Abstract
Let On be the semigroup of all order-preserving full transformations of a finite chain, say Xn = {1, 2, ..., n}, and for a given full transformation α: Xn → Xn let f(α) = |{x Xn: xα = x}|. In this note we obtain and discuss formulae for f(n,r,k) = |{α → On: f(α) = r ∧ max(Im α) = k}| and J(n,r,k) = |{α → On: |Im α| = r ∧ max(Im α) = k}|. We also obtain similar results for E(On), the set of idempotents of On.
Original language | English |
---|---|
Pages (from-to) | 51-62 |
Number of pages | 12 |
Journal | Semigroup Forum |
Volume | 72 |
Issue number | 1 |
DOIs | |
Publication status | Published - Feb 2006 |
Externally published | Yes |
Keywords
- Catalan number
- Fibonacci number
- Full transformation
- Idempotent
- Nilpotent
- Order-decreasing
- Order-preserving
- Partial transformation
- Semigroup
ASJC Scopus subject areas
- Algebra and Number Theory