Erdős–Ko–Rado theorems for permutations and set partitions

Let Sym ( [ n ] ) denote the collection of all permutations of [ n ] = { 1 , … , n } . Suppose A ⊆ Sym ( [ n ] ) is a family of permutations such that any two of its elements (when written in its cycle decomposition) have at least t cycles in common. We prove that for sufficiently large n, | A | ⩽ ( n − t ) ! with equality if and only if A is the stabilizer of t fixed points. Similarly, let B ( n ) denote the collection of all set partitions of [ n ] and suppose A ⊆ B ( n ) is a family of set partitions such that any two of its elements have at least t blocks in common. It is proved that, for sufficiently large n, | A | ⩽ B n − t with equality if and only if A consists of all set partitions with t fixed singletons, where B n is the nth Bell number.