5,171 research outputs found
Variations on a result of Bressoud
The well-known Rogers-Ramanujan identities have been a rich source of mathematical study over the last fifty years. In particular, Gordonâs generalization in the early 1960s led to additional work by Andrews and Bressoud in subsequent years. Unfortunately, these results lacked a certain amount of uniformity in terms of combinatorial interpretation. In this work, we provide a single combinatorial interpretation of the series sides of
these generating function results by using the concept of cluster parities. This unifies the aforementioned results of Andrews and Bressoud and also allows for a strikingly broader family of qâseries results to be obtained. We close the paper by proving congruences for a âdegenerate caseâ of Bressoudâs theorem
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 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 , and . An identity of Ramanujan is proved
combinatorially. Several new identities are also established.Comment: 19 page
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