Combinatorics

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Free Access Combinatorics† Lajos Takács, Lajos TakácsSearch for more papers by this author Lajos Takács, Lajos TakácsSearch for more papers by this author First published: 29 September 2014 https://doi.org/10.1002/9781118445112.stat02223 †This article was originally published online in 2006 in Encyclopedia of Statistical Sciences, © John Wiley & Sons, Inc. and republished in Wiley StatsRef: Statistics Reference Online, 2014. AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat The oldest combinatorial problems of any importance are connected with the notion of binomial coefficients. For any a the kth (k = 1, 2, …) binomial coefficient is defined as ( a k ) = a ( a − 1 ) · · · ( a − k + 1 ) k ! = a ( k ) k ! . and ( a 0 ) = 1 . The notion of binomial coefficients originates from that of figurate numbers F n k ( n ≥ 0 , k ≥ 1 ) . The numbers F 0 k , F 1 k , … can be obtained from the sequence F 0 1 = 1 , F 1 1 = 1 , … by repeated summations, as illustrated below: F n 1 : 1 , 1 , 1 , 1 , 1 , 1 , … F n 2 : 1 , 2 , 3 , 4 , 5 , 6 , … F n 3 : 1 , 3 , 6 , 10 , 15 , 21 , … Here F n 1 = 1 for n ≥ 0 and F n k + 1 = F 0 k + F 1 k + … + F n k for k ≥ 1 and n ≥ 0. The formation of triangular numbers, F n 3 ( n ≥ 0 ) , goes back to Pythagoras (ca. 580–500 B.C.). Pyramidal numbers, F n 4 ( n ≥ 0 ) , and others were studied by Nicomachus of Gerasa, who lived in the first century. (see Dickson 16). Obviously, F n k = ( n + k − 1 n ) and F n k can be interpreted as the number of different ways in which n pearls (or other indistinguishable objects) can be distributed in k boxes. It has been known for a long time that n coins can be arranged in 2n ways in a row if each coin may be either head up or tail up. If k coins are head up and n − k coins are tail up, the number of possible arrangements is ( n k ) . In a different context and for n ≤ 6, the foregoing results are in a 3000-year-old Chinese book, "I Ching" (Book of Changes). A tenth-century commentator, Halāyudha, derived the numbers ( n k ) ( 0 ≤ k ≤ n ) from a passage in the manuscript "Chandah-sūtra" by Piṅgala (ca. 200 B.C.), a Hindu writer on metrical forms. The numbers ( n k ) , 0 ≤ k ≤ n , gained importance when it was recognized that they appear as coefficients in the expansion of (1 + x)n. Apparently, this was known to Omar Khayyam, a Persian poet and mathematician who lived in the eleventh century, (see Woepcke 49). He did not give a law, but in his "Algebra" (ca. 1100) he mentioned that the law can be found in another work by him, which work, if it ever existed, has been lost. In 1303, Shih-chieh Chu 39 refers to the numbers ( n k ) , 0 ≤ k ≤ n , as an old invention and he mentions several surprising identities for these numbers (see Mikami 27, Needham 29, and Takács 43). The numbers ( n k ) . 0 ≤ k ≤ n , arranged in the form of a triangular array, appeared in 1303 at the front of the book of Shih-chieh Chu. Unfortunately, the original book was lost, but in 1839 Shin-lin Lo discovered a copy made in Korea in 1660. The triangular array first appeared in print in 1527 on the title page of the book of P. Apianus (see Smith [40, p. 509]). In 1544, M. Stifel showed that in the binomial expansion ( 1 + x ) n = ∑ k = 0 n C n k x k . the coefficients C n k ( 0 ≤ k ≤ n ) can be obtained by the recurrence equation C n + 1 k + 1 = C n k + 1 + C n k , where C n 0 = C n n = 1 for n = 0, 1, 2, …. He arranged the coefficients C n k ( 0 ≤ k ≤ n ) in the following triangular array, which is known as the arithmetic triangle: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 · · · · · · In 1556, N. Tartaglia claimed this triangular array as his own invention. The numbers C n k ( 0 ≤ k ≤ n ) appeared in the seventeenth century in connection with combinations. The number of combinations without repetition of n objects taken k at a time can be expressed as C n k . In 1634, P. Hérigone knew