Chapter 2

See P. H. H. Massy inGeog. Journ.(Sept. 1905); E. Chantre,Mission en Cappadocie(1898).

(D. G. H.)

COMANA(mod.Gumenek), an ancient city of Pontus, said to have been colonized from Comana in Cappadocia. It stood on the river Iris (Tozanli Su or Yeshil Irmak), and from its central position was a favourite emporium of Armenian and other merchants. The moon-goddess was worshipped in the city with a pomp and ceremony in all respects analogous to those employed in the Cappadocian city. The slaves attached to the temple alone numbered not less than 6000. St John Chrysostom died there on the way to Constantinople from his exile at Cocysus in the Anti-Taurus. Remains of Comana are still to be seen near a village called Gumenek on the Tozanli Su, 7 m. from Tokat, but they are of the slightest description. There is a mound; and a few inscriptions are built into a bridge, which here spans the river, carrying the road from Niksar to Tokat.

(D. G. H.)

COMANCHES,a tribe of North American Indians of Shoshonean stock, so called by the Spaniards, but known to the French as Padoucas, an adaptation of their Sioux name, and among themselvesnimenim(people). They number some 1400, attached to the Kiowa agency, Oklahoma. When first met by Europeans, they occupied the regions between the upper waters of the Brazos and Colorado on the one hand, and the Arkansas and Missouri on the other. Until their final surrender in 1875 the Comanches were the terror of the Mexican and Texan frontiers, and were always famed for their bravery. They were brought to nominal submission in 1783 by the Spanish general Anza, who killed thirty of their chiefs. During the 19th century they were always raiding and fighting, but in 1867, to the number of 2500, they agreed to go on a reservation. In 1872 a portion of the tribe, the Quanhada or Staked Plain Comanches, had again to be reduced by military measures.

COMAYAGUA,the capital of the department of Comayagua in central Honduras, on the right bank of the river Ulua, and on the interoceanic railway from Puerto Cortes to Fonseca Bay. Pop. (1900) about 8000. Comayagua occupies part of a fertile valley, enclosed by mountain ranges. Under Spanish rule it was a city of considerable size and beauty, and in 1827 its inhabitants numbered more than 18,000. A fine cathedral, dating from 1715, is the chief monument of its former prosperity, for most of the handsome public buildings erected in the colonial period have fallen into disrepair. The present city chiefly consists of low adobe houses and cane huts, tenanted by Indians. The university founded in 1678 has ceased to exist, but there is a school of jurisprudence. In the neighbourhood are many ancient Indian ruins (seeCentral America:Archaeology).

Founded in 1540 by Alonzo Caceres, who had been instructedby the Spanish government to find a site for a city midway between the two oceans, Valladolid la Nueva, as the town was first named, soon became the capital of Honduras. It received the privileges of a city in 1557, and was made an episcopal see in 1561. Its decline dates from 1827, when it was burned by revolutionaries; and in 1854 its population had dwindled to 2000. It afterwards suffered through war and rebellion, notably in 1872 and 1873, when it was besieged by the Guatemalans. In 1880 Tegucigalpa (q.v.), a city 37 m. east-south-east, superseded it as the capital of Honduras.

COMB(a word common in various forms to Teut. languages, cf. Ger.Kamm, the Indo-Europ. origin of which is seen inγόμφος, a peg or pin, and Sanskrit,gambhas, a tooth), a toothed article of the toilet used for cleaning and arranging the hair, and also for holding it in place after it has been arranged; the word is also applied, from resemblance in form or in use, to various appliances employed for dressing wool and other fibrous substances, to the indented fleshy crest of a cock, and to the ridged series of cells of wax filled with honey in a beehive. Hair combs are of great antiquity, and specimens made of wood, bone and horn have been found in Swiss lake-dwellings. Among the Greeks and Romans they were made of boxwood, and in Egypt also of ivory. For modern combs the same materials are used, together with others such as tortoise-shell, metal, india-rubber and celluloid. There are two chief methods of manufacture. A plate of the selected material is taken of the size and thickness required for the comb, and on one side of it, occasionally on both sides, a series of fine slits are cut with a circular saw. This method involves the loss of the material cut out between the teeth. The second method, known as “twinning” or “parting,” avoids this loss and is also more rapid. The plate of material is rather wider than before, and is formed into two combs simultaneously, by the aid of a twinning machine. Two pairs of chisels, the cutting edges of which are as long as the teeth are required to be and are set at an angle converging towards the sides of the plate, are brought down alternately in such a way that the wedges removed from one comb form the teeth of the other, and that when the cutting is complete the plate presents the appearance of two combs with their teeth exactly inosculating or dovetailing into each other. In india-rubber combs the teeth are moulded to shape and the whole hardened by vulcanization.

