A Bijective Proof of the Alladi-Andrews-Gordon Partition Theorem

Zhao, James J. Y.

doi:10.37236/4561

Cited by 2 publications

(3 citation statements)

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“…For any positive integer n, the number of partitions of n into distinct parts congruent to 2, 4, 5 mod 6 is equal to the number of partitions of n into parts different from 1 and 3, and where parts differ by at least six with equality only if parts are congruent to 2, 4, 5 mod 6. Like Schur's theorem, Göllnitz's identity can be proved using q-difference equations [5] and an elegant Bressoud-style bijection [8,10].…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…For example, with n = 49, the partitions of the first kind are (35, 10, 4), (34, 11, 4), (28, 11, 10), (23, 22, 4), (23, 16, 10), (22,16,11) and (16,11,10,8,4) and the partitions of the second kind are (35, 14), (34, 15), (33, 16), (45, 4), (39, 10), (38, 11) and (27, 18, 4) • Corollary 1.1 may be compared with Theorem 3 of [1], which is Theorem 1.5 transformed by (1.18) but with the dilation q → q 15 instead of q → q 12 . The paper is organized as follows.…”

Section: Statement Of Resultsmentioning

confidence: 99%

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Beyond Göllnitz' Theorem I: A Bijective Approach

Konan

2019

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In 2003, Alladi, Andrews and Berkovich proved an identity for partitions where parts occur in eleven colors: four primary colors, six secondary colors, and one quaternary color. Their work answered a longstanding question of how to go beyond a classical theorem of Göllnitz, which uses three primary and three secondary colors. Their main tool was a deep and difficult four parameter q-series identity. In this paper we take a different approach. Instead of adding an eleventh quaternary color, we introduce forbidden patterns and give a bijective proof of a ten-colored partition identity lying beyond Göllnitz' theorem. Using a second bijection, we show that our identity is equivalent to the identity of Alladi, Andrews, and Berkovich. From a combinatorial viewpoint, the use of forbidden patterns is more natural and leads to a simpler formulation. In fact, in Part II of this series we will show how our method can be used to go beyond Göllnitz' theorem to any number of primary colors.

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Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Section: Statement Of Resultsmentioning

confidence: 99%

Beyond Göllnitz' Theorem I: A Bijective Approach

Konan

2019

Preprint

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“…Example 1.4. For example, with n = 49, the partitions of the first kind are (35,10,4), (34,11,4), (28,11,10), (23,22,4), (23,16,10), (22,16,11) and (16,11,10,8,4) and the partitions of the second kind are (35,14), (34,15), (33,16), (45,4), (39, 10), (38,11) and (27, 18, 4) •…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Beyond Göllnitz' Theorem II: arbitrarily many primary colors

Konan¹

2019

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In 2003, Alladi, Andrews and Berkovich proved a four parameter partition identity lying beyond a celebrated identity of Göllnitz. Since then it has been an open problem to extend their work to five or more parameters. In part I of this pair of papers, we took a first step in this direction by giving a bijective proof of a reformulation of their result. We introduced forbidden patterns, bijectively proved a ten-colored partition identity, and then related, by another bijection, our identity to the Alladi-Andrews-Berkovich identity. In this second paper, we state and bijectively prove an n(n+1) 2 -colored partition identity beyond Göllnitz' theorem for any number n of primary colors, along with the full set of the n(n−1) 2 secondary colors as the product of two distinct primary colors, generalizing the identity proved in the first paper. Like the ten-colored partitions, our family of n(n+1) 2-colored partitions satisfy some simple minimal difference conditions while avoiding forbidden patterns. Furthermore, the n(n+1) 2-colored partitions have some remarkable properties, as they can be uniquely represented by oriented rooted forests which record the steps of the bijection.

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A Bijective Proof of the Alladi-Andrews-Gordon Partition Theorem

Abstract: Based on the combinatorial proof of Schur's partition theorem given by Bressoud, and the combinatorial proof of Alladi's partition theorem given by Padmavathamma, Raghavendra and Chandrashekara, we obtain a bijective proof of a partition theorem of Alladi, Andrews and Gordon.

Cited by 2 publications

References 16 publications

Beyond Göllnitz' Theorem I: A Bijective Approach

Beyond Göllnitz' Theorem I: A Bijective Approach

Beyond Göllnitz' Theorem II: arbitrarily many primary colors

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