On flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we fi...

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Bibliographic Details
Published in:European journal of combinatorics 2012-05, Vol.33 (4), p.663-678
Main Author: Liu, Xiao-Chuan
Format: Article
Language:English
Subjects:
Combinatorial analysis
Flushing
Partitions
Proving
Repetition
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Summary:In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews’ partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order f(q), ϕ(q) and ψ(q). An identity of Ramanujan is proved combinatorially. Several new identities are also established. ► We give a combinatorial proof of a Sylvester’s problem on flushed partitions. ► We give combinatorial proofs for two generating functions. ► We define an involution and apply it to prove several different identities. ► We prove combinatorially an identity of Ramanujan.
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2011.12.004