COMBACONUM,orKumbakonam, a city of British India, in the Tanjore district of Madras, in the delta of the Cauvery, on the South Indian railway, 194 m. from Madras. Pop. (1901) 59,623, showing an increase of 10% in the decade. It is a large town with wide and airy streets, and is adorned with pagodas, gateways and other buildings of considerable pretension. The greatgopuram, or gate-pyramid, is one of the most imposing buildings of the kind, rising in twelve stories to a height of upwards of 100 ft., and ornamented with a profusion of figures of men and animals formed in stucco. One of the water-tanks in the town is popularly reputed to be filled with water admitted from the Ganges every twelve years by a subterranean passage 1200 m. long; and it consequently forms a centre of attraction for large numbers of devotees. The city is historically interesting as the capital of the Chola race, one of the oldest Hindu dynasties of which any traces remain, and from which the whole coast of Coromandel, or more properly Cholamandal, derives its name. It contains a government college. Brass and other metal wares, silk and cotton cloth and sugar are among the manufactures.

COMBE, ANDREW(1797-1847), Scottish physiologist, was born in Edinburgh on the 27th of October 1797, and was a younger brother of George Combe. He served an apprenticeship in a surgery, and in 1817 passed at Surgeons’ Hall. He proceeded to Paris to complete his medical studies, and whilst there he investigated phrenology on anatomical principles. He became convinced of the truth of the new science, and, as he acquired much skill in the dissection of the brain, he subsequently gave additional interest to the lectures of his brother George, by his practical demonstrations of the convolutions. He returned to Edinburgh in 1819 with the intention of beginning practice; but being attacked by the first symptoms of pulmonary disease, he was obliged to seek health in the south of France and in Italy during the two following winters. He began to practise in 1823, and by careful adherence to the laws of health he was enabled to fulfil the duties of his profession for nine years. During that period he assisted in editing thePhrenological Journaland contributed a number of articles to it, defended phrenology before the Royal Medical Society of Edinburgh, published hisObservations on Mental Derangement(1831), and prepared the greater portion of hisPrinciples of Physiology Applied to Health and Education, which was issued in 1834, and immediately obtained extensive public favour. In 1836 he was appointed physician to Leopold I., king of the Belgians, and removed to Brussels, but he speedily found the climate unsuitable and returned to Edinburgh, where he resumed his practice. In 1836 he published hisPhysiology of Digestion, and in 1838 he was appointed one of the physicians extraordinary to the queen in Scotland. Two years later he completed hisPhysiological and Moral Management of Infancy, which he believed to be his best work and it was his last. His latter years were mostly occupied in seeking at various health resorts some alleviation of his disease; he spent two winters in Madeira, and tried a voyage to the United States, but was compelled to return within a few weeks of the date of his landing at New York. He died at Gorgie, near Edinburgh, on the 9th of August 1847.

His biography, written by George Combe, was published in 1850.

COMBE, GEORGE(1788-1858), Scottish phrenologist, elder brother of the above, was born in Edinburgh on the 21st of October 1788. After attending Edinburgh high school and university he entered a lawyer’s office in 1804, and in 1812 began to practise on his own account. In 1815 theEdinburgh Reviewcontained an article on the system of “craniology” of F. J. Gall and K. Spurzheim, which was denounced as “a piece of thorough quackery from beginning to end.” Combe laughed like others at the absurdities of this so-called new theory of the brain, and thought that it must be finally exploded after such an exposure; and when Spurzheim delivered lectures in Edinburgh, in refutation of the statements of his critic, Combe considered the subject unworthy of serious attention. He was, however, invited to a friend’s house where he saw Spurzheim dissect the brain, and he was so far impressed by the demonstration that he attended the second course of lectures. Investigating the subject for himself, he became satisfied that the fundamental principles of phrenology were true—namely “that the brain is the organ of mind; that the brain is an aggregate of several parts, each subserving a distinct mental faculty; and that the size of the cerebral organ is,caeteris paribus, an index of power or energy of function.” In 1817 his first essay on phrenology was published in theScots Magazine; and a series of papers on the same subject appeared soon afterwards in theLiterary and Statistical Magazine; these were collected and published in 1819 in book form asEssays on Phrenology, which in later editions becameA System of Phrenology. In 1820 he helped to found the Phrenological Society, which in 1823 began to publish aPhrenological Journal. By his lectures and writings he attracted public attention to the subject on the continent of Europe and in America, as well as at home; and a long discussion with Sir William Hamilton in 1827-1828 excited general interest.

