Andrews, G. E. (1998) The Theory of Partitions. Cambridge University Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....class [ The latter can be determined by the MurnagamaNakayama rule [M,JK] Denote by the conjugate partition 1 2 : where j = #fi : i jg. Clearly, Let p(n) be the number of partitions n. It is well known that p(n) exp(O( n) see e.g. [An]) 2. Main results The main result of this paper is establishing cuto for random walks on Sn generated by the single cycle conjugacy classes. Theorem 2.1 Consider a sequence of single cycle conjugacy classes Cn = r n 1 ] where mn = n r n n=2. Then a sequence of random walks W(Sn ; Cn ) ....

G. Andrews, The Theory of Partitions, Addison-Wesley, New York, 1976.

....the number of partitions n of the integer n and P (q) p(n)q the corresponding generating function. When is a partition of n then l( is the length (number of parts) of . We consider the integers l(n) l( the total length for n. Using the conjugacy map on partitions (see e.g. [1], 5] l(n) is also the sum of the rst parts of all n. It is convenient to use the exponential notation for partitions. Write = 1 ; 2 ; where a i ( is the multiplicity of i as a part in the partition . Thus l( a i ( The following result is known (see [9] but we ....

G.E. Andrews, The theory of partitions, Addison-Wesley 1976.

....degree in that order, seeking them in the hash table, and adding them to the output list. To enumerate monomials in this order, we first must enumerate the partitions of the the given degree with at most a certain number of parts (the number of variables) For this I use Hindenburg s algorithm [And76] for each possible number of parts. Now suppose there are 5 variables a, b, c, d, and e. A fixed partition such as 0 1 1 2 2 = 6 can be encoded as (0 , 1 , 2 ) Then each partition of the set of variables into subsets C, L, and Q where C = 1, L = 2, and Q = 2 ....

George Andrews, The Theory of Partitions, 1976.

....to a generalization of the Gauss identity. Finally, we further extend the Gauss identity in which is replaced by any root of unity. Keywords: involution, Ferrers diagrams, Gauss identity, Gauss coe#cients, Schur function, plethysm. 1. Introduction We follow the standard notation on q series [1, 2]. The q shifted factorials (a; q) n are defined by (a; q) n = 1, n = 0, a) 1 aq) aq n 1 ) n = 1, 2, The q binomial coe#cients, or the Gauss coe#cients, are given by or (q; q) n (q; q) k (q; q) n k . Note that the parameter q is often omitted in the notation of ....

....n k . Note that the parameter q is often omitted in the notation of Gauss coefficients when no confusion arises. The following classical identity is due to Gauss: Theorem 1.1 (Gauss) We have q) 1 m 1 ) if m is even. 1. 1) There have been several proofs of this identity [1, 2]. Goldman and Rota [3] find a proof by using a linear operator. Kupershmidt [4] obtains a generalization of the Gauss identity with an additional variable x. In fact, Gauss identity is a by product of results of Littlewood [6] on the evaluation of symmetric functions at roots of unity, and ....

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G.E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Co., Mass-London-Amsterdam, 1976.

.... 1 If we consider 0 compositions (a 1 a 2 : which actually means partitions, and perform the same analysis, we find the functional equation ; and upon iteration 1 F (z; 1) 1 1 Gamma oe(z) 2 ) Since Y we find in this way one of Euler s partition identities, [1]. This time, there is no dominant pole, since the equation oe(z) 1 has no solution inside the unit circle, and the asymptotics are harder. The interested reader can find the asymptotics (originally by Hardy, Ramanujan, and Rademacher) in the book [1] Here, one can also find information about ....

....this way one of Euler s partition identities, 1] This time, there is no dominant pole, since the equation oe(z) 1 has no solution inside the unit circle, and the asymptotics are harder. The interested reader can find the asymptotics (originally by Hardy, Ramanujan, and Rademacher) in the book [1]. Here, one can also find information about the famous Rogers Ramanujan identities which are close in spirit to 1 compositions. A further paper where such generating functions appear, is [13] however, we don t intend to be encyclopedic. 7. Concluding Remarks It should be clear by now that ....

G. E. Andrews, The Theory of Partitions, Addison--Wesley, 1976.

.... bases of the vector space [q ] if we define [x] n = q (n; k) x] we should have S q (n; k)s q (k; m) ffi mn for m;n 2 : It follows that (4:4) x] n ) q (n; k) x] q (n; k) S q (k; l)a Recall that the two classical q analogs of the exponential e [An, GR] are defined by e q (x) and E q (x) Note that (e q (x) E q ( Gammax) Hence (4:5) x] n ) a n k = k] n Since f[x] n g is a basis of [q ] and linear, we obtain the following result. PROPOSITION 6. For any polynomial P (x) of q , we have ....

