A NEW REFINEMENT OF FINE’S PARTITION THEOREM

2021 ◽  
pp. 1-9
Author(s):  
JIAYU KANG ◽  
RUNQIAO LI ◽  
ANDREW Y. Z. WANG
Keyword(s):  
Partition Identities ◽  
Partition Theorem

Abstract We find a new refinement of Fine’s partition theorem on partitions into distinct parts with the minimum part odd. As a consequence, we obtain two companion partition identities. Both analytic and combinatorial proofs are provided.

scholarly journals Bijective Proofs of Partition Identities of MacMahon, Andrews, and Subbarao

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Shishuo Fu ◽  
James Sellers
Keyword(s):  
Partition Identities ◽  
Partition Theorem ◽  
Bijective Proof ◽  
International Audience

International audience We revisit a classic partition theorem due to MacMahon that relates partitions with all parts repeated at least once and partitions with parts congruent to $2,3,4,6 \pmod{6}$, together with a generalization by Andrews and two others by Subbarao. Then we develop a unified bijective proof for all four theorems involved, and obtain a natural further generalization as a result. Nous revisitons un théorème de partitions d'entiers dû à MacMahon, qui relie les partitions dont chaque part est répétée au moins une fois et celles dont les parts sont congrues à $2, 3, 4, 6 \pmod{6}$, ainsi qu'une généralisation par Andrews et deux autres par Subbarao. Ensuite nous construisons unepreuve bijective unifiée pour tous les quatre théorèmes ci-dessus, et obtenons de plus une généralisation naturelle.

scholarly journals Bijective Proofs of Partition Identities of MacMahon, Andrews, and Subbarao

10.37236/3907 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Shishuo Fu ◽  
James Allen Sellers
Keyword(s):  
Partition Identities ◽  
Partition Theorem ◽  
Bijective Proof

We revisit a classic partition theorem due to MacMahon that relates partitions with all parts repeated at least once and partitions with parts congruent to $2,3,4,6 \pmod 6$, together with a generalization by Andrews and two others by Subbarao. Then we develop a unified bijective proof for all four theorems involved, and obtain a natural further generalization as a result.

scholarly journals Proofs of some partition identities conjectured by Kanade and Russell

2021 ◽  
Author(s):  
Hjalmar Rosengren
Keyword(s):  
Lie Algebra ◽  
Partition Identities ◽  
Partition Identity ◽  
Affine Lie Algebra ◽  
Level 2

AbstractKanade and Russell conjectured several Rogers–Ramanujan-type partition identities, some of which are related to level 2 characters of the affine Lie algebra $$A_9^{(2)}$$ A 9 ( 2 ) . Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey–Wilson and Rogers polynomials. We also obtain some related results, including a partition identity conjectured by Capparelli and first proved by Andrews.

scholarly journals New proof to Somos’s Dedekind eta-function identities of level 10

2021 ◽  
Author(s):  
B. R. Srivatsa Kumar ◽  
Shruthi
Keyword(s):  
Dedekind Eta Function ◽  
Partition Identities

AbstractMichael Somos used PARI/GP script to generate several Dedekind eta-function identities by using computer. In the present work, we prove two new Dedekind eta-function identities of level 10 discovered by Somos in two different methods. Also during this process, we give an alternate method to Somos’s Dedekind eta-function identities of level 10 proved by B. R. Srivatsa Kumar and D. Anu Radha. As an application of this, we establish colored partition identities.

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

2008 ◽  
Vol 115 (6) ◽  
pp. 967-996 ◽  
Author(s):  
Carla D. Savage ◽  
Ae Ja Yee
Keyword(s):  
Partition Theorem

scholarly journals Partition identities II. The results of Bateman and Erdős

2006 ◽  
Vol 117 (1) ◽  
pp. 160-190
Author(s):  
Jason P. Bell ◽  
Stanley N. Burris
Keyword(s):  
Partition Identities
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