His most popular work,The Constitution of Man, was published in 1828, and in some quarters brought upon him denunciations as a materialist and atheist. From that time he saw everything by the light of phrenology. He gave time, labour and money to help forward the education of the poorer classes; he established the first infant school in Edinburgh; and he originated a series of evening lectures on chemistry, physiology, history and moral philosophy. He studied the criminal classes, and tried to solve the problem how to reform as well as to punish them; and he strove to introduce into lunatic asylums a humane system of treatment. In 1836 he offered himself as a candidate for the chair of logic at Edinburgh, but was rejected in favour of Sir William Hamilton. In 1838 he visited America and spent about two years lecturing on phrenology, education and thetreatment of the criminal classes. On his return in 1840 he published hisMoral Philosophy, and in the following year hisNotes on the United States of North America. In 1842 he delivered, in German, a course of twenty-two lectures on phrenology in the university of Heidelberg, and he travelled much in Europe, inquiring into the management of schools, prisons and asylums. The commercial crisis of 1855 elicited his remarkable pamphlet onThe Currency Question(1858). The culmination of the religious thought and experience of his life is contained in his workOn the Relation between Science and Religion, first publicly issued in 1857. He was engaged in revising the ninth edition of theConstitution of Manwhen he died at Moor Park, Farnham, on the 14th of August 1858. He married in 1833 Cecilia Siddons, a daughter of the great actress.

COMBE, WILLIAM(1741-1823), English writer, the creator of “Dr Syntax,” was born at Bristol in 1741. The circumstances of his birth and parentage are somewhat doubtful, and it is questioned whether his father was a rich Bristol merchant, or a certain William Alexander, a London alderman, who died in 1762. He was educated at Eton, where he was contemporary with Charles James Fox, the 2nd Baron Lyttelton and William Beckford. Alexander bequeathed him some £2000—a little fortune that soon disappeared in a course of splendid extravagance, which gained him the nickname of Count Combe; and after a chequered career as private soldier, cook and waiter, he finally settled in London (about 1771), as a law student and bookseller’s hack. In 1776 he made his first success in London withThe Diaboliad, a satire full of bitter personalities. Four years afterwards (1780) his debts brought him into the King’s Bench; and much of his subsequent life was spent in prison. His spuriousLetters of the Late Lord Lyttelton1(1780) imposed on many of his contemporaries, and a writer in theQuarterly Review, so late as 1851, regarded these letters as authentic, basing upon them a claim that Lyttelton was “Junius.” An early acquaintance with Lawrence Sterne resulted in hisLetters supposed to have been written by Yorick and Eliza(1779). Periodical literature of all sorts—pamphlets, satires, burlesques, “two thousand columns for the papers,” “two hundred biographies”—filled up the next years, and about 1789 Combe was receiving £200 yearly from Pitt, as a pamphleteer. Six volumes of aDevil on Two Sticks in Englandwon for him the title of “the English le Sage”; in 1794-1796 he wrote the text for Boydell’sHistory of the River Thames; in 1803 he began to write forThe Times. In 1809-1811 he wrote for Ackermann’sPolitical Magazinethe famousTour of Dr Syntax in search of the Picturesque(descriptive and moralizing verse of a somewhat doggerel type), which, owing greatly to Thomas Rowlandson’s designs, had an immense success. It was published separately in 1812 and was followed by two similarTours, “in search of Consolation,” and “in search of a Wife,” the first Mrs Syntax having died at the end of the firstTour. Then cameSix Poemsin illustration of drawings by Princess Elizabeth (1813),The English Dance of Death(1815-1816),The Dance of Life(1816-1817),The Adventures of Johnny Quae Genus(1822)—all written for Rowlandson’s caricatures; together withHistoriesof Oxford and Cambridge, and of Westminster Abbey for Ackermann;Picturesque Toursalong the Rhine and other rivers,Histories of Madeira,Antiquities of York, texts forTurner’s Southern Coast Views, and contributions innumerable to theLiterary Repository. In his later years, notwithstanding a by no means unsullied character, Combe was courted for the sake of his charming conversation and inexhaustible stock of anecdote. He died in London on the 19th of June 1823.

Brief obituary memoirs of Combe appeared in Ackermann’sLiterary Repositoryand in theGentleman’s Magazinefor August 1823; and in May 1859 a list of his works, drawn up by his own hand, was printed in the latter periodical. See alsoDiary of H. Crabb Robinson,Notes and Queries for 1869.

1Thomas, 2nd Baron Lyttelton (1744-1779), commonly known as the “wicked Lord Lyttelton,” was famous for his abilities and his libertinism, also for the mystery attached to his death, of which it was alleged he was warned in a dream three days before the event.