that C n k = n ( n − 1 ) … ( n − k + 1 ) / k ! . The same formula appeared also in 1665 in a treatise by B. Pascal 33. In those early days no mathematical notation was used for the binomial coefficients. It seems that L. Euler (1707–1783) was the first (1781) who used ( a k ) and later [ a k ] for ( a k ) . The notation ( a k ) , which is a slight modification of Euler's first notation, was introduced in 1827 by A. V. Ettingshausen. The combinatorial formula C n k = ( n k ) gained significance in the middle of the seventeenth century, when P. Fermat (1601–1665) and B. Pascal (1623–1662) solved the classical problem of points (division problem). In this problem two players A and B play a series of games. In each game, independently of the others, either A wins a point with probability 1 2 , or B wins a point with probability 1 2 . The players agree to continue the games until one has won a predetermined number of games. However, the match has to stop when A still needs a points and B still needs b points to win the series. In what proportion should the stakes be divided? According to Ore 31 it seems likely that the problem is of Arabic origin. He found some particular versions of this problem in Italian manuscripts dating from as early as 1380. The problem appears for the first time in printed form in 1494 in the book of Lucas de Burgo Pacioli [32, p. 197]. In Pacioli's version a = 1 and b = 3. L. Pacioli (1494), N. Tartaglia (1556), F. Peverone (1558), and others tried to solve this problem without success. In 1654, Antoine Gombauld Chevalier de Méré, a distinguished philosopher and a prominent figure at the court of Louis XIV, called the attention of Blaise Pascal to the problem of points. It seems that Pascal started working on a solution and communicated the problem to Pierre de Fermat. In reply, Fermat found a remarkably elegant solution. He demonstrated that P ( a , b ) = 1 2 a + b − 1 ∑ k = a a + b − 1 ( a + b − 1 k ) is the probability that A wins the series. As a second solution, Fermat also proved that P ( a , b ) = ∑ n = a − 1 a + b − 2 ( n a − 1 ) 1 2 n + 1 . In the meantime Pascal discovered the recurrence formula P ( a , b ) = 1 2 P ( a − 1 , b ) + 1 2 P ( a , b − 1 ) and found the same probabilities as Fermat. The term combinatorics was introduced by G. W. Leibniz (1646–1716) in 1666 23 and he gave a systematic study of the subject. The basic combinatorial problems are concerned with the enumeration of the possible arrangements of several objects under various conditions. The number of ordered arrangements of n distinct objects, numbered 1, 2, …, n, without repetition is n! = 1 · 2…n. Each such arangement is called a permutation without repetition. If only k objects are chosen among n distinct objects, numbered 1, 2, …, n, the number of ordered arrangements without repetition is n(n − 1) ⋯ (n − k + 1), and each arrangement is called a variation without repetition. The number of unordered arrangements of k objects chosen among n distinct objects, numbered 1, 2, …, n, is ( n k ) , and each arrangement is called a combination of size k of n elements without repetition. Permutations, variations, and combinations with repetition are defined in a similar way. In about 1680 4 Jacob Bernoulli (1654–1705) found that the probability that in n independent and identical trials, a given event, which has probability p, occurs exactly k times is given by P ( n , k ) = ( n k ) p k ( 1 − p ) n − k for k = 0, 1, …, n. In 1693, John Wallis (1616–1703) published a textbook of algebra 48 and devoted a considerable part of the book to combinatorial problems. At the beginning of the eighteenth century the applications of combinatorial methods expanded with the rapid growth of probability theory. The works of P. R. Montmort (1678–1719), Nicolas Bernoulli (1687–1759), and A. De Moivre (1667–1754) contain many significant contributions to combinatorics. For a detailed account of their works, see