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Euler's Partition Theorem

2015
*
Formalized Mathematics
*

In this article we prove the

doi:10.1515/forma-2015-0009
fatcat:ndz4y2pdyffgxk4jb2vidql3pa
*Euler's**Partition**Theorem*which states that the number of integer*partitions*with odd parts equals the number of*partitions*with distinct parts. ...*Euler's**Partition**Theorem*is listed as item #45 from the "Formalizing 100*Theorems*" list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [27]. ... Observe that there exists a*partition*of n which is odd-valued and there exists a*partition*of n which is one-to-one. Let us observe that sethood property holds for*partitions*of n. ...##
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A Unification of Two Refinements of Euler's Partition Theorem
[article]

2009
*
arXiv
*
pre-print

We obtain a unification of two refinements of

arXiv:0812.2826v3
fatcat:a5v3auriazfpppuqrfuwuvyl2m
*Euler's**partition**theorem*respectively due to Bessenrodt and Glaisher. ... A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson*partition*identity is used in our combinatorial construction. ...*Euler's**partition**theorem*reads as follows. ...##
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Euler's Partition Theorem with Upper Bounds on Multiplicities
[article]

2012
*
arXiv
*
pre-print

For m=0, our result reduces to Bessenrodt's refinement of

arXiv:1111.1489v2
fatcat:27wht7puirbf5okd2ouiow4g2u
*Euler's**Theorem*. ... If the alternating sum and the number of odd parts are not taken into account, we are led to a connection to a generalization of*Euler's**theorem*, which can be deduced from a*theorem*of Andrews on equivalent ... For the case m = 0, our result reduces to a refinement of*Euler's**partition**theorem*due to Bessenrodt [6] . ...##
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Combinatorial proofs and generalizations of conjectures related to Euler's partition theorem
[article]

2018
*
arXiv
*
pre-print

In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between

arXiv:1801.06815v1
fatcat:5ao2cuqy45fcjflr5ckffynpim
*partitions*of a positive integer into odd parts and*partitions*into ... Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck's conjecture relating the gap-free*partitions*and distinct*partitions*with odd length. ... G ′ I (12) G λ ⊆ G(12) G 0 (12) (12) , G ′ I (12) and G(12) G 0 (12)*Theorem*1.1 (*Euler's**partition**theorem*) The number of distinct*partitions*of n equals the number of odd*partitions*of n. ...##
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The complexity of Euler's integer partition theorem

2012
*
Theoretical Computer Science
*

*Euler's*integer

*partition*

*theorem*, which states that the number of

*partitions*of an integer into odd integers is equal to the number of

*partitions*into distinct integers, ranks 16 in Wells' list of the ... In this paper, we use the algorithmic method to evaluate the complexity of mathematical statements developed in Calude et al. (2006) [5] and Calude (2009, 2010) [6,7] and to show that

*Euler's*

*theorem*is ... To conclude, we cite a question posed by a referee: How does the complexity of

*Euler's*integer

*partition*

*theorem*established in this paper compare with the complexity of related integer

*partition*

*theorems*...

##
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Core partitions into distinct parts and an analog of Euler's theorem
[article]

2016
*
arXiv
*
pre-print

This simple but curious analog of

arXiv:1601.07161v1
fatcat:qrflzkqi7jcixm4nhyjwyjorfe
*Euler's**theorem*appears to be missing from the literature on*partitions*. ... As a by-product of our results, we obtain a bijection between*partitions*into distinct parts and*partitions*into odd parts, which preserves the perimeter (that is, the largest part plus the number of parts ... I thank Tewodros Amdeberhan for introducing me to (s, t)-core*partitions*and his conjecture, as well as for many interesting discussions, comments and suggestions. I am also grateful to George E. ...##
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A bijection for Euler's partition theorem in the spirit of Bressoud
[article]

2018
*
arXiv
*
pre-print

splitting

arXiv:1803.11104v1
fatcat:skdlpxxn4naprcgjydj6lhgg5q
*partitions*of n. ... For each positive integer n, we construct a bijection between the odd*partitions*and the distinct*partitions*of n which extends Bressoud's bijection between the odd-and-distinct*partitions*of n and the ... Introduction*Euler's**Theorem*is that the number of odd*partitions*of a positive integer n equals the number of distinct*partitions*of n. ...##
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Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities
[article]

2018
*
arXiv
*
pre-print

We generalise

arXiv:1608.03635v2
fatcat:lxqvgw6o3bfsnnqjakbiz2sj5i
*Euler's**partition**theorem*involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan- Andrews-Gordon identities related to this*theorem*. ... Introduction In the theory of*partitions*,*Euler's**partition**theorem*involving odd parts and different parts is one of the famous*theorems*. ... We can think of*Euler's**theorem*as a*theorem*on*partitions*involving modulus two by interpreting odd parts as parts ≡ 1 (mod 2). ...##
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Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities
[article]