COMBE,orCoomb, a term particularly in use in south-western England for a short closed-in valley, either on the side of a down or running up from the sea. It appears in place-names as a termination,e.g.Wiveliscombe, Ilfracombe, and as a prefix,e.g.Combemartin. The etymology of the word is obscure, but “hollow” seems a common meaning to similar forms in many languages. In English “combe” or “cumb” is an obsolete word for a “hollow vessel,” and the like meaning attached to Teutonic formskummandkumme. The Welshcwm, in place-names, means hollow or valley, with which may be comparedcumin many Scots place-names. The Greekκύμβηalso means a hollow vessel, and there is a French dialect wordcombemeaning a little valley.

COMBERMERE, STAPLETON COTTON,1st Viscount(1773-1865), British field-marshal and colonel of the 1st Life Guards, was the second son of Sir Robert Salusbury Cotton of Combermere Abbey, Cheshire, and was born on the 14th of November 1773, at Llewenny Hall in Denbighshire. He was educated at Westminster School, and when only sixteen obtained a second lieutenancy in the 23rd regiment (Royal Welsh Fusiliers). A few years afterwards (1793) he became by purchase captain in the 6th Dragoon Guards, and he served in this regiment during the campaigns of the duke of York in Flanders. While yet in his twentieth year, he joined the 25th Light Dragoons (subsequently 22nd) as lieutenant-colonel, and, while in attendance with his regiment on George III. at Weymouth, he became a great favourite of the king. In 1796 he went with his regiment to India, taking parten routein the operations in Cape Colony (July-August 1796), and in 1799 served in the war with Tippoo Sahib, and at the storming of Seringapatam. Soon after this, having become heir to the family baronetcy, he was, at his father’s desire, exchanged into a regiment at home, the 16th Light Dragoons. He was stationed in Ireland during Emmett’s insurrection, became colonel in 1800, and major-general five years later. From 1806 to 1814 he was M.P. for Newark. In 1808 he was sent to the seat of war in Portugal, where he shortly rose to the position of commander of Wellington’s cavalry, and it was here that he most displayed that courage and judgment which won for him his fame as a cavalry officer. He succeeded to the baronetcy in 1809, but continued his military career. His share in the battle of Salamanca (22nd of July 1812) was especially marked, and he received the personal thanks of Wellington. The day after, he was accidentally wounded. He was now a lieutenant-general in the British army and a K.B., and on the conclusion of peace (1814) was raised to the peerage under the style of Baron Combermere. He was not present at Waterloo, the command, which he expected, and bitterly regretted not receiving, having been given to Lord Uxbridge. When the latter was wounded Cotton was sent for to take over his command, and he remained in France until the reduction of the allied army of occupation. In 1817 he was appointed governor of Barbadoes and commander of the West Indian forces. From 1822 to 1825 he commanded in Ireland. His career of active service was concluded in India (1826), where he besieged and took Bhurtpore—a fort which twenty-two years previously had defied the genius of Lake and was deemed impregnable. For this service he was created Viscount Combermere. A long period of peace and honour still remained to him at home. In 1834 he was sworn a privy councillor, and in 1852 he succeededWellingtonas constable of the Tower and lord lieutenant of the Tower Hamlets. In 1855 he was made a field-marshal and G.C.B. He died at Clifton on the 21st of February 1865. An equestrian statue in bronze, the work of Baron Marochetti, was raised in his honour at Chester by the inhabitants of Cheshire. Combermere was succeeded by his only son, Wellington Henry (1818-1891), and the viscountcy is still held by his descendants.

See Viscountess Combermere and Captain W. W. Knollys,The Combermere Correspondence(London, 1866).

COMBES, [JUSTIN LOUIS] ÉMILE(1835- ), French statesman, was born at Roquecourbe in the department of the Tarn. He studied for the priesthood, but abandoned the idea before ordination, and took the diploma of doctor of letters (1860),then he studied medicine, taking his degree in 1867, and setting up in practice at Pons in Charente-Inférieure. In 1881 he presented himself as a political candidate for Saintes, but was defeated. In 1885 he was elected to the senate by the department of Charente-Inférieure. He sat in the Democratic left, and was elected vice-president in 1893 and 1894. The reports which he drew up upon educational questions drew attention to him, and on the 3rd of November 1895 he entered the Bourgeois cabinet as minister of public instruction, resigning with his colleagues on the 21st of April following. He actively supported the Waldeck-Rousseau ministry, and upon its retirement in 1903 he was himself charged with the formation of a cabinet. In this he took the portfolio of the Interior, and the main energy of the government was devoted to the struggle with clericalism. The parties of the Left in the chamber, united upon this question in theBloc republicain, supported Combes in his application of the law of 1901 on the religious associations, and voted the new bill on the congregations (1904), and under his guidance France took the first definite steps toward the separation of church and state. He was opposed with extreme violence by all the Conservative parties, who regarded the secularization of the schools as a persecution of religion. But his stubborn enforcement of the law won him the applause of the people, who called him familiarlyle petit père. Finally the defection of the Radical and Socialist groups induced him to resign on the 17th of January 1905, although he had not met an adverse vote in the Chamber. His policy was still carried on; and when the law of the separation of church and state was passed, all the leaders of the Radical parties entertained him at a noteworthy banquet in which they openly recognized him as the real originator of the movement.