ANDREWS (G.). --- The Theory of Partitions. --- Addison-Wesley, 1976.

.... l p i ; 0.9) minfs; tg; 1 s; t k; and for p 2 Z ; q) p : 1 Gamma q) 1 Gamma q ) q) 0 : 1: In complete analogy with the level one particular case (cf. GeI] Proposition 0.1) the above formula follows from (0. 6) and the Durfee rectangle combinatorial identity [A] 1 (q) 1 l0 (1 Gamma q a;b0 a Gammab=const ab (q) a (q) b : 0.10) 5 The basis underlying the above expression will be discussed in details somewhere else. In the simplest particular case n = k = 1; the right hand side of the above character reduces to p 1 ;p Gamma1 0 ....

G.E. Andrews, The theory of partitions, Addison-Wesley, 1976.

....will not increase as a result of this procedure. A subset X in the compressed form corresponds to a partition of the integer jXj contained in the m n rectangle. We give below the de nitions and properties of partitions that we will use in our proof of Theorem 1. The reader is referred to [2] for further details. Partitions A partition of an integer N is a sequence ( 1 ; 2 ; of positive integers (called parts) satisfying 1 2 and 1 2 = N . We put j j = N . The Ferrers diagram of is a 2 dimensional array of unit cells (or ....

....cannot increase and by the induction hypothesis j 1 X i j j d 1 X i j is smallest for each subgraph i, where X i is the set of elements of X that are in subgraph i. Not surprisingly, this means that candidate extremal sets in higher dimensions are among higher dimensional partitions (see [2], Ch. 11) which are contained in the d dimensional parallelepiped k 1 k 2 k d . Now we repeat the same steps for subgraphs along dimension d 1 as well. This step is illustrated by phase (ii) in Figure 5. Consider the (d 2) dimensional Hamming subgraphs of H when dimensions d and d ....

G.E. Andrews. The Theory of Partitions, Addison{Wesley Publishing Company, Reading, MA, 1976.

....the spectrum of the HS model and the vertex model was first indicated by Bernard [4] in the sl 2 case. where h i s are the standard basis of the Cartan subalgebra of sl n . It is shown in [21] N (q; x) q ; x) 7. 5) where HN (q; x) is a generalization of the Rogers Szego polynomial [1], q) k 1 (q) k 2 Delta Delta Delta (q) kn ; q) k = 1 Gamma q (7.6) Again there is an action of Y (sl n ) on , which commutes with HP [12] Based on a numerical study, it was conjectured in [12] that 1. As a Y (sl n ) module, the spin space decomposes exactly in ....

....1 (q; x) x i ; F 2 (q; x) qs (x) s (x) H 2 (q; x) i (1 q) 1i jn x i x j : Following [12] we consider the recursion relation for FN (q; x) and HN (q; x) Let 1 (tx 1 ; q) 1 (tx 2 ; q) 1 Delta Delta Delta (tx n ; q) 1 ; a; q) 1 = 1 Gamma aq Lemma A. 2 ([1]) The function G(q; x; t) is the generating function of HN (q; x) HN (q; x) Proof. It easily follows from the identity [1] 1 (t; q) 1 Lemma A.3 ( 1, 12] The functions HN (q; x) satisfy the following recursion relation: For any N 1, i 1 (q) N Gamma1 e i (x)HN Gammai (q; ....

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G. E. Andrews, The theory of partitions, Addison-Wesley Publ. 1976.

..... p a b , p 1 . p a =k . 2) For example U 0 (2,2) 00 , U 1 (2,2) 01 , U 2 (2,2) 11, 02 , U 3 (2,2) 12 , U 4 (2,2) 22 , whose number of elements are 1, 1, 2, 1, 1 respectively, the same as the coefficients of G(2,2) The proof of this combinatorial interpretation is as follows ([2],3.2) U k (b ,a ) can be partitioned into two disjoint subsets: those elements p for which p a b, whose number is equal to the cardinality of U k (b 1,a ) and those elements p for which p a =b , whose number is equal to the cardinality of U k b (b ,a 1) Thus ( for any finite set A, A will ....

Andrews, George, "The Theory of Partitions", Addison-Wesley, Reading, Mass., 1976.

.... representing the character of L( 0 ) for g = sl(2; C ) appeared also in [M2] as a limit of certain finite fermionic sums) The proof of the general formula in the above Proposition is analogous to the proof of the special case [FS] Using the Durfee rectangle combinatorial identity [A] 1 (q) 1 l0 (1 Gamma q a;b0 a Gammab=const ab (q) a (q) b ; 0.10) one immediately checks that (0.9) equals the well known character expression (coming from the homogeneous vertex operator construction [K] fi fi fi fi fi 0 ) 1 fi2Q 2 hfi;fii ; 0.11) ....

