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### Euler's Partition Theorem

Karol Pąk
2015 Formalized Mathematics
In this article we prove the 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/ .  ...  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.  ...

### A Unification of Two Refinements of Euler's Partition Theorem [article]

William Y. C. Chen, Henry Y. Gao, Kathy Q. Ji, Martin Y. X. Li
2009 arXiv   pre-print
We obtain a unification of two refinements of 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.  ...

### Euler's Partition Theorem with Upper Bounds on Multiplicities [article]

William Y. C. Chen, Ae Ja Yee, Albert J. W. Zhu
2012 arXiv   pre-print
For m=0, our result reduces to Bessenrodt's refinement of 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  .  ...

### Combinatorial proofs and generalizations of conjectures related to Euler's partition theorem [article]

Jane Y.X. Yang
2018 arXiv   pre-print
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between 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.  ...

### The complexity of Euler's integer partition theorem

Cristian S. Calude, Elena Calude, Melissa S. Queen
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)  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  ...

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

Armin Straub
2016 arXiv   pre-print
This simple but curious analog of 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.  ...

### A bijection for Euler's partition theorem in the spirit of Bressoud [article]

John Murray
2018 arXiv   pre-print
splitting 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.  ...

### Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities [article]

XinHua Xiong, William J. Keith
2018 arXiv   pre-print
We generalise 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).  ...

### Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities [article]

Xinhua Xiong, William J. Keith
2020 arXiv   pre-print
In this paper, we give a conjecture, which generalises 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).  ...

### Harmonic Partitions of Positive Integers and Bosonic Extension of Euler's Pentagonal Number Theorem [article]

Masao Jinzenji
2019 arXiv   pre-print
In this note, we first propose a cohomological derivation of the celebrated 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  ...

### A unification of two refinements of Euler's partition theorem

William Y. C. Chen, Henry Y. Gao, Kathy Q. Ji, Martin Y. X. Li
2010 The Ramanujan journal
We obtain a unification of two refinements of 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.  ...

### Euler's partition theorem and the combinatorics of ℓ-sequences

Carla D. Savage, Ae Ja Yee
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.  ...

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

Armin Straub
2016 European journal of combinatorics (Print)
This simple but curious analog of 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.  ...

### Euler's Partition Theorem with Upper Bounds on Multiplicities

William Chen, Ja Ae, Yee, Albert Zhu
2012 the electronic journal of combinatorics   unpublished
For m = 0, our result reduces to Bessenrodt's refinement of 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  .  ...

### Weighted forms of Euler's theorem

William Y.C. Chen, Kathy Q. Ji
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 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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