Combinatorial Proofs of Two Euler-Type Identities Due to Andrews
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Let a(n) be the number of partitions of n, such that the set of even parts has exactly one element, b(n) be the difference between the number of parts in all odd partitions of n and the number of parts in all distinct partitions of n, and c(n) be the number of partitions of n in which exactly one part is repeated. Beck conjectured that \(a(n)=b(n)\) and Andrews, using generating functions, proved that \(a(n)=b(n)=c(n)\). We give a combinatorial proof of Andrews’ result. Our proof relies on bijections between a set and a multiset, where the partitions in the multiset are decorated with bit strings. We prove combinatorially Beck’s second conjecture, which was also proved by Andrews using generating functions. Let \(c_1(n)\) be the number of partitions of n, such that there is exactly one part occurring three times, while all other parts occur only once and let \(b_1(n)\) be the difference between the total number of parts in the partitions of n into distinct parts and the total number of different parts in the partitions of n into odd parts. Then, \(c_1(n)=b_1(n)\).
KeywordsPartitions Euler’s identity Bit strings Overpartitions
Mathematics Subject Classification05A17 11P81 11P83
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