On a conjecture of H. Gupta

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.