Cross-intersecting families of permutations

For positive integers r and n with r ⩽ n , let P r , n be the family of all sets { ( 1 , y 1 ) , ( 2 , y 2 ) , … , ( r , y r ) } such that y 1 , y 2 , … , y r are distinct elements of [ n ] = { 1 , 2 , … , n } . P n , n describes permutations of [ n ] . For r < n , P r , n describes permutations of r-element subsets of [ n ] . Families A 1 , A 2 , … , A k of sets are said to be cross-intersecting if, for any distinct i and j in [ k ] , any set in A i intersects any set in A j . For any r, n and k ⩾ 2 , we determine the cases in which the sum of sizes of cross-intersecting sub-families A 1 , A 2 , … , A k of P r , n is a maximum, hence solving a recent conjecture (suggested by the author).