Combinatorial results for order-preserving and order-decreasing transformations

Authors : Gonca AYIK

Pages : 617-625

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Publication Date : 0000-00-00

Article Type : Research Paper

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 insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(a);=Y} and Cn,Y={a\in Cn: \fix insert ignore into journalissuearticles values(a );=Y}. Moreover, we find the numbers of transformations of On and Cn with r fixed points.
Keywords : Order preserving transformations, order decreasing transformations, nilpotent, Catalan number