# derivation of the generating series for the Stirling numbers of the second kind

The derivation of the generating series is much simpler if one makes use of the composition^{} lemma for exponential generating series. We are looking for the generating series for sets of nonempty sets, so in the notation of Jackson and Goulden, we have the set decomposition:

$$\mathcal{A}\stackrel{~}{\u27f6}\mathcal{U}\u229b(\mathcal{U}\backslash \{\mathrm{\varnothing}\})$$ |

where $\mathcal{U}$ is the set of all canonical unordered sets, $\mathcal{A}$ is the set which we are interested in counting, and $\u229b$ is star-composition of sets of labelled combinatorial objects.

The set $\mathcal{U}$ has one object in it of each weight, and so has exponential generating series:

$${[(\mathcal{U},\omega )]}_{e}(x)=\sum _{n\ge 0}\frac{{x}^{n}}{n!}={e}^{x}$$ |

The set $\mathcal{U}\backslash \{\mathrm{\varnothing}\}$ then has generating series:

$${[(\mathcal{U}\backslash \{\mathrm{\varnothing}\},\omega )]}_{e}(x)={e}^{x}-1$$ |

So, by the star composition lemma and the above decomposition,

${[(\mathcal{A},\omega )]}_{e}(x)$ | $=$ | ${[(\mathcal{U}\u229b(\mathcal{U}\backslash \{\mathrm{\varnothing}\}),\omega )]}_{e}(x)$ | ||

$=$ | $\left({[(\mathcal{U},\omega )]}_{e}\circ {[(\mathcal{U}\backslash \{\mathrm{\varnothing}\},\omega )]}_{e}\right)(x)$ | |||

$=$ | ${e}^{{e}^{x}-1}$ |

By tensoring the weight function $\omega $ with a weight function $\lambda $ counting the number of parts each set partition^{} contains, we get

$${[(\mathcal{A},\omega \otimes \lambda )]}_{e,o}(x,t)={e}^{t({e}^{x}-1)}$$ |

using a derivation similar^{} to the one above.

Title | derivation of the generating series for the Stirling numbers of the second kind |
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Canonical name | DerivationOfTheGeneratingSeriesForTheStirlingNumbersOfTheSecondKind |

Date of creation | 2013-03-22 15:13:35 |

Last modified on | 2013-03-22 15:13:35 |

Owner | cgibbard (959) |

Last modified by | cgibbard (959) |

Numerical id | 6 |

Author | cgibbard (959) |

Entry type | Derivation |

Classification | msc 05A15 |