In this note we obtain and discuss formulae for the number of partial one-one transformations (of an n-element set) of height (equivalently, width) r and having exactly k fixed points. Moreover, we consider a generalization of the 'Hat Problem' or 'Probleme des Rencontres'.
- Exponential generating function
- Full transformation
- Laguerre polynomial
- Ordinary generating function
- Partial one-one transformation
- Partial transformation
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