 | ECS #350: Stirling numbers of the second kind |
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1. Description
Partitions of a set of size n into 7 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) ,{\rm Sequence}
\left( {\rm Union} \left( Z,Z,Z,Z,Z,Z,Z \right) \right) \right)
\right\}
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3. Coefficients
3.1. First terms
3.2. Recurrence
\displaystyle
\left\{ -5040\,f \left( n \right) +8028\,f \left( n+1 \right) -5104\,f
\left( n+2 \right) +1665\,f \left( n+3 \right) -295\,f \left( n+4
\right) +27\,f \left( n+5 \right) -f \left( n+6 \right) +1=0,f \left( 0
\right) =1,f \left( 1 \right) =28,f \left( 2 \right) =462,f \left( 3
\right) =5880,f \left( 4 \right) =63987,f \left( 5 \right) =627396
\right\}
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3.3. Closed form
\displaystyle
{\frac {1}{720}}+{\frac {117649}{720}}\,{7}^{n}+{\frac {15625}{48}}\,{5
}^{n}-{\frac {1024}{9}}\,{4}^{n}-{\frac {1944}{5}}\,{6}^{n}+{\frac {243
}{16}}\,{3}^{n}-{\frac {8}{15}}\,{2}^{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) \left( -1+7\,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).