Plane Partitions and a Problem of Josephus

Mircea Merca ()
Additional contact information
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

Mathematics, 2023, vol. 11, issue 24, 1-15

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.

Keywords: Josephus problem; partitions; plane partitions; divisors; binomial coefficients (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/11/24/4996/pdf (application/pdf)
https://www.mdpi.com/2227-7390/11/24/4996/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:11:y:2023:i:24:p:4996-:d:1302397

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().