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{{Cleanup|date=November 2009}} {{Cleanup|date=November 2009}}
The '''history of ]''' is an area of study within the ]. Its focus ranges from antiquity to modern times. The '''history of ]''' is an area of study within the ]. Its focus ranges from antiquity to modern times.


==Earliest records== ==Earliest records==
The earliest recorded use of combinatorial techniques comes from problem 79 of the ] for the implementation of a geometric series. In 1882, Leon Rodet noticed similarities between problem 79 and Fibonacci's problem of counting the number of compositions of 1s and 2s that sum to a given total <math>n</math> <ref name="Biggs" />. A result of ] of Chalcedon (396-314 BC) possibly represents the first attempt on record to solve a difficult problem in permutations and combinatorics
The earliest known connection to combinatorics comes from the ], problem 79, for the implementation of a geometric series.
⚫ <ref>{{cite book|last1=Heath|first1=Sir Thomas|title=A history of Greek mathematics|date=1981|publisher=Dover|location=New York|isbn=0486240738|edition=Reprod. en fac-sim.}}</ref>. The ] had the first mention of a combinatorics problem; the problem asked how many possible combinations of tastes were possible from selecting tastes in ones, twos, threes, etc. from a selection of six different tastes (sweet, pungent, astringent, sour, salt, and bitter). The Bhagavati is also the first text to mention the ].<ref name="India">{{cite web|title=India|url=http://binomial.csueastbay.edu/India.html|accessdate=2008-03-05}}</ref> In the second century BC, ] included an enumeration problem in the ] (also Chandahsutra) which asked how many ways a six-syllable meter could be made from short and long notes.<ref name="Hall">{{cite journal|first=Rachel|last=Hall|title=Math for Poets and Drummers-The Mathematics of Meter|date=2005-02-16|url=http://www.sju.edu/~rhall/Rhythms/poets.pdf|accessdate=2008-03-05|format=PDF}}</ref><ref name="Kulkarni">{{cite journal|last=Kulkarni|first=Amba|title=Recursion and Combinatorial Mathematics in Chandashāstra|arxiv=math/0703658}}</ref> Pingala found the number of meters that had <math>n</math> long notes and <math>k</math> short notes; this is equivalent to finding the ].


⚫ The ideas of the Bhagavati were generalized by the Indian mathematician ] in 850 AD, and Pingala's work on ] was expanded by ]<ref name="India" /><ref name="Lilavati">{{cite web
⚫ A ] text, the ], had the first mention of a combinatorics problem; it asked how many ways one could take six tastes one, two, or three tastes at a time. The Bhagavati Sutra was written around 300 BC, and was the first book to mention the ].<ref name="India">{{cite web|title=India|url=http://binomial.csueastbay.edu/India.html|accessdate=2008-03-05}}</ref> The next ideas of Combinatorics came from ], who was interested in ]. Specifically, he wanted to know how many ways a six-syllable meter could be made from short and long notes. He wrote this problem in the Chanda sutra (also Chandahsutra) in the second century BC.<ref name="Hall">{{cite journal|first=Rachel|last=Hall|title=Math for Poets and Drummers-The Mathematics of Meter|date=2005-02-16|url=http://www.sju.edu/~rhall/Rhythms/poets.pdf|accessdate=2008-03-05|format=PDF}}</ref><ref name="Kulkarni">{{cite journal|last=Kulkarni|first=Amba|title=Recursion and Combinatorial Mathematics in Chandashāstra|arxiv=math/0703658}}</ref> In addition, he also found the number of meters that had n long notes and k short notes, which is equivalent to finding the ].

