Talk:Bell number

Untitled

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 moved—jiy (talk) 18:40, 8 January 2006 (UTC)

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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 (talk • contribs).

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)

A minor snafu for Dobinski's formula when n = 0 : the first summand in the infinite summation contains a 0 to the 0th power which is undefined. Kdpw (talk) 21:27, 23 December 2017 (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, B_n, 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, B_n, 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)

Another problem is that the current formatting does not allow to see at first glance the value for, say, B₁₂. I tried to edit with a tabular display providing n on the first row and B_n on the second row while keeping the numbers from 0 to 19, but @Deacon Vorbis: undid the change as the table was too wide. I'd see three options:

1. truncate the sequence to the first 10 or so Bell numbers to reduce the table width
2. use an enumeration with B₀=1, B₁ = 1, B₂=2, B₃=5, B₄=15, B₄=15, etc
3. format the same enumeration in a table with B₀ to B₉ on first line, B₁₀ to B₁₉ on second line

Recommendation welcome. Gcaumon (talk) 18:09, 8 January 2020 (UTC)

My two cents: I would be very happy to see the list of numbers in the introduction shortened to 10 (that's what references are for). To me, a table breaks the flow of the text in a way that a list does not. I don't have strong feelings about writing "B₀=1, B₁ = 1, B₂=2, B₃=5, B₄=15". --JBL (talk) 18:24, 8 January 2020 (UTC)

I'm not sure 10 is the right number but the current listing is definitely too long. I would prefer just the bare list of numbers, not the annotated sequence of which number is which. —David Eppstein (talk) 18:45, 8 January 2020 (UTC)

I agree that the list is too long (I think I'd chop it after 4140). It's also a little awkward that we say "B₀ = B₁ = 1" before we introduce the notation B_n. Maybe we could go with something like,

The Bell numbers are denoted B_n, where n is an integer greater than or equal to zero. Starting with B₀, the first few Bell numbers are

1, 1, 2, 5, 15, 52, 203, 877, 4140, ... (sequence A000110 in the OEIS).

The Bell number B_n counts the number of different ways to partition a set that has exactly n elements ...

Thoughts? XOR'easter (talk) 19:16, 8 January 2020 (UTC)

Sounds good to me. –Deacon Vorbis (carbon • videos) 19:28, 8 January 2020 (UTC)

To me too. —David Eppstein (talk) 19:54, 8 January 2020 (UTC)

To me three, although I would put in a vote for preserving "starting with B_0 = B_1 = 1" to eliminate any possibility of ambiguity. --JBL (talk) 20:33, 8 January 2020 (UTC)

OK, I think that's enough agreement to go ahead and implement the change (without ruling out the possibility of further improvements, of course). XOR'easter (talk) 20:48, 8 January 2020 (UTC)

Passive aggressive introduction

Is it just me or does the introduction seem mildly passive aggressive about the name coming from Bell, despite the much older history of the numbers? I might also be completely misreading it. LudwikSzymonJaniuk (talk) 21:01, 3 November 2019 (UTC)

You think it would be an improvement to leave out the word "but"? To remove some of this information from the lead? Or maybe to link to Stigler's law of eponymy to clarify that this kind of misnomer happens all the time? —David Eppstein (talk) 21:25, 3 November 2019 (UTC)

Maybe reversing the two phrases? "Bell numbers are named after Eric Temple Bell, who wrote about them in the 1930s, although they have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan." But removing "but" (and splitting the second sentence into 2) also works well. --JBL (talk) 22:09, 3 November 2019 (UTC)

The introduction seems nicely written to me as it is now. In any case, it would be very interesting to know how and when the name "Bell" was attached to this sequence. My guess is that this is relatively recent, and that originated after Martin Gardner's nice divulgative articles about 50 years ago, where he refers to E.T.Bell and his 1930s papers.pma 22:24, 10 January 2020 (UTC)

The history of that name is at the end of the article. According to our source (after correcting the date in the source) it's from a paper published 30 years earlier than Gardner. —David Eppstein (talk) 07:20, 11 January 2020 (UTC)