Turkish Journal of Mathematics
Abstract
Let O_n and C_n be the semigroup of all order-preserving transformations and of all order-preserving and order-decreasing transformations on the finite set X_n={1,2,\ldots ,n}, respectively. Let \fix (\alpha )={x\in X_n:x\alpha =x} for any transformation \alpha. In this paper, for any Y\subseteq X_n, we find the cardinalities of the sets O_{n,Y}={\alpha\in O_n:\fix (\alpha)=Y} and C_{n,Y}={\alpha\in C_n: \fix (\alpha )=Y}. Moreover, we find the numbers of transformations of O_n and C_n with r fixed points.
DOI
10.3906/mat-1010-432
Keywords
Order-preserving transformations, order-decreasing transformations, nilpotent, Catalan number
First Page
617
Last Page
625
Recommended Citation
AYIK, G, AYIK, H, & KOÇ, M (2011). Combinatorial results for order-preserving and order-decreasing transformations. Turkish Journal of Mathematics 35 (4): 617-625. https://doi.org/10.3906/mat-1010-432
