Abstract   We consider the problem of finding short strings that contain all permutations of order  k  over an alphabet of size  n , with   k  ⩽  n  . We show constructively that   k  (  n  −  2  )  +  3   is an upper bound on the length of shortest such strings, for   n  ⩾  k  ⩾  10  . Consequently, for   n  ⩾  10  , the shortest strings that contain all permutations of order  n  have length at most    n  2   −  2  n  +  3  . These two new upper bounds improve with one unit the previous known upper bounds.
['Eugen Zălinescu']
Shorter strings containing all k-element permutations