Stirling Numbers of the Second Kind/Examples/5th Power

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Example of Stirling Numbers of the Second Kind

$x^5 = x^{\underline 5} + 10 x^{\underline 4} + 25 x^{\underline 3} + 15 x^{\underline 2} + x^{\underline 1}$

and so:

$x^5 = 120 \dbinom x 5 + 240 \dbinom x 4 + 150 \dbinom x 3 + 30 \dbinom x 2 + \dbinom x 1$

Proof

From the definition of Stirling numbers of the second kind:

$\displaystyle x^n = \sum_k \left\{ {n \atop k}\right\} x^{\underline k}$

Reading the values directly from Stirling's triangle of the second kind:

$\displaystyle \left\{ {5 \atop 5}\right\} = 1$
$\displaystyle \left\{ {5 \atop 4}\right\} = 10$
$\displaystyle \left\{ {5 \atop 3}\right\} = 25$
$\displaystyle \left\{ {5 \atop 4}\right\} = 15$
$\displaystyle \left\{ {5 \atop 4}\right\} = 1$

By definition of binomial coefficient:

$x^{\underline 5} = 5! \dbinom x 5 = 120 \dbinom x 5$
$x^{\underline 4} = 4! \dbinom x 4 = 24 \dbinom x 4$
$x^{\underline 3} = 3! \dbinom x 3 = 6 \dbinom x 3$
$x^{\underline 2} = 2! \dbinom x 2 = 2 \dbinom x 2$
$x^{\underline 1} = 1! \dbinom x 1 = \dbinom x 1$

Hence:

$x^5 = 120 \dbinom x 5 + 240 \dbinom x 4 + 150 \dbinom x 3 + 30 \dbinom x 2 + \dbinom x 1$

$\blacksquare$