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单词 DerivationOfTheGeneratingSeriesForTheStirlingNumbersOfTheSecondKind
释义

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 compositionMathworldPlanetmathPlanetmath 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:

𝒜~𝒰(𝒰\\{})

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

The set 𝒰 has one object in it of each weight, and so has exponential generating series:

[(𝒰,ω)]e(x)=n0xnn!=ex

The set 𝒰\\{} then has generating series:

[(𝒰\\{},ω)]e(x)=ex-1

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

[(𝒜,ω)]e(x)=[(𝒰(𝒰\\{}),ω)]e(x)
=([(𝒰,ω)]e[(𝒰\\{},ω)]e)(x)
=eex-1

By tensoring the weight function ω with a weight function λ counting the number of parts each set partitionMathworldPlanetmath contains, we get

[(𝒜,ωλ)]e,o(x,t)=et(ex-1)

using a derivation similarPlanetmathPlanetmath to the one above.

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更新时间:2025/5/4 19:15:20