COMBINATION(Lat.combinare, to combine), a term meaning an association or union of persons for the furtherance of a common object, historically associated with agreements amongst workmen for the purpose of raising their wages. Such a combination was for a long time expressly prohibited by statute. SeeTrade Unions; alsoConspiracyandStrikes and Lock Outs.

COMBINATORIAL ANALYSIS.The Combinatorial Analysis, as it was understood up to the end of the 18th century, was of limited scope and restricted application. P. Nicholson, in hisEssays on the Combinatorial Analysis, publishedHistorical Introduction.in 1818, states that “the Combinatorial Analysis is a branch of mathematics which teaches us to ascertain and exhibit all the possible ways in which a given number of things may be associated and mixed together; so that we may be certain that we have not missed any collection or arrangement of these things that has not been enumerated.” Writers on the subject seemed to recognize fully that it was in need of cultivation, that it was of much service in facilitating algebraical operations of all kinds, and that it was the fundamental method of investigation in the theory of Probabilities. Some idea of its scope may be gathered from a statement of the parts of algebra to which it was commonly applied, viz., the expansion of a multinomial, the product of two or more multinomials, the quotient of one multinomial by another, the reversion and conversion of series, the theory of indeterminate equations, &c. Some of the elementary theorems and various particular problems appear in the works of the earliest algebraists, but the true pioneer of modern researches seems to have been Abraham Demoivre, who first published inPhil. Trans.(1697) the law of the general coefficient in the expansion of the series a + bx + cx² + dx³ + ... raised to any power. (See alsoMiscellanea Analytica, bk. iv. chap. ii. prob. iv.) His work on Probabilities would naturally lead him to consider questions of this nature. An important work at the time it was published was theDe Partitione Numerorumof Leonhard Euler, in which the consideration of the reciprocal of the product (1 - xz) (1 - x²z) (1 - x³z) ... establishes a fundamental connexion between arithmetic and algebra, arithmetical addition being made to depend upon algebraical multiplication, and a close bond is secured between the theories of discontinuous and continuous quantities. (Cf.Numbers, Partition of.) The multiplication of the two powers xa, xb, viz. xa+ xb= xa+b, showed Euler that he could convert arithmetical addition into algebraical multiplication, and in the paper referred to he gives the complete formal solution of the main problems of the partition of numbers. He did not obtain general expressions for the coefficients which arose in the expansion of his generating functions, but he gave the actual values to a high order of the coefficients which arise from the generating functions corresponding to various conditions of partitionment. Other writers who have contributed to the solution of special problems are James Bernoulli, Ruggiero Guiseppe Boscovich, Karl Friedrich Hindenburg (1741-1808), William Emerson (1701-1782), Robert Woodhouse (1773-1827), Thomas Simpson and Peter Barlow. Problems of combination were generally undertaken as they became necessary for the advancement of some particular part of mathematical science: it was not recognized that the theory of combinations is in reality a science by itself, well worth studying for its own sake irrespective of applications to other parts of analysis. There was a total absence of orderly development, and until the first third of the 19th century had passed, Euler’s classical paper remained alike the chief result and the only scientific method of combinatorial analysis.