....means that we could have discovered our basis just from the requirement that the above reductio ad absurdum works Without any further elaborations, we can write down character formulas for the principal subspaces. The only two observations needed are the fundamental combinatorial identity (cf. [A]) 1 (q) r 1 (1 Gamma q) 1 Gamma q = 4.16) number of partitions of m with at most r parts 24 where r 2 Z ; q) 0 : 1; and the simple numerical identity (2(p Gamma 1) 1) 1 3 5 Delta Delta Delta (2r Gamma 1) r : 4.17) From the very Definition 4.1, ....

G.E. Andrews, The theory of partitions, Addison-Wesley, 1976.

....1 (1 x ) 1 x x 2k and so a term in the product obtained by choosing x ik from this expansion corresponds to choosing a partition with the integer k repeated i times. From (6) one can obtain all sorts of information about p(n) such as its asymptotic behavior. See Andrews book [1] for more details. Our main object of study will be the generating function for the Mobius function of a poset P , the so called characteristic polynomial. Let P have a zero and be ranked so that for any x # P all maximal chains from 0tox have the same length denoted #(x) and called the rank of ....

G. Andrews, "The Theory of Partitions," Addison-Wesley, Reading, MA, 1976.

....known that paths in an n cube may be represented with Gray Codes. In fact other combinatorial objects such as permutations can be made to have Gray Code like orderings [13] Also, the n cube may be viewed as the lattice of the power set of the set f0; 1; 2; n 1g ordered by set inclusion [2]. Of course some edges are removed. Since shortest paths in the n cube are the primary focus of this paper it is instructive to de ne a shortest path predicate. s; t; u) is true if and only if there exists a shortest path in the n cube from s to t inclusive of u. Formally, s; t; u) s t ....

G. E. Andrews. The Theory of Partitions. Addison Wesley Publishing Company, 1976.

....the MIT Class of 1922. 1 A beautiful formula of MacMahon [M] says that the number of plane partitions of n with at most a rows, at most b columns, and no part exceeding c (hereafter to be called (a, b, c) partitions ) is given by a 1 i=0 b 1 j=0 c 1 k=0 i j k 2 i j k 1 (see [A] and [S] for definitions of ordinary partitions and plane partitions, and section 2 of [CLP] for a fairly simple self contained proof of MacMahon s formula) Note that in the case c = 1, a plane partition with no part exceeding 1 can be read as the Ferrers diagram of an ordinary partition, and ....

G. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

.... = k 2(m Gamma k) k log F (k) Gamma Furthermore 2c(k) 2c so exp(F (k) 2c k) exp As (24k) we now have S 1 However Q(1 Gamma k=m) Q 2m 2 m 3 and it is well known (see for instance Andrews [1]) that Q(e ) e =6s s=2(1 O(s) 2.5) and hence Q(1 Gamma k=m) 6 f(k; m) if k = xm for fixed x. Note next that if k = o(m) but km 1 then fi 0 so fi and k c ) Thus (m Gamma k t) log = t O = t o(1) so the sum over those t for which t = k O 7=8 ....

G.E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, 1976.

....the number of partitions n of the integer n and P (q) p(n)q the corresponding generating function. When is a partition of n then l( is the length (number of parts) of . We consider the integers l(n) l( the total length for n. Using the conjugacy map on partitions (see e.g. [1], 5] l(n) is also the sum of the first parts of all n. It is convenient to use the exponential notation for partitions. Write = 1 ; 2 ; where a i ( is the multiplicity of i as a part in the partition . Thus l( a i ( The following result is known (see [9] but we ....

G.E. Andrews, The theory of partitions, Addison-Wesley 1976.

....We then add d copies of j to our partition and recursively generate a random partition of n dj. The recursion bottoms out when n dj = 0. It should be pointed out that the study of integer partitions is a mature research area in the interface between number theory and combinatorics (see Andrews [And84] Several deep results are known about the properties of the partition function p(n) In particular, the work of Hardy, Ramanujan and Rademacher produced an exact formula for p(n) in terms of an in nite series that involves , roots of unity, and hyperbolic functions (see [And84] A corollary ....

