 | ECS #31: Cycles Set |
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1. Description
- Permutation with exactly two cycles
- Stirling numbers of first kind
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ S={\rm Set} \left( {\rm Cycle} \left( Z \right) ,{\rm card}=2
\right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( 2\,n+1+{n}^{2} \right) f \left( n+1 \right) + \left( -
2\,n-3 \right) f \left( n+2 \right) +f \left( n+3 \right) =0,f \left( 0
\right) =0,f \left( 1 \right) =0,f \left( 2 \right) =1,f \left( 3
\right) =3 \right\}
other formats
3.3. Closed form
\displaystyle
\Gamma \left( n \right) \left( \Psi \left( n \right) +\gamma \right)
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
1/2\, \left( \ln \left( \left( 1-x \right) ^{-1} \right) \right) ^{2
}
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It satisfies the following differential equation of order 2:
\displaystyle
\left\{ \left( {\frac {d}{dx}}y \left( x \right) \right) \left( -1+
x \right) -1+ \left( 1-2\,x+{x}^{2} \right) {\frac {d^{2}}{d{x}^{2}}}y
\left( x \right) =0,y \left( 0 \right) =0,\mbox {D} \left( y \right)
\left( 0 \right) =0 \right\}
other formats
5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).