 | ECS #29: Cycles set without fixed point |
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1. Description
permutations Sigma without fixed point such that Sigma^5=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}=5 \right) \right) \right) \right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ \left( -{n}^{4}-10\,{n}^{3}-35\,{n}^{2}-50\,n-24 \right) f
\left( n \right) +f \left( n+5 \right) =0,f \left( 0 \right) =1,f
\left( 1 \right) =0,f \left( 2 \right) =0,f \left( 3 \right) =0,f
\left( 4 \right) =0 \right\}
other formats
3.3. Closed form
\displaystyle
\cases{1/2\,{\frac {{5}^{4/5\,n}\Gamma \left( 1/5\,n+1/5 \right) \Gamma \left( 1/5\,n+2/5 \right) \Gamma \left( 1/5\,n+3/5 \right) \Gamma \left( 1/5\,n+4/5 \right) \left( \cos \left( 2/5\,\pi \right) +\cos \left( 1/5\,\pi \right) \right) }{{\pi }^{2}}}&${\rm irem} \left( n,5 \right) =0$\cr 0&${\rm irem} \left( n-1,5 \right) =0$\cr 0&${\rm irem} \left( n-2,5 \right) =0$\cr 0&${\rm irem} \left( n-3,5 \right) =0$\cr 0&${\rm irem} \left( n-4,5 \right) =0$\cr}
other formats
3.4. Asymptotics
4. Exponential generating function
\displaystyle
{{\rm e}^{1/5\,{x}^{5}}}
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5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).