 | ECS #349: Stirling numbers of the second kind |
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1. Description
Partitions of a set of size n into 6 non-empty subsets
2. Specification
This unlabelled structure is specified as S in
\displaystyle
\left\{ S={\rm Prod} \left( {\rm Sequence} \left( Z \right) ,{\rm
Sequence} \left( {\rm Union} \left( Z,Z \right) \right) ,{\rm Sequence
} \left( {\rm Union} \left( Z,Z,Z \right) \right) ,{\rm Sequence}
\left( {\rm Union} \left( Z,Z,Z,Z \right) \right) ,{\rm Sequence}
\left( {\rm Union} \left( Z,Z,Z,Z,Z \right) \right) ,{\rm Sequence}
\left( {\rm Union} \left( Z,Z,Z,Z,Z,Z \right) \right) \right)
\right\}
other formats
3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ -720\,f \left( n \right) +1044\,f \left( n+1 \right) -580\,f
\left( n+2 \right) +155\,f \left( n+3 \right) -20\,f \left( n+4
\right) +f \left( n+5 \right) -1=0,f \left( 0 \right) =1,f \left( 1
\right) =21,f \left( 2 \right) =266,f \left( 3 \right) =2646,f \left(
4 \right) =22827 \right\}
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3.3. Closed form
\displaystyle
-{\frac {1}{120}}+{\frac {256}{3}}\,{4}^{n}-{\frac {81}{4}}\,{3}^{n}+4/
3\,{2}^{n}+{\frac {324}{5}}\,{6}^{n}-{\frac {3125}{24}}\,{5}^{n}
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3.4. Asymptotics
4. Ordinary generating function
\displaystyle
{\frac {1}{ \left( -1+x \right) \left( -1+2\,x \right) \left( -1+3\,x
\right) \left( -1+4\,x \right) \left( -1+5\,x \right) \left( -1+6\,
x \right) }}
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5. References
6. Random structure
Search a combinatorial structure by: (firstTerms should be a sequence of integers, separated by commas).