Combinatorial results for semigroups of orientation-preserving partial transformations

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Let Xn = {1, 2,..., n}. On a partial transformation α: Dom α ⊆ Xn! Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is {pipe} Dom α {pipe}, the height of α is {pipe} Im α {pipe}, and the right (resp., left) waist of α is max(Im α) (resp., min(Im α)). We compute the cardinalities of some equivalences defined by equalities of these parameters on OPn, the semigroup of orientation-preserving full transformations of Xn, POPn the semigroup of orientation-preserving partial transformations of Xn, ORn the semigroup of orientation-preserving/reversing full transformations of Xn, and PORn the semigroup of orientation-preserving/reversing partial transformations of Xn, and their partial one-to-one analogue semigroups, POPIn and PORIn.

Original languageBritish English
JournalJournal of Integer Sequences
Issue number7
StatePublished - 2011


  • Anti-cyclic sequence
  • Breadth
  • Cyclic sequence
  • Full transformation
  • Height
  • Left waist
  • Orientation-preserving transformation
  • Orientation-reversing transformation
  • Partial one-to-one transformation
  • Partial transformation
  • Right waist


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