# Combinatorial Proofs of Two Euler-Type Identities Due to Andrews

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## Abstract

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, \(c_1(n)=b_1(n)\).

## Keywords

Partitions Euler’s identity Bit strings Overpartitions## Mathematics Subject Classification

05A17 11P81 11P83## Notes

## References

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