Let S n denote the set of permutations of n = { 1 , 2 , ? , n } . For a positive integer k , define S n , k to be the set of all permutations of n with exactly k disjoint cycles, i.e.,? S n , k = { π ? S n : π = c 1 c 2 ? c k } , where c 1 , c 2 , ? , c k are disjoint cycles. The size of S n , k is given by n k = ( - 1 ) n - k s ( n , k ) , where s ( n , k ) is the Stirling number of the first kind. A family A ? S n , k is said to be t -cycle-intersecting if any two elements of A have at least t common cycles. A family A ? S n , k is said to be trivially t -cycle-intersecting if A is the stabiliser of t fixed points, i.e.,? A consists of all permutations in S n , k with some t fixed cycles of length one. For 1 ? j ? t , let Q j = { ? ? S n , k : ? ( i ) = i for?all i ? k ? { j } } . For t + 1 ? s ? k , let B s = { ? ? S n , k : ? ( i ) = i for?all i ? t ? { s } } . In this paper, we show that, given any positive integers k , t with k ? 2 t + 3 , there exists an n 0 = n 0 ( k , t ) , such that for all n ? n 0 , if A ? S n , k is non-trivially t -cycle-intersecting, then | A | ? | B | , where B = ? s = t + 1 k B s ? ? j = 1 t Q j . Furthermore, equality holds if and only if A is a conjugate of B , i.e.,? A = β - 1 B β for some β ? S n .
['Cheng Yeaw Ku', 'Kok Bin Wong']
A non-trivial intersection theorem for permutations with fixed number of cycles