Partitions with Prescribed Hook Differences

Andrews, George E.; Baxter, R. J.; Burge, William H.; Viennot, Gérard

doi:10.1016/s0195-6698(87)80041-0

Cited by 61 publications

(103 citation statements)

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“…The polynomials (1.12) generalize the polynomials used by Schur [2] in connection with difference two partitions and is the finitization we use in this paper. They satisfy a simple recursion relation in L and can be interpreted as the generating functions for partitions with prescribed hook differences [35].…”

Section: Introductionmentioning

confidence: 99%

Rogers-Schur-Ramanujan Type Identities for the M ( p , p ′) Minimal Models of Conformal Field Theory

Berkovich¹,

McCoy²,

Schilling³

1998

Communications in Mathematical Physics

132

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Section: Introductionmentioning

confidence: 99%

Rogers-Schur-Ramanujan Type Identities for the M ( p , p ′) Minimal Models of Conformal Field Theory

Berkovich¹,

McCoy²,

Schilling³

1998

Communications in Mathematical Physics

132

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“…Following Atkin, Andrews [8] and Bressoud [13] studied partitions with prescribed bounds for successive ranks. In [10] the notion of successive ranks was generalized to hook differences, with the hook vertices not necessarily on the diagonal of the Durfee square. Also in [10] the succesive rank theorem of Andrews [8] and Bressoud [13] was revised as follows: …”

Section: Successive Ranks With Prescribed Boundsmentioning

confidence: 99%

“…Theorem R is a consequence of Theorem 5 of [10], but we stress here that if A 2k,k (n) is to be interpreted as the number of partitions of n into parts ≡ 0, ±k (mod 2k), then this has to be in the sense of (7.1), where the residue class k (mod 2k) is "deleted twice" because it occurs as both k and −k (mod 2k).…”

Section: Then For 1 ≤ I < K/2 We Havementioning

confidence: 99%

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New weighted Rogers-Ramanujan partition theorems and their implications

Alladi

Berkovich

2002

Trans. Amer. Math. Soc.

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Abstract. This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems -a set of three theorems involving partitions into parts ≡ 0, ±i (mod 6), and a set of three theorems involving partitions into parts ≡ 0, ±i (mod 7), i = 1, 2, 3. IntroductionThe theory of partitions is rich in examples of identities whose combinatorial interpretation yields the equality of partition functions defined in very different ways. Recently Alladi [2], [3], has developed a theory of weighted partition identities which deals with partition functions which are unequal, but where equality is attained by attaching weights. In this paper we obtain several new results by attaching weights to Rogers-Ramanujan partitions, namely, partitions into parts differing by ≥ 2. In all instances the weights are defined multiplicatively. In [2], [3], the results primarily deal with the case where one set of partitions is a subset of the other, and where positive integral weights are attached to the smaller set of partitions. The results in this paper are further significant examples of such weighted identities and also deal with the more general situation of two unequal sets of partitions where weights positive or negative could be attached to either set.In the first part of the paper ( §1- §6), we obtain a reformulation of a deep theorem of Göllnitz [15] and discuss the implications. The new reformulation is stated as Theorem 1 in §1 and is of great interest because it has several important consequences. Special cases of Theorem 1 yield Jacobi's triple product identity for theta functions (see §2), Sylvester's famous refinement of Euler's theorem (see §3), and

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“…Applying C 4 to λ = (9, 8, 8, 7, 7, 6, 5, 4, 4, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1) gives C 4 (λ) = (9,9,8,7,7,6,5,4,4,3,3,3,3, 2, 2, 2, 2, 1, 1). Intermediate steps are ν 1 , ν 2 , ν 3 , ν 4 , α ′ , and β ′ as shown.…”

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confidence: 99%

Dyson’s New Symmetry and Generalized Rogers-Ramanujan Identities

Boulet¹

2010

Ann. Comb.

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Abstract. We present a generalization, which we call (k, m)-rank, of Dyson's notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries bijectively. Using the two symmetries we give a new combinatorial proof of generalized Roger-Ramanujan identities. We also describe the relationship between (k, m)-rank and Garvan's k-rank. IntroductionFirst discovered by Rogers [22] in 1894, the Rogers-Ramanujan identities,are among the most intriguing partition identities.The goal of this paper is to present a new combinatorial proof of the following generalization (which is due to Andrews [2]) of the first Rogers-Ramanujan identity, for k ≥ 1:(1)where N j = n j +n j+1 +· · ·+n k−1 . We use the standard q-series notation and letInstead of attacking this identity directly, we will use two bijections to prove the following family of identities, which we call the generalized Schur identities:with N j = n j + n j+1 + · · · + n k−1 . By using Jacobi's triple product identity,Key words and phrases. Rogers-Ramanujan identity, Schur's identity, successive Durfee squares, Dyson's rank, bijection, integer partition.* Department of Mathematics, Cornell University, Ithaca, NY 14853. Email: cilanne@math.cornell.edu.1 which specializes to1 − q n when we let t = q 2k+1 and z = −q −k , we see that (1) and (2) are equivalent. This application of Jacobi's triple product identity is a standard first step in Rogers-Ramanujan proofs and in particular was used by Schur [23] in his combinatorial proof of the original Rogers-Ramanujan identities. We note that the Jacobi triple product identity has a combinatorial proof due to Sylvester (see [21,25]) and so its application does not change the combinatorial nature of our proof.Before presenting our proof of the generalized Rogers-Ramanujan identities (1), we must outline our notation and review two important ideas. The first is Andrews' notion of successive Durfee squares which gives a combinatorial interpretation to the left hand side of (1) and (2). The second is Dyson's proof of Euler's pentagonal number theorem based on his definition of rank.

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Partitions with Prescribed Hook Differences

Cited by 61 publications

References 9 publications

Rogers-Schur-Ramanujan Type Identities for the M ( p , p ′) Minimal Models of Conformal Field Theory

Rogers-Schur-Ramanujan Type Identities for the M ( p , p ′) Minimal Models of Conformal Field Theory

New weighted Rogers-Ramanujan partition theorems and their implications

Dyson’s New Symmetry and Generalized Rogers-Ramanujan Identities

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