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Plane Partitions and a Problem of Josephus

Author

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  • Mircea Merca

    (Department of Mathematical Methods and Models, Fundamental Sciences Applied in Engineering Research Center, National University of Science and Technology Politehnica Bucharest, RO-060042 Bucharest, Romania
    Academy of Romanian Scientists, RO-050044 Bucharest, Romania)

Abstract

The Josephus Problem is a mathematical counting-out problem with a grim description: given a group of n persons arranged in a circle under the edict that every k th person will be executed going around the circle until only one remains, find the position L ( n , k ) in which you should stand in order to be the last survivor. Let J n be the order in which the first person is executed on counting when k = 2 . In this paper, we consider the sequence ( J n ) n ⩾ 1 in order to introduce new expressions for the generating functions of the number of strict plane partitions and the number of symmetric plane partitions. This approach allows us to express the number of strict plane partitions of n and the number of symmetric plane partitions of n as sums over partitions of n in terms of binomial coefficients involving J n . Also, we introduce interpretations for the strict plane partitions and the symmetric plane partitions in terms of colored partitions. Connections between the sum of the divisors’ functions and J n are provided in this context.

Suggested Citation

  • Mircea Merca, 2023. "Plane Partitions and a Problem of Josephus," Mathematics, MDPI, vol. 11(24), pages 1-15, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:24:p:4996-:d:1302397
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