aligned in the upper-left corner. 2558 KRISHNASWAMI ALLADI AND ALEXANDER BERKOVICH a weighted partition identity connecting partitions into distinct parts and Rogers-Ramanujan partitions (see x4).The proof of Theorem 1 is given in x5andx6, with x5 describing the necessary prerequisites, namely, the method of weighted words of Alladi-Andrews-Gordon [5], and x6 giving the details of the proof. Abstract. Theorem 3.12 suggests that there is a relationship between the right side of the First Rogers{Ramanujan Identity and … For instance, cluding ones for restricted partition functions represented by various identities of Rogers-Ramanujan type. ( The values of this function for M 2010MSC. for a partition of k parts with largest part AN EXTENSION OF THE HARDY-RAMANUJAN CIRCLE METHOD AND APPLICATIONS TO PARTITIONS WITHOUT SEQUENCES KATHRIN BRINGMANN AND KARL MAHLBURG Abstract. n Ramanujan did not actually discover this result, which was actually published by the French mathematician Frénicle de Bessy in 1657. p . Decomposition of an integer as a sum of positive integers, Partitions in a rectangle and Gaussian binomial coefficients, Partition_function_(number_theory) § Approximation_formulas, "Partition identities - from Euler to the present", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, "On the remainder and convergence of the series for the partition function", Fast Algorithms For Generating Integer Partitions, Generating All Partitions: A Comparison Of Two Encodings, https://en.wikipedia.org/w/index.php?title=Partition_(number_theory)&oldid=998750886, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, A Goldbach partition is the partition of an even number into primes (see, This page was last edited on 6 January 2021, at 21:42. and so there are five ways to partition the number 4. Ramanujan-type congruence, 2-color partition triple, modular form. In 1981, S. Barnard and J.M. got large. Abstract. 1. Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (født 22. december 1887, død 26. april 1920) var en indisk matematiker og et af de mest esoteriske matematiske genier i det 20. århundrede.. Han rejste til England i 1914, hvor han blev vejledt og begyndte et samarbejde med G. H. Hardy på University of Cambridge. n n Debnath, Lokenath. J. D. Rosenhouse, Partitions of Integers In this paper, we study arithmetic properties of the partition functions. The theory of partitions has interested some of the best minds since the 18th century. En n nous donnons explained a partition graphically by an array of dots or nodes. The lattice was originally defined in the context of representation theory, where it is used to describe the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. Such partitions are said to be conjugate of one another. Example The partitions of 4 are 4 = 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1: In the Ferrers diagram or Young diagram of a partition of rank r, the r × r square of entries in the upper-left is known as the Durfee square: The Durfee square has applications within combinatorics in the proofs of various partition identities. By taking conjugates, the number pk(n) of partitions of n into exactly k parts is equal to the number of partitions of n in which the largest part has size k. The function pk(n) satisfies the recurrence, with initial values p0(0) = 1 and pk(n) = 0 if n ≤ 0 or k ≤ 0 and n and k are not both zero. Hypergeometric series 101 VIII. One work of Ramanujan (done with G. H. Hardy) is his formula for the number of partitions of a positive integer n, the famous Hardy-Ramanujan Asymptotic Formula for the partition problem. In 1742, Leonhard Euler established the generating function of P(n). n {\displaystyle 2+2} In particular, we have the generating function, (1.1) X1 n=0 P a;b(n)qn= Y1 n=0 1 (1 qan+b): A famous theorem of Hardy and Ramanujan is that when a= b= 1 P 1;1(n) ˘ 1 4n p 3 eˇ p 2n=3 as n !1. 4 (1960), 473-478. [21] It also has some practical significance in the form of the h-index. This question was finally answered quite completely by Hardy, Ramanujan, and Rademacher [11, 16] and their result will be discussed below (see p. 13). The generating function of partitions with repeated (resp. Proof (outline): The crucial observation is that every odd part can be "folded" in the middle to form a self-conjugate diagram: One can then obtain a bijection between the set of partitions with distinct odd parts and the set of self-conjugate partitions, as illustrated by the following example: Among the 22 partitions of the number 8, there are 6 that contain only odd parts: Alternatively, we could count partitions in which no number occurs more than once. JOURNAL OF NUMBER THEORY 38, 135-144 (1991) A Hardy-Ramanujan Formula for Restricted Partitions GERT ALMKVIST Mathematics Institute, University of Lund, Box 118, S-22100 Lund, Sweden AND GEORGE E. ANDREWS Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802 Communicated by Hans … The more precise asymptotic formula. If A possesses positive natural density α then, and conversely if this asymptotic property holds for pA(n) then A has natural density α. We de ned the number of partitions of zero to equal 1 in de nition 3.1 so this is considered a valid partition. This de nition is … [13], One possible generating function for such partitions, taking k fixed and n variable, is, More generally, if T is a set of positive integers then the number of partitions of n, all of whose parts belong to T, has generating function, This can be used to solve change-making problems (where the set T specifies the available coins). . An example of a problem in the theory of integer partitions that remains unsolved, despite a good deal of There is a natural partial order on partitions given by inclusion of Young diagrams. M Following his notation let N(m;n) be the number of {\displaystyle C=\pi {\sqrt {\frac {2}{3}}}.} . counts the partitions of n into exactly M parts of size at most N, and subtracting 1 from each part of such a partition yields a partition of n − M into at most M parts.[20]. One such example is the rst Rogers-Ramanujan identity (1) 1 Q 1 k=0 (1 q5k+1)(1 q5k+4) = 1 + X1 k=1 qk2 1 (1 q)(1 q2) (1 qk): MacMahon’s combinatorial version of (1) uses integer partitions. (q), (q) AND ˚(q) GEORGE E. ANDREWS, ATUL DIXIT, AND AE JA YEE Abstract. One day Ramunjan came to Hardy and said that he wrote another Series. A few significant contributions were multiple formulae to calculate pi with great accuracy to billions of digits (22/7 is only an approximation to pi), partition functions (a partition … will be divisible by 5.[4]. The second video in a series about Ramanujan. A partition of a positive integer n is a representation of n as a sum of positive integers, called parts, the order of which is irrelevant. In particular he discovered what are referred to as the Ramanujan Congruences of p(n). Hirschhorn EastChinaNormal University Shanghai, July 2013 Introduction Proofs of mod 5 congruence Proof ofmod 7 congruence Proof ofmod 11 congruence Crucialidea Introduction Let p(n) be the number of partitions of n. For example, p(4) = 5, since we can write 4 = 4 = 3 +1 = 2 +2 = 2 +1 +1 = 1 +1 +1 +1 Dev. , Remark 3.14. In his 1919 paper, he proved the first two congruences using the following identities (using q-Pochhammer symbol notation): {\displaystyle 1+1+1+1} The number p n is the number of partitions of n. Here are some examples: p 1 = 1 because there is only one partition of 1 p 2 = 2 because there are two partitions of 2, namely 2 = 1 + 1 p [17][18], If A is a finite set, this analysis does not apply (the density of a finite set is zero). Srinivasa Ramanujan (1887-1920) and the theory of partitions of numbers and statistical mechanics. Taxi Number pour le nombre des partitions de n, ” in the Comptes Rendus, January 2nd, 1917 [No. These are appropriately named because Ramanujan was the rst to notice these interesting properties of the partition function, [Ram00b],[Ram00d],[Ram00a],[Ram00c]. El seu pare, K. Srinivasa Iyengar va treballar com a venedor en una botiga sari del districte de Thanjavur. N Nous commen˘cons par donner trois nouvelles preuves du th eor eme de Schur pour les surpartitions. Using Ramanujan’s di erential equations for Eisenstein series and an idea from Ramanu-jan’s unpublished manuscript on the partition function p(n) and the tau function ˝(n), we provide simple proofs of Ramanujan’s congruences for p(n) modulo 5, 7, and 11. , Dans la premi ere, nous etudions des identit es de partitions du type Rogers-Ramanujan. The partition function satis es additional congruences similar to the original ones of Ramanujan. j j:= 1 + 2 + ::: (Size of ). n = O artigo Weighted forms of Euler's theorem de William Y.C. 2 For instance, whenever the decimal representation of We develop a generalized version of the Hardy-Ramanujan \circle method" in order to derive asymptotic series expansions for the products of modular forms and mock theta functions. are: No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. and Ramanujan, S. (1917a), Une Formule Asymptotique Pour le Nombre des Partitions de n [Comptes Rendus, 2 Jan. 1917] (French) [An Asymptotic Formula for the Number of Partitions of n] Collected Papers of Srinivasa Ramanujan, Pages 239–241, American Mathematical Society (AMS) Chelsea Publishing, Providence, RI, 2000. 