Todhunter 46. Here are a few samples. In 1708, A. De Moivre studied the problem of duration of plays in the theory of games of chance and obtained the following significant result in combinatorics: One can arrange k + j letters A and j letters B in L ( j , k ) = k k + 2 j ( k + 2 j j ) ways such that for every r = 1, 2, …, k + 2j among the first r letters, there are more A's than B's. In 1713, P. R. Montmort 28 studied the "problème de rencontre" and obtained an important result in combinatorics. Denote by Q(n, k) the number of permutations of the elements 1, 2, …, n in which exactly k matches occur. If the ith element of a permutation of 1, 2, …, n is i, then it is said that there is a match at the ith place. Montmort proved that Q ( n , k ) = ( n k ) Q ( n − k , 0 ) and if Q(n) = Q(n, 0), then Q(n + 2) = [Q(n + 1) + Q(n)](n + 1), where Q(0) = 1 and Q(1) = 0. Hence it follows that Q ( n ) = n ! ∑ i = 0 n ( − 1 ) i i ! . In 1708 A. De Moivre, and in 1710 P. R. Montmort, discovered that the number of ways in which one can obtain a total number of k points by throwing n dice is C ( n , k ) = ∑ j = 0 [ ( k − n ) / 6 ] ( − 1 ) j ( n j ) ( k − 1 − 6 j n − 1 ) if n ≤ k ≤ 6n. This problem had already been studied by G. Cardano (1501–1576), and Galileo (1564–1642) for n = 2 and n = 3. In 1730, A. De Moivre 14 proved the foregoing formula by observing that C(n, k) is the coefficient of xk in the expansion of (x + x2 + …x6)n. Today De Moivre's method is called the method of generating functions. In the hands of L. Euler (1707–1783), J. L. Lagrange (1736–1813), and P. S. Laplace (1749–1827), the method of generating functions became a powerful tool of combinatorics. In 1730, J. Stirling (1692–1770) introduced the numbers S(n, k)(0 ≤ k ≤ n) and 𝒮 (n, k) (0 ≤ k ≤ n) defined by the equations x [ n ] = ∑ j = 0 n S ( n , j ) x j and x n = ∑ j = 0 n 𝒮 ( n , j ) x ( j ) . (See Difference of Zero.) The numbers S(n, k)(0 ≤ k ≤ n) and 𝒮(n, k)(0 ≤ k ≤ n), which are now called Stirling numbers of the first kind and second kind, respectively, have significant importance in combinatorics (see Ch. Jordan 21). In 1736, L. Euler introduced some numbers A(n, k)(0 ≤ k < n) which are now called Eulerian numbers. In 1883, J. Worpitzky demonstrated that x n = ∑ k = 0 n − 1 A ( n , k ) ( x + k n ) for n ≥ 1 and for every x. In an n × n triangular grid in which the ith row contains i squares (i = 1, 2, …, n), one can put a mark in a square in each row of the grid in A (n, k) ways so that there are exactly k empty columns (see Takács 44). In 1908, P. A. MacMahon, by solving a problem of Simon Newcomb concerning a card game, proved that a deck of n cards numbered 1, 2, …, n may be dealt into k + 1 piles in A (n, k) ways if cards are placed in one pack as long as they are in decreasing order of magnitude. In 1879, D. André proved that if A (n) is the number of such permutations i1, i2, …, in of 1, 2, …, n for which i1 < i2, i2 > i3, i3 < i4, …, then ∑ n = 0 ∞ A ( n ) x n n ! = sec x + tan x . Extending the work of A. M. Legendre (1798) on the method of inclusion and exclusion, in 1867 C. Jordan 20 proved the following general theorem: Let Ω be a finite set and A1, A2, …, An be subsets of Ω. Denote by Hk the set that contains all those elements of Ω that belong to exactly k sets among A1, A2, …, An. If N (A) denotes the number of elements in any set A, then N ( H k ) = ∑ j = k n ( − 1 ) j − k ( j k ) S j for k = 0, 1, …, n, where S0 = N(Ω) and S j = ∑ 1 ≤ i 1 < i 2 < … < i j ≤ n N ( A i 1 ∩ A i 2 ∩ … ∩ A i j ) for 1 ≤ j ≤ n. In the particular case where k = 0, the method of inclusion and exclusion was used by A. M. Legendre (1798), D. A. da Silva (1854), and J. J. Sylvester (1883) in number theory. In combinatorics and in probability theory the method of inclusion and exclusion has been used extensively (see Takács 41). Such problems as the "problème des ménages," which was proposed by E. Lucas in 1891, have an easy solution