Talk:Partition of a set

message: I just included some displayed TeX within in a link, thus:
The  partition
:etc. etc.
of the integer n ...
It seems to work: you can click on the word partition, or on the displayed TeX, or on the following phrase, and get the integer partition page. Michael Hardy 20:07, 5 Feb 2004 (UTC)
thread_title:  Displayed TeX whithin a link 

message: Is the definition of partition entirely correct / undisputed?
specifically I wonder about the requirement that the elements of the partition needing to be non-empty.
The definition given in Potter, M. "Set theory and its philosophy", ( 2004, Oxford University Press ),
on p. 130 is:
"Definition. A collection B of subsets of A is a partition of A if each element
of A belongs to exactly one element of B.
According to this definition, A is a partition of itself, since it certainly is a subset of itself,
and furthermore { A, {null} } is also a partition of A - since there is no restriction on the elements being null.
This does not seem to quite capture the essence of partition to me, but it does seem a commonly used definition - ie see the 
MathWorld definition:
http://mathworld.wolfram.com/SetPartition.html
so I'm wondering whether the article should be updated to clarify this ambiguity - that there are at least two alternate definitions being used.
--Corvi42 04:15, 10 Jun 2005 (UTC)
replies: text: You are wrong about the empty set.  The partition with 0 blocks satisfies all the axioms including Axiom 1.  --Zero 16:55, 10 Jun 2005 (UTC)

text: PS: I just made the necessary change to correct this. — MFH:Talk 13:23, 20 September 2011 (UTC)
thread_title:  Definition correct? 

message: I was looking for a formula for the number of partitions of an n-element set into exactly k subsets each of size n/k, where k evenly divides n.  For example, say you have 12 donuts, each a different kind; how many ways are there to divide them into four sets of three donuts to give to four cops?  I couldn't find the answer, but derived the following, which I think is correct.  I'm suggesting a notation here, too, along the lines of the "n choose k = nCk" notation.  It might be nice to include this formula in the Counting section:
:n\ partition\ k = \ _n\!T_k = \begin{bmatrix}
n \\
k 
\end{bmatrix} = {n! \over k! (({n \over k})!)^k}
The numerator is the number of ways to permute the order of the n-element set members, the first denominator term is the number of ways to permute the order of the resulting k subsets, and the second denominator term is the number of ways to permute the order of the members of the k subsets. Tedtoal (talk) 00:45, 1 July 2010 (UTC)
replies: text: See Multinomial theorem#Ways to put objects into sized boxes.  The connection between partitions and multinomial coefficients needs presentation in this article, though it needs care.  For multinomial coefficients the order of the parts is significant but for partitions it is (usually) not significant.  One can't in general just adjust the count by the factorial of the number of parts, though it is valid if empty parts are forbidden as they are here. Zerotalk 05:18, 3 July 2010 (UTC)
thread_title:  Counting Partitions 

message: I added the line about ME and CE in the introduction.  I see that there has been some discussion about the basic concept of partitioning here, so if this edit violates any of the more formal aspects of the definition please feel free to revert.  capitalist 09:15, 19 October 2005 (UTC)
thread_title:  ME & CE 

message: I've just completed various changes that I think make this article clearer. The only one I think the slightest bit controversial was deleting the material about the Faà di Bruno formula. I reduced it to an entry in See Also for two reasons. First, it was a verbatim repeat from the other article, and second, I couldn't figure out for the life of me what it was even trying to mean (OK, maybe I'm dense...) I'd suggest somebody who does get it ought to clarify the passage in the other article.—PaulTanenbaum (talk) 06:02, 20 March 2009 (UTC)
thread_title:  Improving readability 

message: A partition covers the entire set, so it is exhaustive. If that is the case, we should not waste words and possibly create confusion by using the term "exhaustive partition".192.139.122.42 (talk) 20:54, 23 October 2012 (UTC)
replies: text: Within the framework of set theory, I have never encountered the phrase ‘exhaustive partition’, and the article does not contain that phrase, as far as I can see. It only mentions the word ‘exhaustive’ in the lead section, where it is part of the definition: “More formally, these "cells" are both collectively exhaustive and mutually exclusive with respect to the set being partitioned.” — Tobias Bergemann (talk) 11:35, 24 October 2012 (UTC)
thread_title:  So "exhaustive" is redundant in "exhaustive partition"? 

