On Disjoint Cross Intersecting Families of Permutations

For the positive integers r and n satisfying r ≤ n , let P r,n be the family of partial permutations {{ (1 , x 1 ) , (2 , x 2 ) , . . . , ( r, x r ) } : x 1 , x 2 , . . . , x r are different elements of { 1 , 2 , . . . , n }} . The subfamilies A 1 , A 2 , . . . , A k of P r,n are called cross intersecting if A ∩ B (cid:54) = ∅ for all A ∈ A i and B ∈ A j , where 1 ≤ i (cid:54) = j ≤ k . Also, if A 1 , A 2 , . . . , A k are mutually disjoint, then they are called disjoint cross intersecting subfamilies of P r,n . For the disjoint cross intersecting subfamilies A 1 , A 2 , . . . , A k of P n,n , it follows from the AM-GM inequality that (cid:81) ki =1 |A i | ≤ ( n ! /k ) k . In this paper, we present two proofs of the following statement: (cid:81) ki =1 |A i | = ( n ! /k ) k if and only if n = 3 and k = 2 . permutations; intersecting families; Erd˝os-Ko-Rado Theorem.