Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Partition theorems for Euler pairs

Author: M. V. Subbarao
Journal: Proc. Amer. Math. Soc. 28 (1971), 330-336
MSC: Primary 10.48
MathSciNet review: 0274410
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper generalizes and extends the recent results of George Andrews on Euler pairs. If $ {S_1}$ and $ {S_2}$ are nonempty sets of natural numbers, we define $ ({S_1},{S_2})$ to be an Euler pair of order $ r$ whenever $ {q_r}({S_1};n) = p({S_2};n)$ for all natural numbers $ n$, where $ {q_r}({S_1};n)$ denotes the number of partitions of $ n$ into parts taken from $ {S_1}$, no part repeated more than $ r - 1$ times $ (r > 1)$, and $ p({S_2};n)$ the number of partitions of $ n$ into parts taken from $ {S_2}$. Using a method different from Andrews', we characterize all such pairs, and consider various applications as well as an extension to vector partitions.

References [Enhancements On Off] (What's this?)

  • [1] George E. Andrews, Two theorems of Euler and a general partition theorem, Proc. Amer. Math. Soc. 20 (1969), 499-502. MR 38 #2112. MR 0233791 (38:2112)
  • [2] M. S. Cheema, Vector partitions and combinatorial identities, Math. Comp. 18 (1964), 414-420. MR 29 #4697. MR 0167424 (29:4697)
  • [3] L. E. Dickson, Modern elementary theory of numbers, Univ. of Chicago Press, Chicago, Ill., 1939. MR 1, 65.
  • [4] H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. 225 (1967), 154-190. MR 35 #2848. MR 0211973 (35:2848)
  • [5] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, London, 1960. MR 20 #828.
  • [6] I. J. Schur, Zur additiven Zahlentheorie, S. B. Akad. Wiss. Berlin 1926, 488-495.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10.48

Retrieve articles in all journals with MSC: 10.48

Additional Information

Keywords: Partition, generating function, prime number, quadratic and higher power residues, vector partition
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society