2 The generating function of partitions with repeated (resp. ends in the digit 4 or 9, the number of partitions of 1. The Gaussian binomial coefficient is defined as: The Gaussian binomial coefficient is related to the generating function of p(N, M; n) by the equality. of n into elements of A. k If we flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14: By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. ( , + for example :-whenever the decimal representation of N ends in the digit 4 or 9 the number of partition of N will be divisible 5 and he found similar rules for partition numbers divisible by 7 and 11. New combinatorial interpretations of Ramanujan’s partition congruences mod 5,7 and 11 @article{Garvan1988NewCI, title={New combinatorial interpretations of Ramanujan’s partition congruences mod 5,7 and 11}, author={F. Garvan}, journal={Transactions of the American … A complete asymptotic expansion was given in 1937 by Hans Rademacher. p was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920. In England Ramanujan made further advances, especially in the partition of numbers (the number of ways that a positive integer can be expressed as the sum of positive integers; e.g., 4 can be expressed as 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1). {\displaystyle n} Framework of Rogers-Ramanujan identities: Lecture 2 Some Preliminaries Integer Partitions De nition A partition is a nonincreasing sequence of positive integers := ( 1; 2;:::) with nitely many non-zero terms. − + INTRODUCTION PARTITIONS ASSOCIATED WITH THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ! In the present paper we , {\displaystyle 1+3} ; In the case of the number 4, pa… , Srinivasa Ramanujan (1887-1920) was an Indian mathematician who made great and original contributions to many mathematical fields, including complex analysis, number theory, infinite series, and continued fractions. Such a partition is said to be self-conjugate.[7]. Recently, Andrews, Dixit, and Yee introduced partition functions associated with the Ramanujan/Watson mock theta functions $$\omega (q)$$ω(q) and $$\nu (q)$$ν(q). Introduction A partition of a natural number n … − As two particular cases, one has that the number of partitions of n in which all parts are 1 or 2 (or, equivalently, the number of partitions of n into 1 or 2 parts) is, and the number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n into at most three parts) is the nearest integer to (n + 3)2 / 12. 4 Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts. λ Abstract. Partitions One of Ramanujan and Hardy’s achievements, cited many times in The Man Who Knew Infinity, is a formula for calculating the number of partitions for any integer. Partition formula by Srinivasa Ramanujan. The Indian mathematician Ramanujan 1 II. p represents the number of possible partitions of a non-negative integer In some sources partitions are treated as the sequence of summands, rather than as an expression with plus signs. {\displaystyle n} has the five partitions explained a partition graphically by an array of dots or nodes. n Ramanujan and the theory of prime numbers 22 III. 1 Keywords: Ferrers and Young diagram, generating function, partitions, Ramanujan. , and Srinivasa Ramanujan and G. H. Hardy, Une formule asymptotique pour le nombre de partitions de n J. Riordan, Enumeration of trees by height and diameter , IBM J. Res. When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n).The influence of this manuscript cannot be underestimated. In fact, Ramanujan conjectured, and it was later shown, that such congruences exist modulo arbitrary powers of 5, 7, and 11. Ramanujan’s partition congruences MichaelD. partition. Hardy was trying to find the formulas for over many years. He de ned the rank of a partition ˇto be the biggest part of ˇminus the number of parts in ˇand conjectured that this rank divides partitions of 5n+ 4 and 7n+ 5 into 5 and 7 equinumerous classes. Remark 3.14. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. Child stated that the different types of partitions … ) [16] This result was stated, with a sketch of proof, by Erdős in 1942. Dans la premi ere, nous etudions des identit es de partitions du type Rogers-Ramanujan. Thus, the Young diagram for the partition 5 + 4 + 1 is, while the Ferrers diagram for the same partition is, While this seemingly trivial variation doesn't appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study of symmetric functions and group representation theory: filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects called Young tableaux, and these tableaux have combinatorial and representation-theoretic significance. For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. For example, the partition 2 + 2 + 1 might instead be written as the tuple (2, 2, 1) or in the even more compact form (22, 1) where the superscript indicates the number of repetitions of a term. Rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named after Norman Macleod Ferrers, and as Young diagrams, named after the British mathematician Alfred Young. Using Ramanujan’s dierential equations for Eisenstein series and an idea from Ramanu- jan’s unpublished manuscript on the partition function p(n) and the tau function ˝(n), we provide simple proofs of Ramanujan’s congruences for p(n) modulo 5, 7, and 11. 1 We de ned the number of partitions of zero to equal 1 in de nition 3.1 so this is considered a valid partition.
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