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| In [[combinatorics|combinatorial]] [[mathematics]], the '''Stirling transform''' of a sequence { ''a''<sub>''n''</sub> : ''n'' = 1, 2, 3, ... } of numbers is the sequence { ''b''<sub>''n''</sub> : ''n'' = 1, 2, 3, ... } given by
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| :<math>b_n=\sum_{k=1}^n \left\{\begin{matrix} n \\ k \end{matrix} \right\} a_k,</math>
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| where <math>\left\{\begin{matrix} n \\ k \end{matrix} \right\}</math> is the [[Stirling number]] of the second kind, also denoted ''S''(''n'',''k'') (with a capital ''S''), which is the number of [[partition of a set|partitions]] of a set of size ''n'' into ''k'' parts.
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| The inverse transform is
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| :<math>a_n=\sum_{k=1}^n s(n,k) b_k,</math>
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| where ''s''(''n'',''k'') (with a lower-case ''s'') is a Stirling number of the first kind.
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| Berstein and Sloane (cited below) state "If ''a''<sub>''n''</sub> is the number of objects in some class with points labeled 1, 2, ..., ''n'' (with all labels distinct, i.e. ordinary labeled structures), then ''b''<sub>''n''</sub> is the number of objects with points labeled 1, 2, ..., ''n'' (with repetitions allowed)."
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| If
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| :<math>f(x)=\sum_{n=1}^\infty {a_n \over n!} x^n</math> | |
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| is a [[formal power series]] (note that the lower bound of summation is 1, not 0), and
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| :<math>g(x)=\sum_{n=1}^\infty {b_n \over n!} x^n</math> | |
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| with ''a''<sub>''n''</sub> and ''b''<sub>''n''</sub> as above, then
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| :<math>g(x)=f(e^x-1).\,</math>
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| ==See also==
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| * [[Binomial transform]]
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| * [[List of factorial and binomial topics]]
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| ==References==
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| * {{cite journal| first1=M. |last1=Bernstein |first2=N. J. A. |last2=Sloane
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| |title=Some canonical sequences of integers | journal=Linear Algebra and its Applications
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| |volume=226/228 |year=1995 | pages=57–72 |doi=10.1016/0024-3795(94)00245-9}}.
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| [[Category:Factorial and binomial topics]]
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| [[Category:Transforms]]
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