Intersecting families of transformations

Consider Tn and Sn to represent the full transformation semigroup and the symmetric group on the set Xn = {1, 2, ... ,n}, respectively. A subset S of T-n is called an intersecting family if any two transformations alpha,beta is an element of S coincide at some point i is an element of Xn. An intersecting family S is called maximum if no other intersecting family S' exists with |S'| > |S|. Frankl and Deza [On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22(3) (1977) 352-360] established that the cardinality of a maximum intersecting family of Sn equals (n - 1)!. Subsequent research demonstrated that a maximum intersecting family of Sn forms a coset of a stabilizer of one point, as shown by Cameron and Ku [Intersecting families of permutations, European J. Combin. 24(7) (2003) 881-890] and independently by Larose and Malvenuto [Stable sets of maximal size in Kneser-type graphs, European J. Combin. 25(5) (2004) 657-673]. Define K(n,r) = {alpha is an element of T-n : |im(alpha)|<= r}, where im(alpha) denotes the image set of alpha. This paper presents a formula for the cardinality of a maximum intersecting family of K(n,r). It then characterizes the maximum intersecting families of K(n,r) and concludes by determining the number of maximum intersecting families of K(n,r).