I¹ve been calling an integer sequence X0, X1, X2,... ³gradual² when 0 <= Xn <= n. I was wondering, is a ³standard² (or better) name for this property? Obviously the gradual sequences are closed under min, max and so on. The motivating application was to define a convention for submitting orderings of the integers for the OEIS. Gradual sequences give a natural bijection: Xn is the index of n when you sort [0..n] according to the given ordering relation. Thus 0,1,2,... gives the normal < ordering, while 0,0,0,... is the > ordering (n Xn is the inverted ordering), and so on. (Given gradual X and Y, what ordering is X min Y?) This often enables interesting orderings like lexicographic, Sharkovsky etc to be encoded as a gradual sequence in a more suggestive way than just giving some initial terms in order. Gradual sequences also appear in other contexts, so I¹d like to adopt a standard (or better) name, if there is one. Thanks! --MLB