⚫ The ideas of the Bhagavati were generalised by the Indian mathematician ] in 850 AD, and Pingala's work on prosody was expanded by Bhaskara<ref name="India" /><ref name="Lilavati">{{cite web
|last=Bhaskara |last=Bhaskara
|authorlink=Bhaskara II |authorlink=Bhaskara II
Line 18: Line 17:
|deadurl=yes |deadurl=yes
|df= |df=
}}</ref> and Hemacandra in 1100 AD. Bhaskara was the first known person to find the generalised choice function, although ] may have known earlier.<ref name="Biggs">{{cite book }}</ref> and ] in 1100 AD. Bhaskara was the first known person to find the generalised choice function, although ] may have known earlier.<ref name="Biggs">{{cite book
| last = Biggs | last = Biggs
| first = Norman | first = Norman
Line 32: Line 31:
| pages = 2163–2188 | pages = 2163–2188
| chapter = 44 | chapter = 44
}}</ref> Hemacandra asked how many meters existed of a certain length if a long note was considered to be twice as long as a short note, which is equivalent to finding the Fibonacci numbers.<ref name="Hall" /> }}</ref> Hemacandra asked how many meters existed of a certain length if a long note was considered to be twice as long as a short note, which is equivalent to finding the ].<ref name="Hall" />
]]] ]]]


The ancient Chinese book of divination, ], is about what different hexagrams mean, and to do this one needs to know how many possible hexagrams there were. Since each hexagram is a permutation with repetitions of six lines, where each line can be one of two states, solid or dashed, combinatorics yields the result that there are 2<sup>&nbsp;6</sup>&nbsp;=&nbsp;64 hexagrams. A monk also may have counted the number of configurations to a game similar to ] around 700 AD.<ref name="Dieudonné">{{cite web The ancient Chinese book of divination ] describes a hexagram as a permutation with repetitions of six lines where each line can be one of two states: solid or dashed. In describing hexagrams in this fashion they determine that there are <math>2^6=64</math> possible hexagrams. A Chinese monk also may have counted the number of configurations to a game similar to ] around 700 AD.<ref name="Dieudonné">{{cite web
| last = Dieudonné | last = Dieudonné
| first = J. | first = J.
Line 41: Line 40:
| work = Historia Math | work = Historia Math
| publisher = Truman State University | publisher = Truman State University
| url = http://mtcs.truman.edu/~thammond/history/RhindPapyrus.html | url = http://archive.is/GgX8
| accessdate = 2008-03-06 }}</ref> Although China had relatively few advancements in enumerative combinatorics, they solved a ] problem, the normal ] of order three, around 100 AD, in the ].<ref name="Biggs" /><ref name="Swaney">{{cite web |last=Swaney |first=Mark |title=Mark Swaney on the History of Magic Squares |url=http://www.netmastersinc.com/secrets/magic_squares.htm | accessdate = 2008-03-06 }}</ref> Although China had relatively few advancements in enumerative combinatorics, around 100 AD they solved the ] which is the ] problem of the normal ] of order three.<ref name="Biggs" /><ref name="Swaney">{{cite web |last=Swaney |first=Mark |title=Mark Swaney on the History of Magic Squares |url=http://www.netmastersinc.com/secrets/magic_squares.htm
⚫ |archiveurl=https://web.archive.org/web/20040807015853/http://www.netmastersinc.com/secrets/magic_squares.htm |archivedate=2004-08-07}}</ref> Magic squares remained an interest of China, and they began to generalize their original <math>3\times3</math> square between 900 and 1300 AD. China corresponded with the Middle East about this problem in the 13th century.<ref name="Biggs" /> The Middle East also learned about binomial coefficients from Indian work and found the connection to polynomial expansion.<ref name="Middle East">{{cite web|title=Middle East|url=http://binomial.csueastbay.edu/MidEast.html|accessdate=2008-03-08}}</ref> The work of Hindus influenced Arabs as seen in the work of al-Halil Ibn-Ahmad who considered the possible arrangements of letters to form syllables. His calculations show an understanding of permutations and combinations. In a passage from the work of Arab mathematician Umar al-Khayyami that dates to around 1100, it is corroborated that the Hindus had knowledge of binomial coefficients, but also that their methods reached the middle east.
|archiveurl=https://web.archive.org/web/20040807015853/http://www.netmastersinc.com/secrets/magic_squares.htm |archivedate=2004-08-07}}</ref>


