Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.5

Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations

A. Umar
Department of Mathematics and Statistics
Sultan Qaboos University
Al-Khod, PC 123
Sultanate of Oman


Let $ X_n = \{1, 2, \ldots , n\}$. On a partial transformation $ \alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow$   Im $ \alpha
\subseteq X_n$ of $ X_n$ the following parameters are defined: the breadth or width of $ \alpha$ is $ \mid {\rm Dom}\ \alpha\mid$, the height of $ \alpha$ is $ \mid$   Im $ \alpha\mid$, and the right (resp., left) waist of $ \alpha$ is $ \max($Im $ \alpha)$ (resp., $ \min($Im $ \alpha)$). We compute the cardinalities of some equivalences defined by equalities of these parameters on $ {\cal
OP}_n$, the semigroup of orientation-preserving full transformations of $ X_n$, $ {\cal POP}_n$ the semigroup of orientation-preserving partial transformations of $ X_n$, $ {\cal OR}_n$ the semigroup of orientation-preserving/reversing full transformations of $ X_n$, and $ {\cal POR}_n$ the semigroup of orientation-preserving/reversing partial transformations of $ X_n$, and their partial one-to-one analogue semigroups, $ {\cal POPI}_n$ and $ {\cal PORI}_n$.

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(Concerned with sequences A000225 A000290 A002061 A066524 A163102.)

Received September 19 2010; revised version received July 9 2011. Published in Journal of Integer Sequences, September 5 2011.

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