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If you start out with six tastes (e.g., six different spices which you can use singly or in combination with each other), if you mix two together you get 30 combinations (i.e., six choices for the first one times five choices for the second equals 30 different mixtures). If, as the text suggests, you mix three together, you get 120 different mixtures. So how did the ancient one arrive at 63, and why has no one questioned this before? 71.251.136.49 ( talk) 22:24, 11 December 2010 (UTC)
The opening sentence in the lead has been stable for many years, but was a bit complex for anyone not familiar with the field. A recent edit attempted to fix this problem by replacing it with the statement that combinatorics is the mathematical theory of counting and citing it to an introductory probability text. I applaud the attempt, but it really goes a bit too far and leaves out major aspects of the subject, thus giving a false impression of the topic. This is understandable given the nature of the source, which I would not call authoritative, since the only aspect of combinatorics that is relevant to introductory probability is counting. To fix this I added the phrase "at its core" to indicate that while not incorrect, the statement was not complete, thus keeping the simplified form without giving the false impression. The next sentence would then have to say something about the broader definition, so I brought back the original sentence, slightly simplified. Another reason to bring back the sentence is that it contained the appropriate links to help make sense of the remainder of the paragraph and the rest of the article. A new problem now arises in that the simplified and broader definitions do not seem to be related (again, for a reader with no knowledge of the topic), so I added the phrase "in which counting is a primary tool" to make this connection clear. While my changes could perhaps be improved upon, a revert clearly needs to be explained and discussed. -- Bill Cherowitzo ( talk) 19:04, 26 October 2017 (UTC)
Perhaps I didn't make myself quite clear. I agree with the editor who felt that the stable version was too opaque. If I thought that returning to it was the best way to handle the situation, I would have reverted to it myself. I think that my revised version is better than either the stable version or the attempt by that editor. Of course there is no universally agreed upon definition, but an encyclopedia needs to give something that a reader can understand and that isn't too far off the mark. Throwing one's hands up in the air is not a good option. We might want to indicate the lack of agreement of a definition, but only after giving some skeletal outline of the topic. What I have suggested comes fairly close to the Princeton Companion treatment if it needs a pedigree. -- Bill Cherowitzo ( talk) 03:51, 27 October 2017 (UTC)
Ok. I will do that, although, I'm afraid, given the nature of the beast, you are going to have to accept some ambiguity in the lead. If this could be done without ambiguity there would be a clear-cut definition of the field, which we know does not exist. -- Bill Cherowitzo ( talk) 17:20, 27 October 2017 (UTC)
Here is a first pass. This is meant to replace the first paragraph. The Mazur quote is from Combinatorics : A Guided Tour and the Ryser reference is to Combinatorial Mathematics.
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This looks good to me. A few copyediting changes and a modification of the first sentence of the second paragraph are all that I think are needed. We can reference Pak's quotations for the addition. I think that this added phrase (or its equivalent) is needed to bring clarity to why we are addressing the definition question in the way that we are. I don't think that there was anything wrong with the algebra statement, as I phrased it, considering that up until very recently combinatorics was dealt with by the NSF in its algebra section. However, this is not a deal breaker and I'll be happy to let it slide. -- Bill Cherowitzo ( talk) 05:07, 30 October 2017 (UTC)
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The following discussion has been closed. Please do not modify it. |
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As we seem to be converging on this change, let's try to get some other comments before finalizing.
Soliciting comments for change of lead ( Joel B. Lewis— Will Orrick— David Eppstein) -- Bill Cherowitzo ( talk) 19:10, 30 October 2017 (UTC)
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc.
To fully understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon. [2] According to H. J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions. [3] In so far as an area can be described by the types of problems it addresses, combinatorics is involved with
- the enumeration (counting) of specified structures, sometimes referred to as arrangements or configurations in a very general sense, associated with finite systems,
- the existence of such structures that satisfy certain given criteria,
- the construction of these structures, perhaps in many ways, drawing upon ideas from several areas of mathematics, and
- optimization, finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other optimality criterion.
Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained." [4] One way to define combinatorics is, perhaps, to describe its subdivisions with their problems and techniques. This is the approach that is used below. However, there are also purely historical reasons for including or not including some topics under the combinatorics umbrella. [5] Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, countable) but discrete setting.
Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, [6] as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. [7] One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.
A mathematician who studies combinatorics is called a combinatorialist.
References
... combinatorial theory has been the mother of several of the more active branches of today's mathematics, which have become independent ... . The typical ... case of this is algebraic topology (formerly known as combinatorial topology)
In my opinion, combinatorics is now growing out of this early stage.