In Greece, ] wrote that ] discovered the number of different syllables possible in the Greek language. This, however, is unlikely because this is one of the few mentions of Combinatorics in Greece. The number they found, 1.002&nbsp;&times;&nbsp;10<sup>&nbsp;12</sup> also seems too round to be more than a guess.<ref name="Dieudonné" /><ref name="Gow">{{cite book In Greece, ] wrote that Xenocrates discovered the number of different syllables possible in the Greek language. While unlikely, this is one of the few mentions of Combinatorics in Greece. The number they found, 1.002&nbsp;&times;&nbsp;10<sup>&nbsp;12</sup>, also seems too round to be more than a guess.<ref name="Dieudonné" /><ref name="Gow">{{cite book
| last = Gow | last = Gow
| first = James | first = James
Line 55: Line 54:
| isbn =0-8284-0218-3 }} | isbn =0-8284-0218-3 }}
</ref> </ref>

⚫ Magic squares remained an interest of China, and they began to generalise their original 3×3 square between 900 and 1300 AD. China corresponded with the Middle East about this problem in the 13th century.<ref name="Biggs" /> The Middle East also learned about binomial coefficients from Indian work, and found the connection to polynomial expansion.<ref name="Middle East">{{cite web|title=Middle East|url=http://binomial.csueastbay.edu/MidEast.html|accessdate=2008-03-08}}</ref>


] (c.953-1029) wrote on the binomial theorem and Pascal's triangle. In a now lost work known only from subsequent quotation by ], ] introduced the idea of argument by mathematical induction. ] (c.953-1029) wrote on the binomial theorem and Pascal's triangle. In a now lost work known only from subsequent quotation by ], ] introduced the idea of argument by mathematical induction.
Line 65: Line 62:


==Combinatorics in the West== ==Combinatorics in the West==
Combinatorics came to Europe in the 13th century through two mathematicians, ] and ]. Fibonacci's ] introduced many of the Arabian and Indian ideas to Europe, including that of the Fibonacci numbers.<ref name="Devlin">{{cite web Combinatorics came to Europe in the 13th century through mathematicians ] and ]. Fibonacci's ] introduced many of the Arabian and Indian ideas to Europe, including that of the Fibonacci numbers.<ref name="Devlin">{{cite web
|url=http://www.maa.org/devlin/devlin_10_02.html |url=http://www.maa.org/devlin/devlin_10_02.html
|title= The 800th birthday of the book that brought numbers to the west |title= The 800th birthday of the book that brought numbers to the west
Line 82: Line 79:
}}</ref> Jordanus was the first person to arrange the binomial coefficients in a triangle, as he did in proposition 70 of ''De Arithmetica''. This was also done in the Middle East in 1265, and China around 1300.<ref name="Biggs" /> Today, this triangle is known as ]. }}</ref> Jordanus was the first person to arrange the binomial coefficients in a triangle, as he did in proposition 70 of ''De Arithmetica''. This was also done in the Middle East in 1265, and China around 1300.<ref name="Biggs" /> Today, this triangle is known as ].


]'s contribution to the triangle that bears his name comes from his work on formal proofs about it, in addition to his connection between it and probability.<ref name="Biggs" /> Together with Leibniz and his ideas about partitions in the 17th century,<ref>Leibniz habilitation thesis '']'' was published as a book in 1666 and reprinted later''</ref> they are considered the founders of modern combinatorics.<ref name="Dickson">{{cite book ]'s contribution to the triangle that bears his name comes from his work on formal proofs about it, and the connections he made between Pascal's triangle and probability.<ref name="Biggs" /> From a letter ] sent to ] we learn that Leibniz was formally studying the mathematical theory of ] in the 17th century, although no formal work was published. Together with Leibniz, Pascal published ] in 1666 which was reprinted later. <ref>Leibniz's habilitation thesis '']'' was published as a book in 1666 and reprinted later''</ref>. Pascal and Leibniz are considered the founders of modern combinatorics.<ref name="Dickson">{{cite book
| last = Dickson | last = Dickson
| first = Leonard | first = Leonard
Line 96: Line 93:
}}</ref> }}</ref>


