Combinatorial Proofs of Two Euler-Type Identities Due to Andrews

Ballantine, Cristina; Bielak, Richard

doi:10.1007/978-3-030-57050-7_9

Cited by 8 publications

(13 citation statements)

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“…Beck's conjecture was quickly proved analytically by Andrews [2], who additionally showed that this excess also equals the number of partitions of n with exactly one part repeated (and all other parts distinct). The conjecture was also proved combinatorially by Yang [11] and Ballantine-Bielak [4] independently. This work was followed by generalizations and Beck-type companions to other well known identities (e.g., [3], [5], [9], [11]).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 84%

On a Partition Identity of Lehmer

Ballantine¹,

Burson²,

Folsom³

et al. 2021

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Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called "Beck-type" companions to other identities.In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 84%

On a Partition Identity of Lehmer

Ballantine¹,

Burson²,

Folsom³

et al. 2021

Preprint

Self Cite

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“…By looking at matrices such as those constructed in [5], in this paper we give quick elementary expositions of combinatorial proofs of Theorem 1, Theorem 2, and Theorem 3 but which essentially reduce to being variants of the proofs earlier given in [2,3,4]. We then state and prove two new results, one of which extends Theorem 2 and the other extends Theorem 3.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…After Andrews' analytical proof of these results, combinatorial proofs of the above conjectures were given in the papers of Yang [2], Fu and Tang [3], and Ballantine and Bielak [4]. Another direction in which we could study a variant of the statement of Theorem 1 is to see if we can allow more than one even part.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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On certain partition bijections related to Euler's partition problem

Aritro¹

2020

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“…Certain generalizations and combinatorial proofs appeared in [6] and [11]. Combinatorial proofs of the original conjectures were also given in [5]. Several additional similar identities were proved in the last two years.…”

Section: Introductionmentioning

confidence: 99%

Beck-type identities for Euler pairs of order $r$

Ballantine¹,

Welch²

2020

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Partition identities are often statements asserting that the set P X of partitions of n subject to condition X is equinumerous to the set P Y of partitions of n subject to condition Y . A Beck-type identity is a companion identity to |P X | = |P Y | asserting that the difference b(n) between the number of parts in all partitions in P X and the number of parts in all partitions in P Y equals a c|P X ′ | and also c|P Y ′ |, where c is some constant related to the original identity, and X ′ , respectively Y ′ , is a condition on partitions that is a very slight relaxation of condition X, respectively Y . A second Beck-type identity involves the difference b ′ (n) between the total number of different parts in all partitions in P X and the total number of different parts in all partitions in P Y . We extend these results to Beck-type identities accompanying all identities given by Euler pairs of order r (for any r ≥ 2). As a consequence, we obtain many families of new Beck-type identities. We give analytic and bijective proofs of our results.

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Combinatorial Proofs of Two Euler-Type Identities Due to Andrews

Cited by 8 publications

References 1 publication

On a Partition Identity of Lehmer

On a Partition Identity of Lehmer

On certain partition bijections related to Euler's partition problem

Beck-type identities for Euler pairs of order $r$

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