On disjoint cross intersecting families of permutations

For the positive integers r and n satisfying r ≤ n, let Pr,n be the family of partial permutations {{(1, x1),(2, x2), . . . ,(r, xr)} : x1, x2, . . . , xr are different elements of {1, 2, . . . , n}}. The subfamilies A1, A2, . . . , Ak of Pr,n are called cross intersecting if A ∩ B 6= ∅ for all A ∈ Ai and B ∈ Aj , where 1 ≤ i 6= j ≤ k. Also, if A1, A2, . . . , Ak are mutually disjoint, then they are called disjoint cross intersecting subfamilies of Pr,n. For the disjoint cross intersecting subfamilies A1, A2, . . . , Ak of Pn,n, it follows from the AM-GM inequality that Qk i=1 |Ai| ≤ (n!/k) k . In this paper, we present two proofs of the following statement: Qk i=1 |Ai| = (n!/k) k if and only if n = 3 and k = 2.