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