Both Pascal and Leibniz understood that algebra and combinatorics corresponded (] was equivalent to the ]). This was expanded by De Moivre, who found the expansion of a multinomial.<ref name="De Moivre">{{cite book Both Pascal and Leibniz understood that the ] was equivalent to the ]. The notion that algebra and combinatorics corresponded was expanded by De Moivre, who found the expansion of a multinomial.<ref name="De Moivre">{{cite book
| last = Hodgson | last = Hodgson
| first = James |author2=William Derham |author3=Richard Mead | first = James |author2=William Derham |author3=Richard Mead
Line 106: Line 103:
| year = 1708 | year = 1708
| pages = 183–191 | pages = 183–191
}}</ref> De Moivre also found the formula for derangements using the principle of inclusion-exclusion, a method different from Nikolaus Bernoulli, who had found them previously.<ref name="Biggs" /> He managed to approximate the ]s and ]. Finally, he found a closed form for the Fibonacci numbers by inventing ].<ref name="O'Connor"> }}</ref> De Moivre also found the formula for derangements using the principle of ], a method different from Nikolaus Bernoulli, who had found it previously.<ref name="Biggs" /> De Moivre also managed to approximate the ]s and ], and found a closed form for the Fibonacci numbers by inventing ].<ref name="O'Connor">
{{cite web {{cite web
|url= http://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.html |url= http://www-history.mcs.st-andrews.ac.uk/Biographies/De_Moivre.html
Line 135: Line 132:
}}</ref> }}</ref>


In the 18th century, ] worked on problems of combinatorics. In addition to working on several problems of probability which is linked to combinatorics, he worked on the ], ], ], and others. He also invented graph theory by solving the ] problem, which also led to the formation of ]. Finally, he broke ground with ] by the use of ].<ref name="Euler Archive">{{cite web In the 18th century, ] worked on problems of combinatorics, and several problems of probability which are linked to combinatorics. Problems Euler worked on include the ], ], ], and others. To solve the ] problem he invented graph theory, which also led to the formation of ]. Finally, he broke ground with ] by the use of ].<ref name="Euler Archive">{{cite web
| title = Combinatorics and probability | title = Combinatorics and probability
| url = http://math.dartmouth.edu/~euler/ | url = http://math.dartmouth.edu/~euler/
Line 141: Line 138:
}}</ref> }}</ref>


==Contemporary Combinatorics==
In the 19th century, the subject of ]s and ] originated in the work of ], ], and ]. However, it was ]'s seminal work in his book ''Lattice Theory'' published in 1967, <ref>{{cite book|last1=Birkhoff|first1=Garrett|title=Lattice theory|date=1984|publisher=American Mathematical Society|location=Providence, R.I.|isbn=978-0821810255|edition=3d ed., reprinted with corrections.}}</ref> and the work of ] that truly established the subjects. <ref name = "Stanley" >{{cite book|last1=Stanley|first1=Richard P.|title=Enumerative combinatorics.|date=2012|publisher=Cambridge University Press|location=Cambridge|isbn=1107602629|page=391-393|edition=2nd ed.}}</ref> In the 1930s, ] (1936) and ] (1935) independently stated the general Möbius inversion formula. <ref>{{cite journal|last1=Bender|first1=Edward A.|last2=Goldman|first2=J.&nbsp;R.|title=On the applications of Möbius inversion in combinatorial analysis|journal=Amer. Math. Monthly|volume=82|year=1975|pages=789–803|url=http://www.maa.org/programs/maa-awards/writing-awards/on-the-applications-of-m-bius-inversion-in-combinatorial-analysis|doi=10.2307/2319793}}</ref> In 1964, ] ''On the Foundations of Combinatorial Theory I. Theory of Miibius Functions '' introduced poset and lattice theory as theories in Combinatorics. <ref name="Stanley"/> ] has had a big impact in contemporary combinatorics for his work in matroid theory, <ref name="matroid">{{cite journal|last1=Stanley|first1=Richard|title=An introduction to hyperplane arrangements|journal=Geometric Combinatorics|date=2007|volume=13|issue=IAS/Park City Mathematics Series|pages=389–496}}</ref> for introducing Zeta polynomials,<ref>{{cite journal|last1=Stanley|first1=Richard|title=Combinatorial reciprocity theorems|journal=Advances in Math|date=1974|volume=14|pages=194–253}}</ref> for explicitly defining Eulerian posets,<ref name="eulerianposets">{{cite journal|last1=Stanley|first1=Richard|title=Some aspects of groups acting on finite posets|journal=J. Combinatorial Theory|date=1982|volume=Ser. A 32|pages=132–161}}</ref>, developing the theory of binomial posets along with Rota and Peter Doubilet,<ref>{{cite journal|last1=Stanley|first1=Richard|title=Binomial posets, M¨obius inversion, and permutation enumeration|journal=J. Combinatorial Theory|date=1976|volume=Ser. A 20|pages=336–356}}</ref> and more.
==Notes== ==Notes==
{{reflist|30em}} {{reflist|30em}}
Line 150: Line 149:
* Rashed, R. (1994). ''The development of Arabic mathematics: between arithmetic and algebra''. London. * Rashed, R. (1994). ''The development of Arabic mathematics: between arithmetic and algebra''. London.
* ] and Watkins, J. (2013). ''Combinatorics: Ancient & Modern''. Oxford. * ] and Watkins, J. (2013). ''Combinatorics: Ancient & Modern''. Oxford.
* Stanley, Richard (2012). ''Enumerative combinatorics (2nd ed. ed.)'', 2nd Edition. Cambridge University Press. {{ISBN| 1107602629}}.

