Distribution of the full rank in residue classes for odd moduli

Keith, William J.

doi:10.1016/j.disc.2009.02.032

Cited by 6 publications

(4 citation statements)

References 16 publications

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“…The identities which we will obtain in Theorem 5.1 (1) were proven in Theorem 17 of Andrews [5]. The remaining identities in Theorem 5.1 were proven (with a different method) by Keith [23]. However, we include this case for completeness as well as to exhibit this method of constructing identities.…”

Section: Atkin and Swinnerton-dyer Type Identitiesmentioning

confidence: 96%

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Inequalities for full rank differences of 2-marked Durfee symbols

Bringmann

Kane

2012

Journal of Combinatorial Theory, Series A

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In this paper, we obtain infinitely many non-trivial identities and inequalities between full rank differences for 2-marked Durfee symbols, a generalization of partitions introduced by Andrews. A certain strict inequality, which almost always holds, shows that identities for Dyson's rank, similar to those proven by Atkin and Swinnerton-Dyer, are quite rare. By showing an analogous strict inequality, we show that such non-trivial identities are also rare for the full rank, but on the other hand we obtain an infinite family of non-trivial identities, in contrast with the partition theoretic case.Dyson conjectured that the congruence for 5n + 4 is explained by the fact that the rank modulo 5 divides the partitions of 5n + 4 into 5 equally sized classes, namely for every r, s ∈ Z N (r, 5; 5n + 4) = N (s, 5, 5n + 4)( 1.2) holds for all n ∈ N 0 . This implies the above congruence, since by (1.2) p (5n + 4) = 5N (0, 5, 5n + 4) ≡ 0 (mod 5).

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Section: Atkin and Swinnerton-dyer Type Identitiesmentioning

confidence: 96%

“…We note that in the case t = 5, the identities in Section 5 were proven by Keith in Theorems 1 and 2 of [23]. He considers general k, but restricts himself to the special case t = 2k + 1 and exploits identities of the type…”

Section: Some Remarksmentioning

confidence: 99%

Inequalities for full rank differences of 2-marked Durfee symbols

Bringmann

Kane

2012

Journal of Combinatorial Theory, Series A

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“…The full rank function NF k (r, t; n) have been extensively studied and they posses many congruence properties, see for example, [5][6][7][8]14]. Recently, Bringmann, Garvan and Mahlburg [6] used the automorphic properties of the generating functions of NF k (r, t; n) to prove the existence of infinitely many congruences for NF k (r, t; n).…”

Section: Introductionmentioning

confidence: 99%

k-Marked Dyson Symbols and Congruences for Moments of Cranks

Chen¹,

Ji²,

Shen³

2013

Preprint

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By introducing k-marked Durfee symbols, Andrews found a combinatorial interpretation of 2k-th symmetrized moment η 2k (n) of ranks of partitions of n. Recently, Garvan introduced the 2k-th symmetrized moment µ 2k (n) of cranks of partitions of n in the study of the higher-order spt-function spt k (n). In this paper, we give a combinatorial interpretation of µ 2k (n). We introduce k-marked Dyson symbols based on a representation of ordinary partitions given by Dyson, and we show that µ 2k (n) equals the number of (k + 1)-marked Dyson symbols of n. We then introduce the full crank of a k-marked Dyson symbol and show that there exist an infinite family of congruences for the full crank function of k-marked Dyson symbols which implies that for fixed prime p ≥ 5 and positive integers r and k ≤ (p − 1)/2, there exist infinitely many non-nested arithmetic progressions An + B such that µ 2k (An + B) ≡ 0 (mod p r ).

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“…They used the automorphic properties to prove the existence of infinitely many congruences for k-marked Durfee symbols. Also, William J. Keith did some work on the congruential properties of k-marked Durfee symbols in [15,16]. He mainly showed that k-marked Durfee symbols of n equally populate the residue classes a and b (mod 2k + 1) if gcd(a, 2k + 1) = gcd(b, 2k + 1).…”

Section: Theorem 13 (Andrews) the Number Of Ordinary Partitions Of mentioning

confidence: 99%

The combinatorics of $k$-marked Durfee symbols

2011

Trans. Amer. Math. Soc.

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Abstract. George E. Andrews recently introduced k-marked Durfee symbols which are connected to moments of Dyson's rank. By these connections, Andrews deduced their generating functions and some combinatorial properties and left their purely combinatorial proofs as open problems. The primary goal of this article is to provide combinatorial proofs in answer to Andrews' request. We obtain a partition identity, which gives a relation between k-marked Durfee symbols and Durfee symbols by constructing bijections, and all identities on k-marked Durfee symbols given by Andrews could follow from this identity. In a similar manner, we also prove the identities due to Andrews on k-marked odd Durfee symbols combinatorially, which resemble ordinary k-marked Durfee symbols with a modified subscript and with odd numbers as entries.

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Distribution of the full rank in residue classes for odd moduli

Cited by 6 publications

References 16 publications

Inequalities for full rank differences of 2-marked Durfee symbols

Inequalities for full rank differences of 2-marked Durfee symbols

k-Marked Dyson Symbols and Congruences for Moments of Cranks

The combinatorics of $k$-marked Durfee symbols

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