 | ECS #33: Cycles Set |
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1. Description
- Permutation with exactly 4 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}=4
\right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( 4\,n+1+6\,{n}^{2}+4\,{n}^{3}+{n}^{4} \right) f \left(
n+1 \right) + \left( -18\,{n}^{2}-4\,{n}^{3}-15-28\,n \right) f \left(
n+2 \right) + \left( 25+6\,{n}^{2}+24\,n \right) f \left( n+3 \right) +
\left( -4\,n-10 \right) f \left( n+4 \right) +f \left( n+5 \right) =0,
f \left( 0 \right) =0,f \left( 1 \right) =0,f \left( 2 \right) =0,f
\left( 3 \right) =0,f \left( 4 \right) =1,f \left( 5 \right) =10
\right\}
other formats
3.3. Asymptotics
4. Exponential generating function
\displaystyle
1/24\, \left( \ln \left( \left( 1-x \right) ^{-1} \right) \right) ^{
4}
other formats
It satisfies the following differential equation of order 4:
\displaystyle
\left\{ \left( {\frac {d}{dx}}y \left( x \right) \right) \left( -1+
x \right) + \left( 7-14\,x+7\,{x}^{2} \right) {\frac {d^{2}}{d{x}^{2}}}
y \left( x \right) + \left( -6+18\,x-18\,{x}^{2}+6\,{x}^{3} \right) {
\frac {d^{3}}{d{x}^{3}}}y \left( x \right) -1+ \left( 1-4\,x+6\,{x}^{2}
-4\,{x}^{3}+{x}^{4} \right) {\frac {d^{4}}{d{x}^{4}}}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, \left( D^{ \left( 3 \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).