{{History of science}} {{History of science}}


Revision as of 23:48, 4 March 2018

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The history of combinatorics is an area of study within the history of mathematics. Its focus ranges from antiquity to modern times.

Earliest records

The earliest recorded use of combinatorial techniques comes from problem 79 of the Rhind papyrus for the implementation of a geometric series. In 1882, Leon Rodet noticed similarities between problem 79 and Fibonacci's problem of counting the number of compositions of 1s and 2s that sum to a given total n {\displaystyle n} . A result of Xenocrates of Chalcedon (396-314 BC) possibly represents the first attempt on record to solve a difficult problem in permutations and combinatorics . The Bhagavati Sutra had the first mention of a combinatorics problem; the problem asked how many possible combinations of tastes were possible from selecting tastes in ones, twos, threes, etc. from a selection of six different tastes (sweet, pungent, astringent, sour, salt, and bitter). The Bhagavati is also the first text to mention the choose function. In the second century BC, Pingala included an enumeration problem in the Chanda Sutra (also Chandahsutra) which asked how many ways a six-syllable meter could be made from short and long notes. Pingala found the number of meters that had n {\displaystyle n} long notes and k {\displaystyle k} short notes; this is equivalent to finding the binomial coefficients.

The ideas of the Bhagavati were generalized by the Indian mathematician Mahavira in 850 AD, and Pingala's work on prosody was expanded by Bhāskara II and Hemacandra in 1100 AD. Bhaskara was the first known person to find the generalised choice function, although Brahmagupta may have known earlier. Hemacandra asked how many meters existed of a certain length if a long note was considered to be twice as long as a short note, which is equivalent to finding the Fibonacci numbers.

A hexagram

The ancient Chinese book of divination I Ching describes a hexagram as a permutation with repetitions of six lines where each line can be one of two states: solid or dashed. In describing hexagrams in this fashion they determine that there are 2 6 = 64 {\displaystyle 2^{6}=64} possible hexagrams. A Chinese monk also may have counted the number of configurations to a game similar to Go around 700 AD. Although China had relatively few advancements in enumerative combinatorics, around 100 AD they solved the Lo Shu Square which is the combinatorial design problem of the normal magic square of order three. Magic squares remained an interest of China, and they began to generalize their original 3 × 3 {\displaystyle 3\times 3} square between 900 and 1300 AD. China corresponded with the Middle East about this problem in the 13th century. The Middle East also learned about binomial coefficients from Indian work and found the connection to polynomial expansion. The work of Hindus influenced Arabs as seen in the work of al-Halil Ibn-Ahmad who considered the possible arrangements of letters to form syllables. His calculations show an understanding of permutations and combinations. In a passage from the work of Arab mathematician Umar al-Khayyami that dates to around 1100, it is corroborated that the Hindus had knowledge of binomial coefficients, but also that their methods reached the middle east.

In Greece, Plutarch wrote that Xenocrates discovered the number of different syllables possible in the Greek language. While unlikely, this is one of the few mentions of Combinatorics in Greece. The number they found, 1.002 × 10, also seems too round to be more than a guess.

