Let Snβ denote the set of permutations of [n]={1,2,β¦,n}. For a
positive integer k, define Sn,kβ to be the set of all permutations of
[n] with exactly k disjoint cycles, i.e., Sn,kβ={ΟβSnβ:Ο=c1βc2ββ―ckβ}, where c1β,c2β,β¦,ckβ are disjoint cycles.
The size of Sn,kβ is given by [nkβ]=(β1)nβks(n,k), where s(n,k) is the Stirling number of
the first kind. A family AβSn,kβ is said to be t-{\em
intersecting} if any two elements of A have at least t common
cycles. In this paper, we show that, given any positive integers k,t with
kβ₯t+1, there exists an integer n0β=n0β(k,t), such that for all nβ₯n0β, if AβSn,kβ is t-intersecting, then β£Aβ£β€[nβtkβtβ], with
equality if and only if A is the stabiliser of t fixed points.Comment: 8 page
Let N0β be the set of non-negative integers, and let P(n,l)
denote the set of all weak compositions of n with l parts, i.e., P(n,l)={(x1β,x2β,β¦,xlβ)βN0lβΒ :Β x1β+x2β+β―+xlβ=n}. For any
element u=(u1β,u2β,β¦,ulβ)βP(n,l), denote its ith-coordinate
by u(i), i.e., u(i)=uiβ. A family AβP(n,l) is said to be t-intersecting if β£{iΒ :Β u(i)=v(i)}β£β₯t for all u,vβA. A family
AβP(n,l) is said to be trivially t-intersecting if there
is a t-set T of {1,2,β¦,l} and elements ysββN0β (sβT) such that A={uβP(n,l)Β :Β u(j)=yjβΒ forallΒ jβT}. We prove that given any positive integers l,t with
lβ₯2t+3, there exists a constant n0β(l,t) depending only on l and t,
such that for all nβ₯n0β(l,t), if AβP(n,l) is
non-trivially t-intersecting then \begin{equation} \vert \mathcal{A}
\vert\leq {n+l-t-1 \choose l-t-1}-{n-1 \choose l-t-1}+t.\notag \end{equation}
Moreover, equality holds if and only if there is a t-set T of
{1,2,β¦,l} such that \begin{equation} \mathcal A=\bigcup_{s\in
\{1,2,\dots, l\}\setminus T} \mathcal A_s\cup \left\{ \mathbf q_i\ :\ i\in T
\right\},\notag \end{equation} where \begin{align} \mathcal{A}_s & =\{\mathbf
u\in P(n,l)\ :\ \mathbf u(j)=0\ {\rm for all}\ j\in T\ {\rm and}\ \mathbf
u(s)=0\}\notag \end{align} and qiββP(n,l) with qiβ(j)=0
for all jβ{1,2,β¦,l}β{i} and qiβ(i)=n.Comment: 19 page