Annals of Combinatorics

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

Partitions and the Minimal Excludant

  • George E. AndrewsEmail author
  • David Newman


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.


Minimal excludant MEX Partitions Two color partitions 

Mathematics Subject Classification

11A63 11P81 05A19 



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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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