Combinatorial results for semigroups of order-preserving or order-reversing subpermutations
Abdallah Laradji, Abdullahi Umar · The Journal of Difference Equations and Applications · 2015
Let Xn={1,2,…,n}. For a partial one–one transformation (or subpermutation) α of Xn, the following parameters are defined: the heighth(α)=|Im α|, the waistw(α)= max(Im α), and the fixf(α)=|{x∈Xn:α(x)=x}|. We compute the cardinalities of some equivalence classes defined by equalities of these parameters on ℐ𝒪n and 𝒫𝒪𝒟ℐn, the semigroups of order-preserving and of order-preserving or order-reversing subpermutations of Xn, respectively. As a consequence, we obtain several formulae and generating functions for the number of nilpotents in ℐ𝒪n and 𝒫𝒪𝒟ℐn. We also prove that, for large n, a randomly chosen order-preserving (resp. order-reversing) subpermutation of Xn has probability 8/n (resp. 96/n) of being nilpotent.