by using the method of inclusion and exclusion. The "problème des ménages" is as follows. At a dinner party n married couples are seated in 2n seats at a round table according to the following pattern: the women take alternate seats and the men choose the remaining seats. The problem is to determine the number of sitting arrangements in which exactly k men are sitting next to their wives. Here are a few famous problems that originated in the eighteenth and nineteenth centuries. In 1751, L. Euler discovered that the number of different ways of dissecting a convex polygon of n sides into n − 2 triangles by n − 3 nonintersecting diagonals is D n = ( 2 n − 3 n − 2 ) 1 2 n − 3 for n ≥ 3. This formula was proved by J. A. de Segner in 1758. Other proofs and generalizations were given by N. v. Fuss (1793), G. Lamé (1838), E. Catalan (1838), O. Rodrigues (1838), J. Binet (1839), J. A. Grunert (1841), J. Liouville (1843), E. Schröder (1870), A. Cayley (1890), and others. In several papers, T. P. Kirkman (1857, 1860) gave far-reaching generalizations of Euler's problem. Some recent developments of the subject are discussed by T. Motzkin (1948), W. T. Tutte (47, ref. 47), W. G. Brown (1964; 1965, ref. 10), and others. In 1838, E. Catalan proved that the number of ways a product of n factors can be calculated by pairs is Dn+1. This result was extended by J. H. M. Wedderburn (1923), I. M. H. Etherington (1937), G. N. Raney (1960), and others. In the years 1859–1889, A. Cayley solved several combinatorial problems concerning some special graphs which he appropriately called trees (see Tree‐structured methods). His results made it possible to determine the number of possible isomers of certain hydrocarbons. Cayley's work was an inspiration to Pólya 34, who in 1937 worked out a general method for the enumeration of trees, graphs, groups and chemical structures (see also de Bruijn 13). In 1748, L. Euler proved that if p (n) is the number of ways in which n can be represented as a sum of any number of positive integers, then 1 + ∑ n = 1 ∞ p ( n ) x n = ∏ n = 1 ∞ ( 1 − x n ) − 1 for |x| < 1. By using this equation G. H. Hardy and S. Ramanujan (1918) expressed p (n) in the form of an asymptotic series which made it possible to calculate p (n) for every n. In 1937, H. Rademacher expressed p (n) in the form of a convergent series. In 1779, L. Euler proposed the following problem. n officers of n different ranks are chosen from each of n different regiments. It is required to arrange them in a square array so that no officer of the same rank or of the same regiment shall be in the same row or in the same column. In the case of n = 6, this problem is known as "the problem of the 36 officers." Euler conjectured that the problem of the n2 officers has no solution if n ≡ 2 (mod 4). If n = 2, then obviously there is no solution. In 1901, G. Tarry proved that there is no solution for n = 6. In 1959, Bose and Shrikhande 6 demonstrated that Euler's conjecture is false for n = 22, and in 1960, Bose et al. 9 proved that Euler's conjecture is false for n > 6. Euler's problem led to the problems of orthogonal Latin squares and combinatorial designs which have great importance in the theory of design of experiments. A particularly important topic is that of balanced incomplete block designs. A balanced incomplete block design is an arrangement of v distinct objects (varieties) into b blocks (sets) in such a way that each block contains exactly k objects, each object occurs in exactly r different blocks, and every pair of distinct objects occurs together in exactly λ blocks. Such a design is a (b, v, r, k, λ)-configuration. The integers b, v, r, k, λ cannot be chosen arbitarily. They should satisfy the requirements λ > 0, v − 1 > k, r(k − 1) = λ(v − 1) and bk = vr. The main problem is to find conditions for the existence of various types of configurations and to give methods for the construction of an actual design. The