bell numbers

**Sampras** · Sep 28th 2009, 07:51 AM

So the bell numbers satisfy the following recurrence: $b(n) = \sum_{k} \binom{n-1}{k} b(k)$ , $n \geq 1$ , $b(0) = 1$ .

Is there a combinatorial interpretation of this?

**Plato** · Sep 28th 2009, 08:20 AM

Originally Posted by Sampras

So the bell numbers satisfy the following recurrence: $b(n) = \sum_{k} \binom{n-1}{k} b(k)$ , $n \geq 1$ , $b(0) = 1$ .
Is there a combinatorial interpretation of this?

In point of fact, the Bell numbers are given by $B(n) = \sum\limits_{k = 0}^n {S_2 (n,k)}$ .
Now ${S_2 (n,k)}$ are the Stirling numbers of the second kind.
That is the number of ways to partition a set of n elements into k nonempty subsets.
To calculate: ${S_2 (n,k)}= \frac{1}{{k!}}\left[ {\sum\limits_{k = 0}^n {\left( { - 1} \right)^j \binom{k}{j}\left( {k - j} \right)^n } } \right]$ .

So the Bell numbers count the total number of ways to partition a set.

**Sampras** · Sep 28th 2009, 09:32 AM

Originally Posted by Plato

In point of fact, the Bell numbers are given by $B(n) = \sum\limits_{k = 0}^n {S_2 (n,k)}$ .
Now ${S_2 (n,k)}$ are the Stirling numbers of the second kind.
That is the number of ways to partition a set of n elements into k nonempty subsets.
To calculate: ${S_2 (n,k)}= \frac{1}{{k!}}\left[ {\sum\limits_{k = 0}^n {\left( { - 1} \right)^j \binom{k}{j}\left( {k - j} \right)^n } } \right]$ .

So the Bell numbers count the total number of ways to partition a set.

Yes but that is the generating function combinatorial interpretation. But can we also interpret the recurrence relation solely? In other words can generating function combinatorial interpretations be interpreted in combinatorially in recurrence form?

**Plato** · Sep 28th 2009, 09:50 AM

Originally Posted by Sampras

Yes but that is the generating function combinatorial interpretation. But can we also interpret the recurrence relation solely? In other words can generating function combinatorial interpretations be interpreted in combinatorially in recurrence form?

I posted an attached PDF file in this thread.
That may be what you are asking.

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