_____________________________________________________________
1) I am thinking "combinatorist" is old fashioned and should be deleted. I tried WP:GOOGLE and found 50K for "combinatorialist" vs 2.2K for "combinatorist", many of which I am sure to this WP page and its mirrors. I say we scrap this version.
2) I find "This approach is not completely satisfactory" to be overly negative. I would put a positive spin on it, as in "To fully understand the definition one needs to see historical reasons for including or not including some topics under the combinatorics umbrella. According to Gian-Carlo Rota, "combinatorial theory has been the mother of several of the more active branches of today's mathematics, which have become independent", singling out algebraic topology as the most successful example." [4] I tried to find a better link but failed. I think it's a chapter of "Discrete thought" by Rota.
P.S. Now that I read algebraic topology, that article includes a section "Notable algebraic topologists". Maybe we also should have a section like that. Category:Combinatorialists has many, of course, but some of those are best known for work in other fields. Mhym ( talk) 20:50, 1 November 2017 (UTC)
_____________________________________________________________
Reopening the discussion, I disagree with the emphasis on counting and restriction to finiteness in the opening sentence of the current lede. A lot of combinatorics is about structure without counting, or structure with some counting as an auxiliary aspect. Also, most combinatorics is finite, but an unmodified restriction is "misledeing" (sic). I suggest a rewrite that puts counting and structure on the same level of importance. I also hope for an improvement in "certain ... structures", where "certain" is too vague. On the other hand, it's very hard to be more specific. Ideas? Zaslav ( talk) 17:34, 29 February 2024 (UTC)
I am researching the actual usage/meaning of Riordan transforms and noticed they seem to be missing from Wikipedia. Would anybody be interested in an entry, and where should it be placed. I can give various published references to it and one project paper by Renzo Sprugnoli (which seems to have disappeared from the internet, but I have a copy). "In the raw" they are a little difficult/confusing to understand. They are related to Riordan Arrays Combinatorial identities "Formal"/"Method of coefficients" mathematical methods https://local.disia.unifi.it/merlini/papers/MofC.pdf etc... In my case I am using them to generate generating functions for Hadamard products of power series, and attempts to form closed partial sums of various functions. But since it's for private usage it won't be published. Rrogers314 ( talk) 21:37, 24 March 2021 (UTC)
This is the
talk page for discussing improvements to the
Combinatorics article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
Archives: 1 |
This
level-5 vital article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
If you start out with six tastes (e.g., six different spices which you can use singly or in combination with each other), if you mix two together you get 30 combinations (i.e., six choices for the first one times five choices for the second equals 30 different mixtures). If, as the text suggests, you mix three together, you get 120 different mixtures. So how did the ancient one arrive at 63, and why has no one questioned this before? 71.251.136.49 ( talk) 22:24, 11 December 2010 (UTC)
The opening sentence in the lead has been stable for many years, but was a bit complex for anyone not familiar with the field. A recent edit attempted to fix this problem by replacing it with the statement that combinatorics is the mathematical theory of counting and citing it to an introductory probability text. I applaud the attempt, but it really goes a bit too far and leaves out major aspects of the subject, thus giving a false impression of the topic. This is understandable given the nature of the source, which I would not call authoritative, since the only aspect of combinatorics that is relevant to introductory probability is counting. To fix this I added the phrase "at its core" to indicate that while not incorrect, the statement was not complete, thus keeping the simplified form without giving the false impression. The next sentence would then have to say something about the broader definition, so I brought back the original sentence, slightly simplified. Another reason to bring back the sentence is that it contained the appropriate links to help make sense of the remainder of the paragraph and the rest of the article. A new problem now arises in that the simplified and broader definitions do not seem to be related (again, for a reader with no knowledge of the topic), so I added the phrase "in which counting is a primary tool" to make this connection clear. While my changes could perhaps be improved upon, a revert clearly needs to be explained and discussed. -- Bill Cherowitzo ( talk) 19:04, 26 October 2017 (UTC)
Perhaps I didn't make myself quite clear. I agree with the editor who felt that the stable version was too opaque. If I thought that returning to it was the best way to handle the situation, I would have reverted to it myself. I think that my revised version is better than either the stable version or the attempt by that editor. Of course there is no universally agreed upon definition, but an encyclopedia needs to give something that a reader can understand and that isn't too far off the mark. Throwing one's hands up in the air is not a good option. We might want to indicate the lack of agreement of a definition, but only after giving some skeletal outline of the topic. What I have suggested comes fairly close to the Princeton Companion treatment if it needs a pedigree. -- Bill Cherowitzo ( talk) 03:51, 27 October 2017 (UTC)
Ok. I will do that, although, I'm afraid, given the nature of the beast, you are going to have to accept some ambiguity in the lead. If this could be done without ambiguity there would be a clear-cut definition of the field, which we know does not exist. -- Bill Cherowitzo ( talk) 17:20, 27 October 2017 (UTC)
Here is a first pass. This is meant to replace the first paragraph. The Mazur quote is from Combinatorics : A Guided Tour and the Ryser reference is to Combinatorial Mathematics.
