Combinatorial results of collapse for order-preserving and order-decreasing transformations

The full transformation semigroup TnTn is defined to consist of all functions from Xn={1,…,n}Xn={1,…,n} itself, under the operation composition. In \cite{JMH1}, for any αα in TnTn, Howie and denoted collapse by c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}. Let OnOn be order-preserving transformations CnCn decreasing on XnXn=under its natural order, respectively. E(On)E(On) set idempotent elements OnOn, E(Cn)E(Cn) N(Cn)N(Cn) sets nilpotent CnCn, UU one {Cn,N(Cn),E(Cn),On,E(On)}{Cn,N(Cn),E(Cn),On,E(On)}. For α∈Uα∈U, we consider \imc(α)={t∈\im(α):|tα−1|≥2}\imc(α)={t∈\im(α):|tα−1|≥2}. positive integers 2≤k≤r≤n2≤k≤r≤n, define U(k)={α∈U:t∈\imc(α) |tα−1|=k},U(k,r)={α∈U(k):∣∣⋃t∈\imc(α)tα−1|=r}.U(k)={α∈U:t∈\imc(α) |tα−1|=k},U(k,r)={α∈U(k):|⋃t∈\imc(α)tα−1|=r}. main objective this paper determine |U(k,r)||U(k,r)|, so |U(k)||U(k)| some values rr kk.