.... (see Andrews [And84] Several deep results are known about the properties of the partition function p(n) In particular, the work of Hardy, Ramanujan and Rademacher produced an exact formula for p(n) in terms of an in nite series that involves , roots of unity, and hyperbolic functions (see [And84] A corollary of this exact formula is the following simpler asymptotic formula for p(n) as n 1: p(n) 1 4n p 3 e p (2n) 3 : The values of p(n) increase very rapidly with n. For instance, p(40) 37338, p(60) 966467, p(80) 15796476, and p(100) 190569292. To implement the ....

G. E. Andrews. The Theory of Partitions. Cambridge U. Press, 1984.

.... (a;b;c;d;i;j) q maj(P) less(Q) q i 2 j 2 8 : 2 4 a i 3 5 2 4 b i 3 5 2 4 c j 3 5 2 4 d j 3 5 Gamma 2 4 a 1 i 1 3 5 2 4 b Gamma 1 i Gamma 1 3 5 2 4 c Gamma 1 j Gamma 1 3 5 2 4 d 1 j 1 3 5 9 = 1) where 2 4 a i 3 5 is the q binomial coefficient (see [1]) In 1955, Narayana [8] recorded such a result when the numbers of turns is ignored and q = 1: Formula (1) is a q analogue of a symmetric form of a distribution in four parameters formulated by Kreweras and Poupard [6, 5] see also [12] for counting Catalan paths with a fixed number of valleys, ....

....P k (n) denote the collection of k element subsets of f1; 2; ng. G(C) X (X 0 ;V 0 ;U 0 ;V 0 )2C q kX 0 k q kY 0 k q kU 0 k q kV 0 k = G(P i 1 (a 1) G(P i Gamma1 (b Gamma 1) G(P j Gamma1 (c Gamma 1) G(P j 1 (d 1) It is not difficult to show (See Andrews [1], Ch. 3) that G(P k (n) q ( k 1 2 ) 2 4 n k 3 5 ; whence Lemma 5 follows. Lemma 4 is proved in the same fashion. 9 Proof of formula ( 1) In this proof we will give a procedure similar to the encoding Gamma of section 4. Unfortunately, now we do not know an explicit formula for a ....

G.E. Andrews, Theory of Partitions, Addison-Wesley Pub. Co., New York, 1976.

....will not increase as a result of this procedure. A subset X 0 in the compressed form corresponds to a partition of the integer jXj contained in the m n rectangle. We give below the de nitions and properties of partitions that we will use in our proof of Theorem 1. The reader is referred to [2] for further details. Partitions A partition of an integer N is a sequence ( 1 ; 2 ; of positive integers (called parts) satisfying 1 2 and 1 2 = N . We put j j = N . The Ferrers diagram of is a 2 dimensional array of unit cells (or ....

....Xj cannot increase and by induction hypothesis j 1 X i j c 1 j d 1 X i j c d 1 is smallest for each subgraph i, where X i is the set of elements of X that are in subgraph i. This means that candidate extremal sets in higher dimensions are among higher dimensional partitions (see [2], Ch. 11) which are contained in the d dimensional parallelpiped k 1 k 2 k d . Now we repeat the same steps for subgraphs along dimension d 1 as well. Consider the (d 2) dimensional Hamming subgraphs of H d when dimensions d and d 1 are xed. We call such a subgraph complete i ....

G.E. Andrews. The Theory of Partitions, Addison{Wesley Publishing Company, Reading, MA, 1976.

....that paths in an n cube may be represented with Gray codes. In fact other combinatorial objects such as permutations can be made to have Gray Code like orderings [14] Also, the n cube may be viewed as the lattice of the power set of the set f0; 1; 2; n Gamma 1g ordered by set inclusion [3]. Since shortest paths in the n cube are the primary focus of this paper it is instructive to define a shortest path predicate. Theta(s; t; u) is true if and only if there exists a shortest path in the n cube from s to t inclusive of u. Formally, Theta(s; t; u) s Phi t = s Phi u bitwise ....

G. E. Andrews. The Theory of Partitions. Addison Wesley Publishing Company, 1976.

....geometic random variables (r.v.s, for short) with parameter 1=2. We then study the number of distinct part sizes, the maximum part size, and the rst empty part in a random composition. 2. 1 Representation of random compositions The following representation of a composition that can be found in [4] is of crucial importance: there is a one to one correspondence between compositions of N and strings of black and white dots of length N with the following provisions: i) the last dot is always black (ii) each of the remaining N 1 dots is black or white; part sizes in a composition are waiting ....

