Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

FIXED-WIDTH PARTITIONS ACCORDING TO THE PARITY OF THE EVEN PARTS

  • John M. Campbell (Department of Mathematics and Statistics York University)
  • Received : 2022.07.02
  • Accepted : 2023.01.26
  • Published : 2023.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

A celebrated result in the study of integer partitions is the identity due to Lehmer whereby the number of partitions of n with an even number of even parts minus the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct odd parts. Inspired by Lehmer's identity, we prove explicit formulas for evaluating generating functions for sequences that enumerate integer partitions of fixed width with an even/odd number of even parts. We introduce a technique for decomposing the even entries of a partition in such a way so as to evaluate, using a finite sum over q-binomial coefficients, the generating function for the sequence of partitions with an even number of even parts of fixed, odd width, and similarly for the other families of fixed-width partitions that we introduce.

Keywords

Acknowledgement

The author wants to thank George Beck for some useful feedback.

References

  1. R. B. J. T. Allenby and A. Slomson, How to Count: An Introduction to Combinatorics, Second Edition, Taylor & Francis, New York, 2011. 
  2. M. Archibald, A. Blecher, C. Brennan, A. Knopfmacher, and T. Mansour, Partitions according to multiplicities and part sizes, Australas. J. Combin. 66 (2016), 104-119. 
  3. A. D. Christopher, Partitions with fixed number of sizes, J. Integer Seq. 18 (2015), no. 11, Article 15.11.5, 14 pp. 
  4. J. E. Friedman, J. T. Joichi, and D. W. Stanton, More monotonicity theorems for partitions, Experiment. Math. 3 (1994), no. 1, 31-37. http://projecteuclid.org/euclid.em/1062621001  1062621001
  5. H. Gupta, Combinatorial proof of a theorem on partitions into an even or odd number of parts, J. Combinatorial Theory Ser. A 21 (1976), no. 1, 100-103. https://doi.org/10.1016/0097-3165(76)90051-0 
  6. S. Kohl, Classifying finite simple groups with respect to the number of orbits under the action of the automorphism group, Comm. Algebra 32 (2004), no. 12, 4785-4794. https://doi.org/10.1081/AGB-200039279