message: Definition of noncrossing in this article is different from that in Noncrossing partition, in a way that they are not describing the same thing. Doyoon1995 (talk) 07:50, 15 July 2015 (UTC)
thread_title:  Noncrossing partition 

message: The partition concept itself is somewhat confusing because it deals with sets on three or four levels: 
# The overall set of elements being considered: X
# The set of elements in a subset of X: S1, S2, ... Si
# The set of subsets making up a partition: P1, P2 etc. where perhaps P1 contains S1 and S4
# The set of all possible partitions: PAll, containing P1, P2...Pn
This article could really benefit from additional distinction between these (levels of) sets.
For example, in the figure "The 52 partitions of a set with 5 elements" it would be helpful to point out certain important subtleties and resolve certain discrepancies:
1. The subsets are indicated by the colored blobs. So for example, in row 10 each partition is a set containing two subsets, where the subsets contain two or three elements. But...
In other rows where some dots are uncolored, are the uncolored dots considered to be an additional subset? That would be needed to fulfill the requirement \bigcup_{A\in P} A = X. However, if that was the case, then, for example, the partitions in row 2 are the same as those in row 10. So is the figure incorrect?
2. The single partitions show in rows 1 and row 12: Are these actually the same partition? Or is row one actually depicting the partition having a single subset which is empty? But then what about \bigcup_{A\in P} A = X ?
And on that score, it would be helpful to explicitly point out which of the following is allowed:
a. A partition having a single empty subset. Perhaps allowed only in the case where X is empty, since otherwise the partition wouldn't cover X?
b. A partition that has no subsets: Again, is this allowed only for empty X? Or never? Or?
c. What does \emptyset \notin P actually mean? P is not allowed to be empty? Or P has no subsets that are empty? I think it's the latter, but then the later statement "The empty set has exactly one partition, namely the empty set" seems to violate that.
Hopefully someone with authoritative knowledge can clarify.  Gwideman (talk) 15:54, 27 February 2016 (UTC)
replies: text: Re c: It means that empty subsets are not allowed as components of the partition. The partition of the empty set has no components, so it meets this requirement. —David Eppstein (talk) 17:06, 27 February 2016 (UTC)
replies: text:  Ah, OK, so a partition having no (sub)sets is fine for an empty set X, but a partition having a single empty (sub)set is not. But I'm still puzzled about what the "52 partitions" figure is trying to depict with its first partition at the top.  Maybe in this figure uncolored dots are supposed to represent subsets with single members? So the top partition contains five single-element subsets?  Gwideman (talk) 17:45, 27 February 2016 (UTC)
replies: text: Yes, in this image, when one of the dots is not surrounded by a colored region, then it is in a set by itself. —David Eppstein (talk) 18:15, 27 February 2016 (UTC)
replies: text:  Thanks, that key not-so-obvious point makes things fall into place. I've noted that in the caption.  — Preceding unsigned comment added by Gwideman (talk • contribs) 19:31, 27 February 2016 (UTC)
thread_title:  Some clarity regarding empty set versus empty subset 

message: In the Examples section, it is written that "The set { 1, 2, 3 } has these five partitions:
* { {1}, {2}, {3} }, sometimes written 1|2|3.
* { {1, 2}, {3} }, or 12|3.
* […]
According to the Definition section above the Examples, however, if A,B \in P and A\neq B then A \cap B = \emptyset. In this case, { {1}, {2}, {3} }   \neq { {1, 2}, {3} } and A \cap B = {3}, which seems to be contradicting the definition.--Sae1962 (talk) 10:23, 10 February 2017 (UTC)
replies: text: Two different items correspond to two different partitions. You cannot check the definition of a partition simultaneously on two different partitions. I have edited the article for clarification. D.Lazard (talk) 11:06, 10 February 2017 (UTC)
thread_title:  Examples should be clarified 

message: What does the red and green coloring mean in the image of the "chapters of the Tale of Genji"? Thanks in advacne. --Backinstadiums (talk) 09:12, 14 March 2018 (UTC)
replies: text: This is explained in the file containing the image (click on the image). I have summarized the explanation in the caption. D.Lazard (talk) 09:28, 14 March 2018 (UTC)
replies: text:  could you please elaborate a bit what is represented? --Backinstadiums (talk) 18:56, 22 October 2018 (UTC)
thread_title:  colors in chapters of the Tale of Genji