Hayrullah AYIK, Gonca AYIK, Metin KOÇ

81122

Combinatorial results for order-preserving and order-decreasing transformations

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.

Anahtar Kelimeler:

Order-preserving transformations, order-decreasing transformations, nilpotent, Catalan number

Combinatorial results for order-preserving and order-decreasing transformations

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.

Keywords:

Order-preserving transformations, order-decreasing transformations, nilpotent, Catalan number,

Turkish Journal of Mathematics-Cover

ISSN: 1300-0098
Yayın Aralığı: 6
Yayıncı: TÜBİTAK

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