Abū Bakr ibn Muḥammad ibn al Ḥusayn Al-Karaji (c.953-1029) wrote on the binomial theorem and Pascal's triangle. In a now lost work known only from subsequent quotation by al-Samaw'al, Al-Karaji introduced the idea of argument by mathematical induction.

The philosopher and astronomer Rabbi Abraham ibn Ezra (c. 1140) counted the permutations with repetitions in vocalization of Divine Name. He also established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematician Levi ben Gerson (better known as Gersonides), in 1321. The arithmetical triangle— a graphical diagram showing relationships among the binomial coefficients— was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.

Combinatorics in the West

Combinatorics came to Europe in the 13th century through mathematicians Leonardo Fibonacci and Jordanus de Nemore. Fibonacci's Liber Abaci introduced many of the Arabian and Indian ideas to Europe, including that of the Fibonacci numbers. Jordanus was the first person to arrange the binomial coefficients in a triangle, as he did in proposition 70 of De Arithmetica. This was also done in the Middle East in 1265, and China around 1300. Today, this triangle is known as Pascal's triangle.

Pascal's contribution to the triangle that bears his name comes from his work on formal proofs about it, and the connections he made between Pascal's triangle and probability. From a letter Leibniz sent to Daniel Bernoulli we learn that Leibniz was formally studying the mathematical theory of partitions in the 17th century, although no formal work was published. Together with Leibniz, Pascal published De Arte Combinatoria in 1666 which was reprinted later. . Pascal and Leibniz are considered the founders of modern combinatorics.

Both Pascal and Leibniz understood that the binomial expansion was equivalent to the choice function. The notion that algebra and combinatorics corresponded was expanded by De Moivre, who found the expansion of a multinomial. De Moivre also found the formula for derangements using the principle of principle of inclusion-exclusion, a method different from Nikolaus Bernoulli, who had found it previously. De Moivre also managed to approximate the binomial coefficients and factorial, and found a closed form for the Fibonacci numbers by inventing generating functions.

In the 18th century, Euler worked on problems of combinatorics, and several problems of probability which are linked to combinatorics. Problems Euler worked on include the Knights tour, Graeco-Latin square, Eulerian numbers, and others. To solve the Seven Bridges of Königsberg problem he invented graph theory, which also led to the formation of topology. Finally, he broke ground with partitions by the use of generating functions.

Contemporary Combinatorics

In the 19th century, the subject of partially ordered sets and lattice theory originated in the work of Dedekind, Peirce, and Schröder. However, it was Garrett Birkhoff's seminal work in his book Lattice Theory published in 1967, and the work of John von Neumann that truly established the subjects. In the 1930s, Hall (1936) and Weisner (1935) independently stated the general Möbius inversion formula. In 1964, Gian-Carlo Rota's On the Foundations of Combinatorial Theory I. Theory of Miibius Functions introduced poset and lattice theory as theories in Combinatorics. Richard P. Stanley has had a big impact in contemporary combinatorics for his work in matroid theory, for introducing Zeta polynomials, for explicitly defining Eulerian posets,, developing the theory of binomial posets along with Rota and Peter Doubilet, and more.