In 1846 Karl G. J. Jacobi studied the partitions of numbers by means of certain identities involving infinite series that are met with in the theory of elliptic functions. The method employed is essentially that of Euler. Interest in England was aroused, in the first instance, by Augustus De Morgan in 1846, who, in a letter to Henry Warburton, suggested that combinatorial analysis stood in great need of development, and alluded to the theory of partitions. Warburton, to some extent under the guidance of De Morgan, prosecuted researches by the aid of a new instrument, viz. the theory of finite differences. This was a distinct advance, and he was able to obtain expressions for the coefficients in partition series in some of the simplest cases (Trans. Camb. Phil. Soc., 1849). This paper inspired a valuable paper by Sir John Herschel (Phil. Trans.1850), who, by introducing the idea and notation of the circulating function, was able to present results in advance of those of Warburton. The new idea involved a calculus of the imaginary roots of unity. Shortly afterwards, in 1855, the subject was attacked simultaneously by Arthur Cayley and James Joseph Sylvester, and their combined efforts resulted in the practical solution of the problem that we have to-day. The former added the idea of the prime circulator, and the latter applied Cauchy’s theory of residues to the subject, and invented the arithmetical entity termed a denumerant. The next distinct advance was made by Sylvester, Fabian Franklin, William Pitt Durfee and others, about the year 1882 (Amer. Journ. Math.vol. v.) by the employment of a graphical method. The results obtained were not only valuable in themselves, but also threw considerable light upon the theory of algebraic series. So far it will be seen that researches had for their object the discussion of the partition of numbers. Other branches of combinatorial analysis were, from any general point of view, absolutely neglected. In 1888 P. A. MacMahon investigated the general problem of distribution, of which the partition of a number is a particular case. He introduced the method of symmetric functions and the method of differential operators, applying both methods to the two important subdivisions, the theory of composition and the theory of partition. He introduced the notion of the separation of a partition, and extended all the results so as to include multipartite as well as unipartite numbers. He showed how to introduce zero and negative numbers, unipartite and multipartite, into the general theory; he extended Sylvester’s graphical method to three dimensions; and finally, 1898, he invented the “Partition Analysis” and applied it to the solution of novel questions in arithmetic and algebra. An important paper by G. B. Mathews, which reduces the problem of compound partition to that of simple partition, should also be noticed. This is the problem which was known to Euler and his contemporaries as “The Problem of the Virgins,” or “the Rule of Ceres”; it is only now, nearly 200 years later, that it has been solved.

The most important problem of combinatorial analysis is connected with the distribution of objects into classes. A number n may be regarded as enumerating n similar objects; it is then said to be unipartite. On the other hand, if theFundamental problem.objects be not all similar they cannot be effectively enumerated by a single integer; we require a succession of integers. If the objects be p in number of one kind, q of a second kind, r of a third, &c., the enumeration is given by the succession pqr... which is termed a multipartite number, and written,

pqr...,

where p + q + r + ... = n. If the order of magnitude of the numbers p, q, r, ... is immaterial, it is usual to write them in descending order of magnitude, and the succession may then be termed a partition of the number n, and is written (pqr...). The succession of integers thus has a twofold signification: (i.) as a multipartite number it may enumerate objects of different kinds; (ii.) it may be viewed as a partitionment into separate parts of a unipartite number. We may say either that the objects are represented by the multipartite numberpqr...,or that they are defined by the partition (pqr...) of the unipartite number n. Similarly the classes into which they are distributed may be m in number all similar; or they may be p1of one kind, q1of a second, r1of a third, &c., where p1+ q1+ r1+ ... = m. We may thus denote the classes either by the multipartite numbersp1q1r1...,or by the partition (p1q1r1...) of the unipartite number m. The distributions to be considered are such that any number of objects may be in any one class subject to the restriction that no class is empty. Two cases arise. If the order of the objects in a particular class is immaterial, the class is termed aparcel; if the order is material, the class is termed agroup. The distribution into parcels is alone considered here, and the main problem is the enumeration of the distributions of objects defined by the partition (pqr...) of the number n into parcels defined by the partition (p1q1r1...) of the number m. (See “Symmetric Functions and the Theory of Distributions,”Proc. London Mathematical Society, vol. xix.) Three particular cases are of great importance. Case I. is the “one-to-one distribution,” in which the number of parcels is equal to the number of objects, and one object is distributed in each parcel. Case II. is that in which the parcels are all different, being defined by the partition (1111...), conveniently written (1m); this is the theory of the compositions of unipartite and multipartite numbers. Case III. is that in which the parcels are all similar, being defined by the partition (m); this is the theory of the partitions of unipartite and multipartite numbers. Previous to discussing these in detail, it is necessary to describe the method of symmetric functions which will be largely utilized.

Let α, β, γ, ... be the roots of the equation

xn- a1xn-1+ a2xn-2- ... = 0

The symmetric function Σαpβqγr..., where p + q + r + ... = n is, in the partition notation, written (pqr...). Let A(pqr...), (p1q1r1...)denote the number of ways of distributingThe distribution function.the n objects defined by the partition (pqr...) into the m parcels defined by the partition (p1q1r1...). The expression

ΣA(pqr...), (p1q1r1...)· (pqr...),

where the numbers p1, q1, r1... are fixed and assumed to be in descending order of magnitude, the summation being for every partition (pqr...) of the number n, is defined to be the distribution function of the objects defined by (pqr...) into the parcels defined by (p1q1r1...). It gives a complete enumeration of n objects of whatever species into parcels of the given species.

1.One-to-One Distribution. Parcelsmin number(i.e.m = n).—Let hsbe the homogeneous product-sum of degree s ofCase I.the quantities α, β, γ, ... so that

(1 - αx. 1 - βx. 1 - γx. ...)-1= 1 + h1x + h2x² + h3x³ + ...

h1= Σα = (1)h2= Σα² + Σαβ = (2) + (1²)h3= Σα³ + Σα²β + Σαβγ = (3) + (21) + (1³).

h1= Σα = (1)

h2= Σα² + Σαβ = (2) + (1²)

h3= Σα³ + Σα²β + Σαβγ = (3) + (21) + (1³).

