Permutations with Arithmetic Constraints

Carl Pomerance ()
Additional contact information
Carl Pomerance: Dartmouth College, Mathematics Department

A chapter in Number Theory in Memory of Eduard Wirsing, 2023, pp 285-298 from Springer

Abstract: Abstract Let S lcm ( n ) $$S_{\mathrm {lcm}}(n)$$ denote the set of permutations π $$\pi $$ of [ n ] = { 1 , 2 , … , n } $$[n]=\{1,2,\dots ,n\}$$ such that lcm [ j , π ( j ) ] ≤ n $$\mathrm {lcm}[j,\pi (j)]\le n$$ for each j ∈ [ n ] $$j\in [n]$$ . Further, let S div ( n ) $$S_{\mathrm {div}}(n)$$ denote the number of permutations π $$\pi $$ of [ n ] $$[n]$$ such that j ∣ π ( j ) $$j\mid \pi (j)$$ or π ( j ) ∣ j $$\pi (j)\mid j$$ for each j ∈ [ n ] $$j\in [n]$$ . Clearly S div ( n ) ⊂ S lcm ( n ) $$S_{\mathrm {div}}(n)\subset S_{\mathrm {lcm}}(n)$$ . We get upper and lower bounds for the counts of these sets, showing they grow geometrically. We also prove a conjecture from a recent paper on the number of “anti-coprime” permutations of [ n ] $$[n]$$ , meaning that each gcd ( j , π ( j ) ) > 1 $$\gcd (j,\pi (j))>1$$ except when j = 1 $$j=1$$ .

Keywords: Permutations; Arithmetic constraints (search for similar items in EconPapers)
Date: 2023
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

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:spr:sprchp:978-3-031-31617-3_19

Ordering information: This item can be ordered from
http://www.springer.com/9783031316173

DOI: 10.1007/978-3-031-31617-3_19

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().