aforementioned problem is of recreational origin and is associated with the triple systems of T. P. Kirkman (1847, 1850) and J. Steiner (1853). Kirkman's 15 schoolgirls problem is famous and it runs as follows. A schoolteacher takes her class of 15 girls on a daily walk. The girls are arranged in 5 rows of 3 each, so that each girl has 2 companions. The problem is to arrange the girls so that for 7 consecutive days no girl will walk with one of her companions in a triplet more than once. The basic paper on block designs is Bose 5. Further references are Bose and Shrikhande 7, Fisher 18, Mann 26, Ryser 38, Hall 19, and several others. The book by Dénes and Keedwell 15 is a comprehensive study of Latin squares. Recreational combinatorial problems are discussed in many books. The two-volume work of Ahrens 1 and the book of Rouse Ball and Coxeter 37 are inspiring. Here is an example for recreational problems. In 1850, F. Nauck proposed the problem of finding the number of ways in which eight queens can be placed on a chessboard so that no queen can take any other. There are 92 solutions which can be obtained from 12 fundamental solutions by rotation and reflection. In 1901 the first textbook on combinatorics by Netto 30 appeared. The book is an excellent collection of classical problems and methods. One of the most active researchers in combinatorics at the turn of the century was MacMahon (1854–1929). His two-volume book Combinatory Analysis 24 appeared in 1915 and 1916. His 1500-page Collected Papers 25 on combinatorics was published in 1978. Symmetric functions, lattice permutations, and partition problems form the bulk of his work in combinatorics. In the twentieth century, combinatorics penetrated many fields of mathematics. In probability theory, random walks, occupancy problems and fluctuation problems require the use of combinatorial methods. In mathematical statistics, order statistics* and theory of sampling are based on combinatorial methods. In graph theory and in number theory, combinatorial methods are abundant. Combinatorial geometry is one of the recent fields of mathematics. The outstanding work of Gian-Carlo Rota and his collaborators on the foundations of combinatorial theory amply demonstrates the importance of combinatorics in all fields of mathematics. Most of the articles in two recent journals, the Journal of Combinatorial Theory (started in 1966) and the Journal of Graph Theory (started in 1977), are devoted to new developments in combinatorics. Related Articles See also Hypergeometric Distributions: Overview; Urn Models; Stirling Numbers and Generalizations; Latin Squares, Latin Cubes, Latin Rectangles; Graeco‐Latin Squares; Orthogonal Designs; Optimal Design of Experiments; Nearly Balanced Designs; Magic square designs; Multinomial Coefficients; Binomial distribution; Trees, probabilistic functional; Probability, History of; Geometric Probability Theory; Foundations of Probability ‐ Historical. References 1Ahrens, W. (1910, 1918). Mathematische Unterhaltungen und Spiele, 2nd ed., Vols. 1 and 2. B. G. Teubner, Leipzig. 2Andrews, G. E. (1976). The Theory of Partitions. Encyclopedia of Mathematics and Its Applications, Vol. 2. Addison-Wesley, Reading Mass. 3 E. F. Beckenbach, ed. (1964). Applied Combinatorial Mathematics. Wiley, New York. 4Bernoulli, J. (1713). Ars Conjectandi (Opus posth.). Basel. (Reprint: Culture et Civilisation, Bruxelles, 1968.) 5Bose, R. C. (1939). Ann. Eugen. (Lond.), 9, 353– 399. 6Bose, R. 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De Algebra Tractatus. Oxford. (Reprinted in John Wallis, Opera Mathematica, Vol. 2. Georg Olms Verlag, Hildesheim, 1972.) 49Woepcke, F. (1851). L'Algèbre d'Omar Alkâyyami (French translation). Duprat, Paris. Wiley StatsRef: Statistics Reference OnlineBrowse other articles of this reference work:BROWSE BY TOPICBROWSE A-Z ReferencesRelatedInformation

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