Journals / Turkish Journal of Mathematics / 2011 / Cilt: 35 - Sayı: 4
Combinatorial results for order-preserving and order-decreasing transformations
- Pages
- 617–625
- DOI
- —
Abstract
Let On and Cn be the semigroup of all order-preserving transformations and of all order-preserving and order-decreasing transformations on the finite set Xn={1,2,\ldots ,n}, respectively. Let \fix (a )={x\in Xn:xa =x} for any transformation a. In this paper, for any Y\subseteq Xn, we find the cardinalities of the sets On,Y={a\in On:\fix (a)=Y} and Cn,Y={a\in Cn: \fix (a )=Y}. Moreover, we find the numbers of transformations of On and Cn with r fixed points.
Özet
Let On and Cn be the semigroup of all order-preserving transformations and of all order-preserving and order-decreasing transformations on the finite set Xn={1,2,\ldots ,n}, respectively. Let \fix (a )={x\in Xn:xa =x} for any transformation a. In this paper, for any Y\subseteq Xn, we find the cardinalities of the sets On,Y={a\in On:\fix (a)=Y} and Cn,Y={a\in Cn: \fix (a )=Y}. Moreover, we find the numbers of transformations of On and Cn with r fixed points.