On a conjecture of H. Gupta

TL;DR

This paper investigates a conjecture about the minimal length of sequences containing all permutations of a set as subsequences, confirming it for even n and providing bounds for odd n.

Contribution

The paper proves Gupta's conjecture for even n and establishes an upper bound for odd n, advancing understanding of permutation-containing sequences.

Findings

Confirmed Gupta's conjecture for even n

Established an upper bound for odd n

Provided new bounds on sequence length r(n)

Abstract

Denote by r(n) the length of a shortest integer sequence on a circle containing all permutations of the set {1,2,...,n} as subsequences. Hansraj Gupta conjectured in 1981 that r(n) <= n^2/2. In this paper we confirm the conjecture for the case where n is even, and show that r(n) < n^2/2 + n/4 -1 if n is odd.