2020
*
arXiv
*
pre-print

In this paper, we give a conjecture, which generalises

arXiv:1607.07583v2
fatcat:mfltrj2z5rdhznh7t375tq7vhy
*Euler's**partition**theorem*involving odd parts and different parts for all moduli. We prove this conjecture for two family*partitions*. ... Introduction In the theory of*partitions*,*Euler's**partition**theorem*involving odd parts and distinct parts is one of the famous*theorems*. ... We can think of*Euler's**theorem*as a*theorem*on*partitions*involving modulus two by interpreting odd parts as parts ≡ 1 (mod 2). ...##
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Harmonic Partitions of Positive Integers and Bosonic Extension of Euler's Pentagonal Number Theorem
[article]

2019
*
arXiv
*
pre-print

In this note, we first propose a cohomological derivation of the celebrated

arXiv:1911.08702v1
fatcat:2263jrpawbao5ooouqcqoicim4
*Euler's*Pentagonal Number*Theorem*. Then we prove an identity that corresponds to a bosonic extension of the*theorem*. ... The proof corresponds to a cohomological re-derivation of*Euler's*another celebrated identity. ... The Case of*Partitions*with Distinct Parts and*Euler's*Pentagonal Number*Theorem*Definition 1 Let us define the*partition*of positive integer n with distinct parts: σ := (n 1 , n 2 , · · · , n ℓ ), (n ...##
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A unification of two refinements of Euler's partition theorem

2010
*
The Ramanujan journal
*

We obtain a unification of two refinements of

doi:10.1007/s11139-008-9156-7
fatcat:trfcnq3xmzabbispwiqo6vs2oq
*Euler's**partition**theorem*respectively due to Bessenrodt and Glaisher. ... A specialization of Bessenrodt's insertion algorithm for a generalization of the Andrews-Olsson*partition*identity is used in our combinatorial construction. ... These two bijections imply refinements of*Euler's**theorem*. ...##
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Euler's partition theorem and the combinatorics of ℓ-sequences

2008
*
Journal of combinatorial theory. Series A
*

*Euler's*

*partition*

*theorem*states that the number of

*partitions*of an integer N into odd parts is equal to the number of

*partitions*of N in which the ratio of successive parts is greater than 1. ... This generalization of

*Euler's*

*theorem*is intrinsically different from the many others that have appeared, as it involves a family of

*partitions*constrained by the ratio of successive parts. ... These sequences λ are called lecture hall

*partitions*. Note that taking limits as n → ∞ in

*Theorem*2 gives

*Euler's*

*theorem*. ...

##
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Core partitions into distinct parts and an analog of Euler's theorem

2016
*
European journal of combinatorics (Print)
*

This simple but curious analog of

doi:10.1016/j.ejc.2016.04.002
fatcat:zqbf6ziutvbsvb2dkau5e75k6u
*Euler's**theorem*appears to be missing from the literature on*partitions*. * ... As a by-product of our results, we obtain a bijection between*partitions*into distinct parts and*partitions*into odd parts, which preserves the perimeter (that is, the largest part plus the number of parts ... I thank Tewodros Amdeberhan for introducing me to (s, t)-core*partitions*and his conjecture, as well as for many interesting discussions, comments and suggestions. I am also grateful to George E. ...##
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Euler's Partition Theorem with Upper Bounds on Multiplicities

2012
*
the electronic journal of combinatorics
*
unpublished

For m = 0, our result reduces to Bessenrodt's refinement of

fatcat:npeqjjkatvd3xd7bqgopws2yhi
*Euler's**partition**theorem*. ... If the alternating sum and the number of odd parts are not taken into account, we are led to a generalization of*Euler's**partition**theorem*, which can be deduced from a*theorem*of Andrews on equivalent ... There is another refinement of*Euler's**partition**theorem*due to Glaisher [12] . ...##
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Weighted forms of Euler's theorem

2007
*
Journal of combinatorial theory. Series A
*

In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan's "lost" notebook, we obtain weighted forms of

doi:10.1016/j.jcta.2006.06.005
fatcat:5ed4emm6ufesjk3pp667up2a2m
*Euler's**theorem*on*partitions*with odd parts and distinct ... This work is inspired by the insight of Andrews on the connection between Ramanujan's identities and*Euler's**theorem*. ... two weighted forms (1.5) and (1.6) of*Euler's**theorem*. ...
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