Notes

  1. ^ Biggs, Norman; Keith Lloyd; Robin Wilson (1995). "44". In Ronald Graham; Martin Grötschel; László Lovász (eds.). Handbook of Combinatorics (Google book). MIT Press. pp. 2163–2188. ISBN 0-262-57172-2. Retrieved 2008-03-08.
  2. Heath, Sir Thomas (1981). A history of Greek mathematics (Reprod. en fac-sim. ed.). New York: Dover. ISBN 0486240738.
  3. ^ "India". Retrieved 2008-03-05.
  4. ^ Hall, Rachel (2005-02-16). "Math for Poets and Drummers-The Mathematics of Meter" (PDF). Retrieved 2008-03-05. {{cite journal}}: Cite journal requires |journal= (help)
  5. Kulkarni, Amba. "Recursion and Combinatorial Mathematics in Chandashāstra". arXiv:math/0703658. {{cite journal}}: Cite journal requires |journal= (help)
  6. Bhaskara. "The Lilavati of Bhaskara". Brown University. Archived from the original on 2008-03-25. Retrieved 2008-03-06. {{cite web}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)
  7. ^ Dieudonné, J. "The Rhind/Ahmes Papyrus - Mathematics and the Liberal Arts". Historia Math. Truman State University. Retrieved 2008-03-06.
  8. Swaney, Mark. "Mark Swaney on the History of Magic Squares". Archived from the original on 2004-08-07.
  9. "Middle East". Retrieved 2008-03-08.
  10. Gow, James (1968). A Short History of Greek Mathematics. AMS Bookstore. p. 71. ISBN 0-8284-0218-3.
  11. The short commentary on Exodus 3:13
  12. History of Combinatorics, chapter in a textbook.
  13. Arthur T. White, ”Ringing the Cosets,” Amer. Math. Monthly 94 (1987), no. 8, 721-746; Arthur T. White, ”Fabian Stedman: The First Group Theorist?,” Amer. Math. Monthly 103 (1996), no. 9, 771-778.
  14. Devlin, Keith (October 2002). "The 800th birthday of the book that brought numbers to the west". Devlin's Angle. Retrieved 2008-03-08.
  15. "Fibonacci Sequence- History". Net Industries. 2008. Retrieved 2008-03-08.
  16. Leibniz's habilitation thesis De Arte Combinatoria was published as a book in 1666 and reprinted later
  17. Dickson, Leonard (2005) . "Chapter III". Diophantine Analysis. History of the Theory of Numbers. Mineola, New York: Dover Publications, Inc. p. 101. ISBN 0-486-44233-0.
  18. Hodgson, James; William Derham; Richard Mead (1708). Miscellanea Curiosa (Google book). Volume II. pp. 183–191. Retrieved 2008-03-08.
  19. O'Connor, John; Edmund Robertson (June 2004). "Abraham de Moivre". The MacTutor History of Mathematics archive. Retrieved 2008-03-09.
  20. Pang, Jong-Shi; Olvi Mangasarian (1999). "10.6 Generating Function". In Jong-Shi Pang (ed.). Computational Optimisation (Google book). Volume 1. Netherlands: Kluwer Academic Publishers. pp. 182–183. ISBN 0-7923-8480-6. Retrieved 2008-03-09.
  21. "Combinatorics and probability". Retrieved 2008-03-08.
  22. Birkhoff, Garrett (1984). Lattice theory (3d ed., reprinted with corrections. ed.). Providence, R.I.: American Mathematical Society. ISBN 978-0821810255.
  23. ^ Stanley, Richard P. (2012). Enumerative combinatorics (2nd ed. ed.). Cambridge: Cambridge University Press. p. 391-393. ISBN 1107602629. {{cite book}}: |edition= has extra text (help)
  24. Bender, Edward A.; Goldman, J. R. (1975). "On the applications of Möbius inversion in combinatorial analysis". Amer. Math. Monthly. 82: 789–803. doi:10.2307/2319793.
  25. Stanley, Richard (2007). "An introduction to hyperplane arrangements". Geometric Combinatorics. 13 (IAS/Park City Mathematics Series): 389–496.
  26. Stanley, Richard (1974). "Combinatorial reciprocity theorems". Advances in Math. 14: 194–253.
  27. Stanley, Richard (1982). "Some aspects of groups acting on finite posets". J. Combinatorial Theory. Ser. A 32: 132–161.
  28. Stanley, Richard (1976). "Binomial posets, M¨obius inversion, and permutation enumeration". J. Combinatorial Theory. Ser. A 20: 336–356.

References

  • N.L. Biggs, The roots of combinatorics, Historia Mathematica 6 (1979), 109-136.
  • Katz, Victor J. (1998). A History of Mathematics: An Introduction, 2nd Edition. Addison-Wesley Education Publishers. ISBN 0-321-01618-1.
  • O'Connor, John J. and Robertson, Edmund F. (1999–2004). MacTutor History of Mathematics archive. St Andrews University.
  • Rashed, R. (1994). The development of Arabic mathematics: between arithmetic and algebra. London.
  • Wilson, R. and Watkins, J. (2013). Combinatorics: Ancient & Modern. Oxford.
  • Stanley, Richard (2012). Enumerative combinatorics (2nd ed. ed.), 2nd Edition. Cambridge University Press. ISBN 1107602629.
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