Form the product hp1hq1hr1...

Any term in hp1may be regarded as derived from p1objects distributed into p1similar parcels, one object in each parcel, since the order of occurrence of the letters α, β, γ, ... in any term is immaterial. Moreover, every selection of p1letters from the letters in αpβqγr... will occur in some term of hp1, every further selection of q1letters will occur in some term of hq1, and so on. Therefore in the product hp1hq1hr1... the term αpβqγr..., and therefore also the symmetric function (pqr ...), will occur as many times as it is possible to distribute objects defined by (pqr ...) into parcels defined by (p1q1r1...) one object in each parcel. Hence

ΣA(pqr...), (p1q1r1...)· (pqr...) = hp1hq1hr1....

This theorem is of algebraic importance; for consider the simple particular case of the distribution of objects (43) into parcels (52), and represent objects and parcels by small and capital letters respectively. One distribution is shown by the scheme

wherein an object denoted by a small letter is placed in a parcel denoted by the capital letter immediately above it. We may interchange small and capital letters and derive from it a distribution of objects (52) into parcels (43); viz.:—

The process is clearly of general application, and establishes a one-to-one correspondence between the distribution of objects (pqr ...) into parcels (p1q1r1...) and the distribution of objects (p1q1r1...) into parcels (pqr ...). It is in fact, in Case I., an intuitive observation that we may either consider an object placed in or attached to a parcel, or a parcel placed in or attached to an object. Analytically we have

Theorem.—“The coefficient of symmetric function (pqr ...) in the development of the product hp1hq1hr1... is equal to the coefficient of symmetric function (p1q1r1...) in the development of the product hphqhr....”

The problem of Case I. may be considered when the distributions are subject to various restrictions. If the restriction be to the effect that an aggregate of similar parcels is not to contain more than one object of a kind, we have clearly to deal with the elementary symmetric functions a1, a2, a3, ... or (1), (1²), (1³), ... in lieu of the quantities h1, h2, h3, ... The distribution function has then the value ap1aq1ar1... or (1p1) (1q1) (1r1) ..., and by interchange of object and parcel we arrive at the well-known theorem of symmetry in symmetric functions, which states that the coefficient of symmetric function (pqr ...) in the development of the product ap1aq1ar1... in a series of monomial symmetric functions, is equal to the coefficient of the function (p1q1r1...) in the similar development of the product apaqar....

The general result of Case I. may be further analysed with important consequences.

Write

X1= (1)x1,X2= (2)x2+ (1²)x1²,X3= (3)x3+ (21)x2x1+ (1³)x1³

X1= (1)x1,

X2= (2)x2+ (1²)x1²,

X3= (3)x3+ (21)x2x1+ (1³)x1³

.......

and generally

Xs= Σ(λμν ...) xλxμxν...

the summation being in regard to every partition of s. Consider the result of the multiplication—

Xp1Xq1Xr1... = ΣP xσ1s1xσ2s2xσ3s3...

To determine the nature of the symmetric function P a few definitions are necessary.

Definition I.—Of a number n take any partition (λ1λ2λ3... λs) and separate it into component partitions thus:—

(λ1λ2) (λ3λ4λ5) (λ6) ...

in any manner. This may be termed aseparationof the partition, the numbers occurring in the separation being identical with those which occur in the partition. In the theory of symmetric functions the separation denotes the product of symmetric functions—

Σ αλ1βλ2Σ αλ3βλ4γλ5Σ αλ6...

The portions (λ1λ2), (λ3λ4λ5), (λ6), ... are termedseparates, and if λ1+ λ2= p1, λ3+ λ4+ λ5= q1, λ6= r1... be in descending order of magnitude, the usual arrangement, the separation is said to have aspeciesdenoted by the partition (p1q1r1...) of the number n.

Definition II.—If in any distribution of n objects into n parcels (one object in each parcel), we write down a number ξ, whenever we observe ξ similar objects in similar parcels we will obtain a succession of numbers ξ1, ξ2, ξ3, ..., where (ξ1, ξ2, ξ3...) is some partition of n. The distribution is then said to have aspecificationdenoted by the partition (ξ1ξ2ξ3...).