Andrews, G. E. The Theory of Partitions, Addison { Wesley, Reading, MA, 1976. 23

....) P n (b) b n = X n0 ( Gamma1) n 1 x n P n (b) b n = X n0 ( Gamma1) n P n (b) since x n = b n = Y n1 (1 Gamma b n ) 10) 4 The last step follows from an identity originally proved by Euler in 1748 and generalised by Heine in the 19th century (see formula 2.2. 6 of Andrews (1976), or formula (1.195) of Johnson, Kotz and Kemp (1993) The identity is X n0 z n P n (b) Y n1 (1 zb n ) 11) valid for jbj 1 and all z. We have used z = Gamma1 to derive (10) For j 1, commencing with (6) and using (7) we have g j (t) g j = X n0 ( Gamma1) n Lh j (x n ) ....

Andrews, G. (1976) The Theory of Partitions. Encyclopedia of Mathematics and Its Applications.

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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

....the notation. We denote partitions of n by = 1 ; 2 ; write n, or j j = n. Let be a conjugate partition to . The largest part and the number of parts of are denoted by a( and ( respectively. We use Young diagrams to represent partitions graphically. See [A3] for standard references, definitions and details. Partitions Into Distinct Parts and Franklin s Involution We start with the following result straight from [F1] THEOREM 1 (Fine) Let D parts, such that the largest part a( 1 is even and odd, respectively. Then: 8 1; if n ....

....classical combinatorial arguments. The proofs of both Theorem 3 and Theorem 4 follows from Sylvester s celebrated bijection, sometimes called a sh hook construction. This bijection is a map between partitions into odd and distinct numbers, and gives a combinatorial proof of Euler s theorem (see [A1,A3]) Sylvester s bijection is another xture in the combinatorics of partitions. It has been restated in many di erent ways (e.g. by using Frobenius coordinates and 2 modular diagrams [A6,B,PP] was used to prove other re nements of Euler s theorem [KY] Had Theorem 3 been better known and not ....

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G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, MA, 1976.

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G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, 1976. (Reissued; Cambridge University Press, Cambridge, 1998).

....a 0 = A] If (a n ) n0 satisfies (1) and (2) then it is the Engel sequence associated to A. In [3] and [4] various classical q series identities are shown to be examples of extended Engel expansions. For instance, in [3] one finds a detailed proof that the celebrated RogersRamanujan identities [2] fit exactly into this pattern. In the present article we show that they form the basis of an infinite collection of extended Engel expansions. Only recently, T. Garrett, M. Ismail, and D. Stanton have found a new and elegant generalization [5, 3.5) It involves an extra parameter the ....

G.E. Andrews, The Theory of Partitions, in "Encyclopedia Math. Appl.", Vol. 2, G.- C. Rota ed., Addison-Wesley, Reading,

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G.E. Andrews, The theory of partitions, Encycl. Math. Appl. vol. 2, Addison-Weley, 1976.

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Andrews, G. E. (1998) The Theory of Partitions. Cambridge University Press.

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G.E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Massachusetts, 1976.

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G. Andrews, The Theory of Partitions, Addison-Wesley Pub. Co., Reading, Mass., 1976.

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G. E. Andrews, "The Theory of Partitions", Encycloped i of Mathemat iN and

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G.E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Massachusetts, 1976.

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G. E. Andrews, The Theory of Partitions, Addison-Wesley 1976

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G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, MA, 1976. 6

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George E. Andrews. The theory of partitions. Addison-Wesley Publishing Co., Reading, Mass.-LondonAmsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2.

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Andrews G.E., (1998), The Theory of Partitions, Cambridge University Press.

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G. E. Andrews, The Theory of Partitions, Addison--Wesley, Reading, MA, 1976.

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G. E. Andrews, The Theory of Partitions, Addison-Wesley 1976

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G.E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

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G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, MA, 1976.

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G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, MA, 1976.

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Andrews, G. E. (1976) The theory of partitions. Encyclopedia of Mathematics and its Applications, Volume 2, Edited by Gian-Carlo Rota Addison-Wesley Publishing Company.

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Andrews, G. E. (1976) The theory of partitions. Encyclopedia of Mathematics and its Applications, Volume 2, Edited by Gian-Carlo Rota Addison-Wesley Publishing Company.

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G. Andrews, The theory of partitions, Addison-Wesley, 1976.

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G. E. Andrews, The theory of partitions, Addison-Wesley Publ. 1976.

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G. E. Andrews, The Theory of Partitions, Addison-Wesley 1976.

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George E. Andrews, "The Theory of Partitions." London, AddisonWesley, 1976 (Encyclopedia of Math. and Its Appl., 2).

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George Andrews, The Theory of Partitions, 1976.

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G.E. Andrews (1977), The Theory of Partitions, Addison-Wesley, Reading, MA.

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G. E. Andrews, `The Theory of Partitions', Encyclopedia of Mathematics and its Applications, Addison--Wesley, Reading 1976.

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