 | ECS #27: Cycles set without fixed point |
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1. Description
permutations Sigma without fixed point such that Sigma^3=Id
2. Specification
This labelled structure is specified as S in
\displaystyle
\left\{ S={\rm Set} \left( {\rm Union} \left( {\rm Cycle} \left( Z,{
\rm card}=3 \right) \right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -{n}^{2}-3\,n-2 \right) f \left( n \right) +f \left( n
+3 \right) =0,f \left( 0 \right) =1,f \left( 1 \right) =0,f \left( 2
\right) =0 \right\}
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3.3. Closed form
\displaystyle
\cases{1/2\,{\frac {{3}^{2/3\,n+1/2}\Gamma \left( 1/3\,n+2/3 \right) \Gamma \left( 1/3\,n+1/3 \right) }{\pi }}&${\rm irem} \left( n,3 \right) =0$\cr 0&${\rm irem} \left( n-1,3 \right) =0$\cr 0&${\rm irem} \left( n-2,3 \right) =0$\cr}
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3.4. Asymptotics
4. Exponential generating function
\displaystyle
{{\rm e}^{1/3\,{x}^{3}}}
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5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).