The number of orientation-preserving full transformations on n points with m fixed points equals binom(2n, n-m) for 2 ≤ m ≤ n.
Umar, Some combinatorial problems in the theory of partial transfor- mation semigroups,Algebra Discrete Math.17(2014) 110–134
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Fixed points of orientation-preserving full transformation
The number of orientation-preserving full transformations on n points with m fixed points equals binom(2n, n-m) for 2 ≤ m ≤ n.