5865

Andrews, G. E.; Stenger, Allen

doi:10.2307/2318562

Cited by 4 publications

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“…which turns out to be related to a problem proposed by Andrews [3] in 1972, see also, Andrews [4, pp. 156-157].…”

Section: Connection To An Identity Of Andrewsmentioning

confidence: 80%

“…As the third application of our involution, we give a combinatorial proof of a numbertheoretic theorem on partitions into distinct parts with smallest part being even derived from Andrews' identity (1.6). Moreover, we note that a special case of this partition theorem is related to an identity of Andrews, first proposed as a problem in [3]. A generating function proof was given by Stenger [9].…”

Section: Introductionmentioning

confidence: 97%

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A Franklin type involution for squares

Chen

Liu

2012

Advances in Applied Mathematics

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We find an involution as a combinatorial proof of a Ramanujan's partial theta identity. Based on this involution, we obtain a Franklin type involution for squares in the sense that the classical Franklin involution provides a combinatorial interpretation of Euler's pentagonal number theorem. This Franklin type involution can be considered as a solution to a problem proposed by Pak concerning the parity of the number of partitions of n into distinct parts with the smallest part being odd. Using a weighted form of our involution, we give a combinatorial proof of a weighted partition theorem derived by Alladi from Ramanujan's partial theta identity. This answers a question of Berndt, Kim and Yee. Furthermore, through a different weight assignment, we find combinatorial interpretations for another partition theorem derived by Alladi from a partial theta identity of Andrews. Moreover, we obtain a partition theorem based on Andrews' identity and provide a combinatorial proof by certain weight assignment for our involution. A specialization of our partition theorem is relate to an identity of Andrews concerning partitions into distinct nonnegative parts with the smallest part being even. Finally, we give a more general form of our partition theorem which in return corresponds to a generalization of Andrews' identity.

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“…which turns out to be related to a problem proposed by Andrews [3] in 1972, see also, Andrews [4, pp. 156-157].…”

Section: Connection To An Identity Of Andrewsmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 97%

A Franklin type involution for squares

Chen

Liu

2012

Advances in Applied Mathematics

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A note on “failures‐before‐first‐success run size” in life testing

Goel

1975

Naval Research Logistics

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Partition-Theoretic Formulas for Arithmetic Densities

Ono

Schneider

Wagner

2017

Springer Proceedings in Mathematics &Amp; Statistics

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In celebration of Krishnaswami Alladi's 60th birthday.Abstract If gcd(r, t) = 1, then Alladi proved the Möbius sum identityHere p min (n) is the smallest prime divisor of n. The right-hand side represents the proportion of primes in a fixed arithmetic progression modulo t. Locus generalized this to Chebotarev densities for Galois extensions. Answering a question of Alladi, we obtain analogs of these results to arithmetic densities of subsets of positive integers using q-series and integer partitions. For suitable subsets S of the positive integers with density d S , we prove thatwhere the sum is taken over integer partitions λ, µ P (λ) is a partition-theoretic Möbius function, |λ| is the size of partition λ, and sm(λ) is the smallest part of λ. In particular, we obtain partition-theoretic formulas for even powers of π when considering power-free integers.

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5865

Cited by 4 publications

References 0 publications

A Franklin type involution for squares

A Franklin type involution for squares

A note on “failures‐before‐first‐success run size” in life testing

Partition-Theoretic Formulas for Arithmetic Densities

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