Intersecting families of transformations

Abstract

Consider [Formula: see text] and [Formula: see text] to represent the full transformation semigroup and the symmetric group on the set [Formula: see text], respectively. A subset [Formula: see text] of [Formula: see text] is called an intersecting family if any two transformations [Formula: see text] coincide at some point [Formula: see text]. An intersecting family [Formula: see text] is called maximum if no other intersecting family [Formula: see text] exists with [Formula: see text]. 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 [Formula: see text] equals [Formula: see text]. Subsequent research demonstrated that a maximum intersecting family of [Formula: see text] 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 [Formula: see text], where [Formula: see text] denotes the image set of [Formula: see text]. This paper presents a formula for the cardinality of a maximum intersecting family of [Formula: see text]. It then characterizes the maximum intersecting families of [Formula: see text] and concludes by determining the number of maximum intersecting families of [Formula: see text].