# Talk:Bell number

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I've added a section on calculating Bell numbers that doesn't use any equations; I think pictures like this are more accessible than descriptions which use a lot of equations (people have also complained about the typeface we use for math; see, for example, the discussion over at JPEG). Samboy 04:38, 16 May 2005 (UTC)

I've put the algorithm after the "theoretical" material; it seems of lesser interest. Michael Hardy 21:03, 16 May 2005 (UTC)
One of my goals when making those additions is to present Bell numbers so that an intelligent elementry school kid can understand and even calculate them. Of course, what I really need to do is add a picture showing the Bell number combining three numbers together (or even have pictures of labelled balls in boxes). I've added some section headers so that people can go from section to section quickly; this hopefully makes the article more readable. Samboy 01:11, 17 May 2005 (UTC)

## Requested move

The following discussion is an archived debate of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the debate was: it was movedjiy (talk) 18:40, 8 January 2006 (UTC)

I know currently more pages link to the plural than the singular. but for pages about kinds of numbers (Fibonacci number, Catalan number, highly totient number, etc.) their usually at the singular. Numerao 18:38, 20 September 2005 (UTC)

• I support this; it has the practical advantage that the plural [[Bell number]]s can link to the singular without piping or redirection; the reverse is not true. Septentrionalis 19:40, 21 September 2005 (UTC)
• I also support the proposed move, for the same reason as ibid. PrimeFan 17:55, 11 October 2005 (UTC)

Such titles should be plural when the sequence of numbers is what is of interest. In the case of prime number or perfect number, the concept is of interest when thinking about a particular number and not just when thinking about a sequence of numbers. This distinction seems fairly clear-cut when thinking about polynomial sequence (it is appropriate that the article title Hermite polynomials is plural while Bernstein polynomial is not). Maybe it's a bit less clear-cut with sequences of numbers, but we need to talk about it. Michael Hardy 00:34, 22 September 2005 (UTC)

From a Handbook of Integer Sequences viewpoint, "Bell numbers" is of greater interest. From a Dictionary of Curious Integers viewpoint, "Bell number" is of greater interest. PrimeFan 17:55, 11 October 2005 (UTC)
The above discussion is preserved as an archive of the debate. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

## Empty set

Surely there can be no partitions of an empty set.

If we permit {} to be a partition of {}, then there are an infinite number of partitions of {X}: { {X} }, { {X}, {} }, { {X}, {}, {} } and so on.

The situation is ananlagous to the reason why we do not count 1 as a prime number, because if we did then there would be more than one way to break any number up into prime factors. Indeed, the factorisations of any number are prescisely the ways of dividing up it's set of prime factors into partitions. —The preceding unsigned comment was added by 203.10.224.60 (talkcontribs).

A member of a partition cannot be empty. (No member of the empty set is non-empty!) Michael Hardy (talk) 05:06, 15 August 2010 (UTC)
Reliable sources define B0 = 1, so we do the same. It makes good sense to me. The single partition of {} is {} as you correctly write. Your then-part apparently makes the incorrect assumption that the partition is {{}}. A partition is a set of subsets, and the partition of the empty set is the empty set (which means it contains no subsets). The partition does not contain the empty set, and no partition is allowed to contain the empy set. Does this make sense to you? PrimeHunter 10:35, 24 July 2007 (UTC)
There are a number of sources that do use or analyse the sequence with B0 = 0. Ramanujan, for instance, studied it as the 2nd iteration of: a(0) = x; a(n) = e^(a(n-1)) - 1. This has the advantage that the summations all deal with no constant terms, so standard formula apply on their exponents that have the standard combinatorial interpretations. That's probably why the combinatorial value of zero works out. See Berndt's Book 1 of Ramanujan's Notebooks, chapter 4. Ex0du5 5u+u7e (talk) 04:26, 15 August 2010 (UTC)

## Tale of Genji connection

There's a connection with 52 of the 54 symbols used to indicate chapters of the Tale of Genji -- see http://www.viewingjapaneseprints.net/texts/topictexts/artist_varia_topics/genjimon7.html etc. I believe this is discussed in one of the old Martin Gardner columns. What we have on Wikipedia now is Japanese incense#Kōdō... -- AnonMoos (talk) 10:02, 20 October 2010 (UTC)

On that page, the symbols for chapter 35 and 42 are equivalent, and the symbol for chapter 54 is a strange variant of that for chapter 53, but the remaining 52 symbols illustrate how the number of groupings of five elements is the Bell number 52... AnonMoos (talk) 05:04, 22 October 2010 (UTC)

## Another Python implementation

I offer another Python implementation that is a little simpler than the current one

def comb(n, k):

``` if k == 0: return 1
if n == 0: return 0
e = n - k
i = n - 1
c = n
while i > e:
c = c*i
i = i - 1
r = (c / factorial(k))
return r

```

def bell(n):

``` if n < 2: return 1
b = [None]*(n+1)
b[0] = 1
b[1] = 1
for i in range(2, n+1):
sum_b = 0
for k in range(i):
sum_b = sum_b + (b[k]* comb(i-1,k))
b[i] = sum_b
return b[n]
```

If you think it useful, please move it to the main page. — Preceding unsigned comment added by 24.6.172.74 (talk) 00:46, 4 September 2011 (UTC)

## The nth of these numbers.

Right after spelling out the sequence of the first Bell numbers, the introductory part of the article states

"The nth of these numbers, Bn, counts the number of different ways to partition a set that has exactly n elements [...]"

Given that the sequence provided just above is

1, 1, 2, 5, 15, ... ,

the previous sentence seems incorrect. For example, the number 15 in the Bell sequence corresponds to the number of possible partitions out of a set with n=4 elements, but the number 15 occupies the fifth position in the sequence rather than the fourth. In other words, the sentence should probably read

"The nth of these numbers, Bn, counts the number of different ways to partition a set that has exactly n-1 elements [...]"

Thegreenfuse (talk) 14:12, 29 September 2015 (UTC)

It says:
"Starting with B0 = B1 = 1, the first few Bell numbers are:
1, 1, 2, 5, 15, ..."
The initial 1 is B0 (the zeroth number, see zero-based numbering), so 15 is B4. PrimeHunter (talk) 14:34, 29 September 2015 (UTC)