Plane Partitions and a Problem of Josephus - Mircea Merca (Mathematics, 2023)
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 kth person will be executed going around the circle until only one remains, find the position <semantics>L(n,k)</semantics> in which you should stand in order to be the last survivor. Let <semantics>Jn</semantics> be the order in which the first person is executed on counting when <semantics>k=2</semantics>. In this paper, we consider the sequence <semantics>(Jn)n⩾1</semantics> 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 <semantics>Jn</semantics>. 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 <semantics>Jn</semantics> are provided in this context.