# A bijective proof and generalization of Siladić's Theorem

@article{Konan2020ABP,
title={A bijective proof and generalization of Siladi{\'c}'s Theorem},
author={Isaac Konan},
journal={Eur. J. Comb.},
year={2020},
volume={87},
pages={103101}
}
• Isaac Konan
• Published 31 May 2019
• Computer Science, Mathematics
• Eur. J. Comb.
In a recent paper, Dousse introduced a refinement of Siladic's theorem on partitions, where parts occur in two primary and three secondary colors. Her proof used the method of weighted words and $q$-difference equations. The purpose of this paper is to give a bijective proof of a generalization of Dousse's theorem from two primary colors to an arbitrary number of primary colors.
5 Citations

## Topics from this paper

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## References

SHOWING 1-10 OF 19 REFERENCES
An iterative-bijective approach to generalizations of Schur's theorem
• Mathematics, Computer Science
Eur. J. Comb.
• 2006
A bijective proof of Schur's theorem due to Alladi and Gordon is started and how a particular iteration of it leads to some very general theorems on colored partitions are described.
A variation on a theme of Sylvester - a smoother road to Göllnitz's (Big) theorem
• Computer Science, Mathematics
Discret. Math.
• 1999
A three-parameter refinement of the bijective proof of the number of partitions of a positive integer n into distinct odd parts is obtained and shown to be equivalent to a deep partition theorem of Gollnitz.
Siladi\'c's theorem: weighted words, refinement and companion
In a previous paper, the author gave a combinatorial proof and refinement of Siladi\'c's theorem, a Rogers-Ramanujan type partition identity arising from the study of Lie algebras. Here we use the
A new bijective proof of a partition theorem of K. Alladi
• Computer Science, Mathematics
Discret. Math.
• 2004
A different bijective proof of the theorem that the number of partitions of a positive integer n into distinct odd parts equals the numberOf partitions of n into parts not equal 2 and differing by greater than or equal to, 6 is given.
Generalizations of Schur's partition theorem
• Mathematics
• 1993
AbstractSchur's partition theorem states that the number of partitions ofn into distinct parts ≡ 1,2 (mod 3) is equal to the number of partitions ofn into parts with minimal difference 3 and no
A combinatorial proof of Schur's 1926 partition theorem
• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published
Twisted sl(3,C)˜-modules and combinatorial identities
The main result of this paper is a combinatorial description of a basis of standard level 1 module for the twisted affine Lie algebra A (2) 2 . This description also gives two new combinatorial
Annihilating ideals of standard modules of ${rm sl}(2, Bbb C)sp sim$ and combinatorial identities
• Mathematics
• 1987
Abstract Introduction Formal Laurent series and rational functions Generating fields The vertex operator algebra $N(k\Lambda_0)$ Modules over $N(k\Lambda_0)$ Relations on standard modules Colored
PRIMC, Annihilating ideals of standard modules of sl(2,C)∼ and combinatorial identities
• Mem. Amer. Math. Soc
• 1999