 | ECS #32: Cycles Set |
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1. Description
- Permutation with exactly 3 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}=3
\right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -3\,n-1-3\,{n}^{2}-{n}^{3} \right) f \left( n+1
\right) + \left( 9\,n+7+3\,{n}^{2} \right) f \left( n+2 \right) +
\left( -3\,n-6 \right) f \left( n+3 \right) +f \left( n+4 \right) =0,f
\left( 0 \right) =0,f \left( 1 \right) =0,f \left( 2 \right) =0,f
\left( 3 \right) =1,f \left( 4 \right) =6 \right\}
other formats
3.3. Closed form
\displaystyle
\Gamma \left( n \right) \left( \gamma\,\Psi \left( n \right) -\gamma+
{\gamma}^{2}+\sum _{{\rm \_n0}=0}^{n-3}{\frac {\Psi \left( {\rm \_n0}+2
\right) }{{\rm \_n0}+2}} \right)
other formats
3.4. Asymptotics
4. Exponential generating function
\displaystyle
1/6\, \left( \ln \left( \left( 1-x \right) ^{-1} \right) \right) ^{3
}
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It satisfies the following differential equation of order 3:
\displaystyle
\left\{ \left( {\frac {d}{dx}}y \left( x \right) \right) \left( -1+
x \right) + \left( 3-6\,x+3\,{x}^{2} \right) {\frac {d^{2}}{d{x}^{2}}}y
\left( x \right) +1+ \left( -1+3\,x-3\,{x}^{2}+{x}^{3} \right) {\frac
{d^{3}}{d{x}^{3}}}y \left( x \right) =0,y \left( 0 \right) =0,\mbox {D}
\left( y \right) \left( 0 \right) =0, \left( D^{ \left( 2 \right) }
\right) \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).