 | ECS #21: derangements |
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1. Description
- Cycles >=2, permutations of n element with no fixed points
- derangements (subfactorial or rencontre numbers)
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ S={\rm Set} \left( {\rm Cycle} \left( Z,2\leq {\rm card}
\right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -n-1 \right) f \left( n \right) + \left( -n-1 \right)
f \left( n+1 \right) +f \left( n+2 \right) =0,f \left( 0 \right) =1,f
\left( 1 \right) =0 \right\}
other formats
3.3. Closed form
\displaystyle
{{\rm e}^{-1}}\Gamma \left( n+1,-1 \right)
other formats
3.4. Asymptotics
4. Exponential generating function
\displaystyle
{{\rm e}^{\ln \left( \left( 1-x \right) ^{-1} \right) -x}}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).