Linked partition ideals and Kanade–Russell conjectures

Chern, Shane; Li, Zhitai

doi:10.1016/j.disc.2020.111876

Cited by 17 publications

(20 citation statements)

References 8 publications

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“…Also, Kurşungöz carried out similar treatments for Capparelli's and Göllnitz-Gordon identities in [20]. Later Li and the author [9] proposed an algebraic method to reprove six Andrews-Gordon type generating function identities related to the Kanade-Russell conjectures and also to discover a number of new identities.…”

Section: Theorem 1 (Rogers-ramanujan Identities)mentioning

confidence: 99%

“…In September 2018, George Andrews communicated to Zhitai Li and the author a conjecture on the generating function for linked partition ideals, which was recorded in [9].…”

Section: Generating Function For Span One Linked Partition Idealsmentioning

confidence: 99%

“…By examining a number of examples in [9,17,19,20], it seems that in some cases the G k (x)'s in Theorem 4 are of a unified form of q-multi-summations. It motivates us to consider a matrix factorization problem involving column functional vectors of certain qmulti-summations.…”

Section: Generating Function For Span One Linked Partition Idealsmentioning

confidence: 99%

“…Further, the algebraic method in [9] of proving generating function identities such as (1) relies heavily on computer algebra (Mathematica packages qMultiSum [23] and qGeneratingFunctions [18]). Now we are able to present a new approach that may simplify the considerations related to computer assistance.…”

Section: Generating Function For Span One Linked Partition Idealsmentioning

confidence: 99%

“…It should be admitted that when a span one linked partition ideal is given, to derive an Andrews-Gordon type generating function identity, one still has to obtain first a conjectural (x, q) sum-side. This then requires an extensive search using the general shape given by Andrews' Conjecture 7; see Section 5 in the work of Li and the author [9] for some discussions.…”

Section: Generating Function For Span One Linked Partition Idealsmentioning

confidence: 99%

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Linked Partition Ideals, Directed Graphs and $q$-Multi-Summations

Chern

2020

Electron. J. Combin.

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In this paper, we start by considering generating function identities for linked partition ideals in the setting of basic graph theory. Then our attention is turned to $q$-difference systems, which eventually lead to a factorization problem of a special type of column functional vectors involving $q$-multi-summations. Using a recurrence relation satisfied by certain $q$-multi-summations, we are able to provide non-computer-assisted proofs of some Andrews--Gordon type generating function identities. These proofs also have an interesting connection with binary trees. Further, we give illustrations of constructing a linked partition ideal, or more loosely, a set of integer partitions whose generating function corresponds to a given set of special $q$-multi-summations.

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Section: Theorem 1 (Rogers-ramanujan Identities)mentioning

confidence: 99%

“…In September 2018, George Andrews communicated to Zhitai Li and the author a conjecture on the generating function for linked partition ideals, which was recorded in [9].…”

Section: Generating Function For Span One Linked Partition Idealsmentioning

confidence: 99%