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What is the cardinality of $N$ where $$N=\{P\mid P \, \text{is a partition of }\, \{0,1,2,3,4,5,6,7\},\text{ and } |P| \text{ is even}\}$$

My ideas:

The number of all partitions is the Bell number $B_8= 4140$.

But how can I find the number of the even partitions?

  • 1
    The answer just depends on the values at $\pm 1$ of the Touchard polynomial (https://en.wikipedia.org/wiki/Touchard_polynomials) $T_8(x)$. – 2017-02-20
  • 0
    If you are asking for the number of partitions into an even number of parts... then just how bell numbers are often calculated you could work with [Stirling Numbers of the second kind](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind). This would be just the sum of the number of parts into $k$ parts with $k$ even. – 2017-02-20
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    Ultimately this boils down to computing $\sum_{k\geq 0}\frac{k^8}{k!}$ (that is rather large) and $\sum_{k\geq 0}\frac{(-1)^k k^8}{k!}$ (that is rather small), so the answer ($\color{red}{2095}$) is pretty close to $\frac{B_8}{2}$. – 2017-02-20
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    Jack D'Aurizio thanks but how do you get 2095? I get different values for the series you given – 2017-02-20
  • 2
    The exponential generating function is $\cosh(e^x-1).$ This is OEIS sequence [A024430](http://oeis.org/A024430). The value for $n=8$ is $2095.$ – 2017-02-20

1 Answers 1

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Here are the numbers I came up with to compare any other result with:

Divide into 2 parts:
$(1,7)$: $\binom 87 =8$
$(2,6)$: $\binom 86 = 28$
$(3,5)$: $\binom 85 = 56$
$(4,4)$: $\binom 84/2 =35$

Divide into 4 parts:
$(1,1,1,5)$: $\binom 85 = 56$
$(1,1,2,4)$: $\binom 84 \binom 42= 420$
$(1,1,3,3)$: $\binom 83\binom 53/2 = 280$
$(1,2,2,3)$: $\binom 83\binom 52\binom 32/2 = 840$
$(2,2,2,2)$: $\binom 82\binom 62\binom 42\binom 22/4! = 105$

Divide into 6 parts:
$(1,1,1,1,1,3)$: $\binom 83 = 56$
$(1,1,1,1,2,2)$: $\binom 82\binom 62/2 = 210$

Divide into 8 parts:
$(1,1,1,1,1,1,1,1)$: $\binom 80 = 1$

Total $2095$

This could probably be written more neatly with multinomial coefficients, but the repeats still need to be divided out.