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Annals of Combinatorics

, Volume 23, Issue 2, pp 249–254 | Cite as

Partitions and the Minimal Excludant

  • George E. AndrewsEmail author
  • David Newman
Article
  • 59 Downloads

Abstract

Fraenkel and Peled have defined the minimal excludant or “\({{\,\mathrm{mex}\,}}\)” function on a set S of positive integers is the least positive integer not in S. For each integer partition \(\pi \), we define \({{\,\mathrm{mex}\,}}(\pi )\) to be the least positive integer that is not a part of \(\pi \). Define \(\sigma {{\,\mathrm{mex}\,}}(n)\) to be the sum of \({{\,\mathrm{mex}\,}}(\pi )\) taken over all partitions of n. It will be shown that \(\sigma {{\,\mathrm{mex}\,}}(n)\) is equal to the number of partitions of n into distinct parts with two colors. Finally the number of partitions \(\pi \) of n with \({{\,\mathrm{mex}\,}}(\pi )\) odd is almost always even.

Keywords

Minimal excludant MEX Partitions Two color partitions 

Mathematics Subject Classification

11A63 11P81 05A19 

Notes

References

  1. 1.
    Andrews, G.E.: The Theory of Partitions. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam (1976) (Reissued: Cambridge University Press, Cambridge (1998))Google Scholar
  2. 2.
    Andrews, G.E.: Concave compositions. Electron. J. Combin. 18(2), #P6 (2011)Google Scholar
  3. 3.
    Andrews, G.E., Newman, D.: The minimal excludant in integer partitions. (submitted)Google Scholar
  4. 4.
    Andrews, G.E., Merca, M.: The truncated pentagonal number theorem. J. Combin. Theory Ser. A 119(8), 1639–1643 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fraenkel, A.S., Peled, U.: Harnessing the unwieldy MEX function. In: Nowakowski, R.J.(ed.) Games of No Chance 4. Math. Sci. Res. Inst. Publ., 63, pp. 77–94. Cambridge Univ. Press, New York (2015)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.The Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Far RockawayNew YorkUSA

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