Version 1 |
---|
The following discussion has been closed. Please do not modify it. |
|
Version 2 |
---|
The following discussion has been closed. Please do not modify it. |
_______________________________________________________________ |
This looks good to me. A few copyediting changes and a modification of the first sentence of the second paragraph are all that I think are needed. We can reference Pak's quotations for the addition. I think that this added phrase (or its equivalent) is needed to bring clarity to why we are addressing the definition question in the way that we are. I don't think that there was anything wrong with the algebra statement, as I phrased it, considering that up until very recently combinatorics was dealt with by the NSF in its algebra section. However, this is not a deal breaker and I'll be happy to let it slide. -- Bill Cherowitzo ( talk) 05:07, 30 October 2017 (UTC)
Version 3 |
---|
The following discussion has been closed. Please do not modify it. |
_____________________________________________________________ |
As we seem to be converging on this change, let's try to get some other comments before finalizing.
Soliciting comments for change of lead ( Joel B. Lewis— Will Orrick— David Eppstein) -- Bill Cherowitzo ( talk) 19:10, 30 October 2017 (UTC)
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc.
To fully understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon. [2] According to H. J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions. [3] In so far as an area can be described by the types of problems it addresses, combinatorics is involved with
- the enumeration (counting) of specified structures, sometimes referred to as arrangements or configurations in a very general sense, associated with finite systems,
- the existence of such structures that satisfy certain given criteria,
- the construction of these structures, perhaps in many ways, drawing upon ideas from several areas of mathematics, and
- optimization, finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other optimality criterion.
Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained." [4] One way to define combinatorics is, perhaps, to describe its subdivisions with their problems and techniques. This is the approach that is used below. However, there are also purely historical reasons for including or not including some topics under the combinatorics umbrella. [5] Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, countable) but discrete setting.
Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, [6] as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. [7] One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.
A mathematician who studies combinatorics is called a combinatorialist.
References
... combinatorial theory has been the mother of several of the more active branches of today's mathematics, which have become independent ... . The typical ... case of this is algebraic topology (formerly known as combinatorial topology)
In my opinion, combinatorics is now growing out of this early stage.
_____________________________________________________________
1) I am thinking "combinatorist" is old fashioned and should be deleted. I tried WP:GOOGLE and found 50K for "combinatorialist" vs 2.2K for "combinatorist", many of which I am sure to this WP page and its mirrors. I say we scrap this version.
2) I find "This approach is not completely satisfactory" to be overly negative. I would put a positive spin on it, as in "To fully understand the definition one needs to see historical reasons for including or not including some topics under the combinatorics umbrella. According to Gian-Carlo Rota, "combinatorial theory has been the mother of several of the more active branches of today's mathematics, which have become independent", singling out algebraic topology as the most successful example." [4] I tried to find a better link but failed. I think it's a chapter of "Discrete thought" by Rota.
P.S. Now that I read algebraic topology, that article includes a section "Notable algebraic topologists". Maybe we also should have a section like that. Category:Combinatorialists has many, of course, but some of those are best known for work in other fields. Mhym ( talk) 20:50, 1 November 2017 (UTC)
_____________________________________________________________
Reopening the discussion, I disagree with the emphasis on counting and restriction to finiteness in the opening sentence of the current lede. A lot of combinatorics is about structure without counting, or structure with some counting as an auxiliary aspect. Also, most combinatorics is finite, but an unmodified restriction is "misledeing" (sic). I suggest a rewrite that puts counting and structure on the same level of importance. I also hope for an improvement in "certain ... structures", where "certain" is too vague. On the other hand, it's very hard to be more specific. Ideas? Zaslav ( talk) 17:34, 29 February 2024 (UTC)
I am researching the actual usage/meaning of Riordan transforms and noticed they seem to be missing from Wikipedia. Would anybody be interested in an entry, and where should it be placed. I can give various published references to it and one project paper by Renzo Sprugnoli (which seems to have disappeared from the internet, but I have a copy). "In the raw" they are a little difficult/confusing to understand. They are related to Riordan Arrays Combinatorial identities "Formal"/"Method of coefficients" mathematical methods https://local.disia.unifi.it/merlini/papers/MofC.pdf etc... In my case I am using them to generate generating functions for Hadamard products of power series, and attempts to form closed partial sums of various functions. But since it's for private usage it won't be published. Rrogers314 ( talk) 21:37, 24 March 2021 (UTC)