Schmidt-type theorems via weighted partition identities

A 1999 theorem of F. Schmidt states that the number of partitions into distinct parts such that the odd indexed parts sum to n , is equal to the number of partitions of n . Recently, using MacMahon’s partition analysis, Andrews and Paule established two further theorems of the Schmidt-type. Here we...

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Bibliographic Details
Published in:The Ramanujan journal 2023-06, Vol.61 (2), p.701-714
Main Author: Alladi, Krishnaswami
Format: Article
Language:English
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Theorems
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Summary:A 1999 theorem of F. Schmidt states that the number of partitions into distinct parts such that the odd indexed parts sum to n , is equal to the number of partitions of n . Recently, using MacMahon’s partition analysis, Andrews and Paule established two further theorems of the Schmidt-type. Here we show that Schmidt’s 1999 theorem is equivalent to a weighted partition identity involving Rogers–Ramanujan partitions that I established in 1997. Using the weighted partition approach, we shall also establish combinatorially the two recent Schmidt-type theorems of Andrews–Paule. We conclude by proving another Schmidt-type theorem using weighted partitions.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-022-00655-4