Dyson's crank and the mex of integer partitions

Hopkins, Brian; Sellers, James A.; Stanton, Dennis

doi:10.1016/j.jcta.2021.105523

Cited by 16 publications

(13 citation statements)

References 12 publications

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“…For example, m 1,2 (n) is the number of partitions of n with odd mex. In [23], it was productive to split the odd mex partitions modulo 4; in our notation, these are m 1,4 (n) and [23].) Next, we show that breaking these counts down further, by parity of partition length, shows new relations and allows for cleaner proofs of previous results.…”

Section: Connecting Mex and Crankmentioning

confidence: 76%

“…Other results of Hopkins, Sellers, and Stanton [23] show unexpected connections between the crank and another representation of integer partitions: The Frobenius symbol of a partition consists of two rows of strictly decreasing nonnegative integers. Given the Ferrers diagram of a partition, the top row of the Frobenius symbol gives the number of boxes to the right of the diagonal entries and the bottom row gives the number of boxes below the diagonal entries.…”

mentioning

confidence: 90%

“…Further results of Hopkins and Sellers [22], followed by Hopkins, Sellers, and Stanton [23], establish additional connections between partitions satisfying certain crank conditions and partitions with certain mex properties. Those proofs primarily use generating function identities.…”

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confidence: 91%

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Combinatorial Perspectives on the Crank and Mex Partition Statistics

Hopkins

Sellers²,

Yee³

2022

Electron. J. Combin.

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Several authors have recently considered the smallest positive part missing from an integer partition, known as the minimum excludant or mex. In this work, we revisit and extend connections between Dyson's crank statistic, the mex, and Frobenius symbols, with a focus on combinatorial proof techniques. One highlight is a generating function expression for the number of partitions with a bounded crank that does not include an alternating sum. This leads to a combinatorial interpretation involving types of Durfee rectangles. A recurring combinatorial technique uses sign reversing involutions on certain triples of partitions to establish a result of Fine and other identities.

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Section: Connecting Mex and Crankmentioning

confidence: 76%

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confidence: 90%

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Combinatorial Perspectives on the Crank and Mex Partition Statistics

Hopkins

Sellers²,

Yee³

2022

Electron. J. Combin.

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“…The novelty of Theorem 1.10 is that it identifies the odd crank enumeration of partitions with those partitions into odd number of parts and self-conjugate partitions through Liouville's function λ. Following the work done in [6,18], Theorem 1.11 springs up rather organically. Here, we count Frobenius symbols with restrictions on the entries and equate them to the enumeration of number of partitions with no parts that equal the size of the Durfee square of that partition, two ideas in the theory of partitions that are very rarely correlated.…”

Section: Definition 14 ([7]mentioning

confidence: 99%

Ramanujan’s Theta Functions and Parity of Parts and Cranks of Partitions

Banerjee

Dastidar²

2022

Ann. Comb.

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In this paper, we explore intricate connections between Ramanujan’s theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation with the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.

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“…Using the following reformulation of a result provided by Hopkins, Sellers and Stanton in [6], we derive the generalization of Theorem 1.3 from Theorem 1.9 and Corollary 1.2.…”

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confidence: 99%

A bijective proof of a generalization of the non-negative crank--odd mex identity

Konan¹

2022

Preprint

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Recent works of Andrews-Newman and Hopkins-Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer n, there are as many partitions of n with non-negative crank as partitions of n with odd mex. In this paper, we provide a generalization of this identity and prove it bijectively. Our method uses an alternative definition of the Durfee decomposition, whose combinatorial link to the crank was recently studied by Hopkins, Sellers, and Yee.

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Dyson's crank and the mex of integer partitions

Cited by 16 publications

References 12 publications

Combinatorial Perspectives on the Crank and Mex Partition Statistics

Combinatorial Perspectives on the Crank and Mex Partition Statistics

Ramanujan’s Theta Functions and Parity of Parts and Cranks of Partitions

A bijective proof of a generalization of the non-negative crank--odd mex identity

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