Now it is clear that P consists of an aggregate of terms, each of which, to a numerical factorprès, is a separation of the partition (sσ11sσ32sσ33...) of species (p1q1r1...). Further, P is the distribution function of objects into parcels denoted by (p1q1r1...), subject to the restriction that the distributions have each of them the specificationdenoted by the partition (sσ11sσ32sσ33...) Employing a more general notation we may write

Xπ1p1Xπ2p2Xπ3p3... = ΣP xσ1s1xσ2s2xσ3s3...

and then P is the distribution function of objects into parcels (pπ11pπ22pπ33...), the distributions being such as to have the specification (sσ11sσ22sσ33...). Multiplying out P so as to exhibit it as a sum of monomials, we get a result—

Xπ1p1Xπ2p2Xπ3p3... = ΣΣθ (λl11λl22λl33...) xσ1s1xσ2s2xσ3s3...

indicating that for distributions of specification (sσ11sσ32sσ33...) there are θ ways of distributing n objects denoted by (λl11λl22λl33...) amongst n parcels denoted by (pπ11pπ22pπ33...), one object in each parcel. Now observe that as before we may interchange parcel and object, and that this operation leaves the specification of the distribution unchanged. Hence the number of distributions must be the same, and if

Xπ1p1Xπ2p2Xπ3p3... = ... + θ (λl11λl22λl33...) xσ1s1xσ2s2xσ3s3... + ...

then also

Xl1λ1Xl2λ2Xl3λ3... = ... + θ (pπ11pπ22pπ33...) xσ1s1xσ2s2xσ3s3... + ...

This extensive theorem of algebraic reciprocity includes many known theorems of symmetry in the theory of Symmetric Functions.

The whole of the theory has been extended to include symmetric functions symbolized by partitions which contain as well zero and negative parts.

2.The Compositions of Multipartite Numbers. Parcels denoted by(Im).—There are here no similarities between the parcels.Case II.

Let (π1π2π3) be a partition of m.

(pπ11pπ22pπ33...) a partition of n.

Of the whole number of distributions of the n objects, there will be a certain number such that n1parcels each contain p1objects, and in general πsparcels each contain psobjects, where s = 1, 2, 3, ... Consider the product hπ1p1hπ2p2hπ3p3... which can be permuted in m! / π1!π2!π3! ... ways. For each of these ways hπ1p1hπ2p2hπ3p3... will be a distribution function for distributions of the specified type. Hence, regarding all the permutations, the distribution function is

and regarding, as well, all the partitions of n into exactly m parts, the desired distribution function is

that is, it is the coefficient of xnin (h1x + h2x² + h3x³ + ... )m. The value ofA(pπ11pπ22pπ33...) is the coefficient of (pπ11pπ22pπ33...)xnin the development of the above expression, and is easily shown to have the value

Observe that when p1= p2= p3= ... = π1= π2= π3... = 1 this expression reduces to the mth divided differences of 0n. The expression gives the compositions of the multipartite numberpπ11pπ22pπ33... into m parts. Summing the distribution function from m = 1 to m = ∞ and putting x = 1, as we may without detriment, we find that the totality of the compositions is given by (h1+ h2+ h3+ ...) / (1 - h1- h2- h3+ ...) which may be given the form (a1- a2+ a3- ...) / [1 - 2(a1- a2+ a3- ...)] Adding ½ we bring this to the still more convenient form

Let F (pπ11pπ22pπ33... ) denote the total number of compositions of the multipartitepπ11pπ22pπ33.... Then ½ · (1 / 1 - 2a) = ½ + Σ F(p)αp, and thence F(p) = 2p - 1. Again ½ · [1 / 1 - 2(α + β - αβ)] = Σ F(p1p2) αp1βp2, and expanding the left-hand side we easily find

We have found that the number of compositions of the multipartitep1p2p3... psis equal to the coefficient of symmetric function (p1p2p3... ps)orof the single term αp11αp22αp22... αpssin the development according to ascending powers of the algebraic fraction

This result can be thrown into another suggestive form, for it can be proved that this portion of the expanded fraction

which is composed entirely of powers of

t1α1, t2α2, t3α3, ... tsαs

has the expression

and therefore the coefficient of αp11αp22... αpssin the latter fraction, when t1, t2, &c., are put equal to unity, is equal to the coefficient of the same term in the product

½ (2α1+ α2+ ... + αs)p1(2α1+ 2α2+ ... + αs)p2... (2α1+ 2α2+ ... + 2αs)ps.

This result gives a direct connexion between the number of compositions and the permutations of the letters in the product αp11αp22... αpss. Selecting any permutation, suppose that the letter aroccurs qrtimes in the last pr+ pr+1+ ... + psplaces of the permutation; the coefficient in question may be represented by ½ Σ2q1+q2+ ... +qs, the summation being for every permutation, and since q1= p1this may be written

2p1-1Σ2q1+q2+ ... +qs.

Ex. Gr.—For the bipartite22, p1= p2= 2, and we have the following scheme:—

Hence

F(22) = 2 (2² + 2 + 2 + 2 + 2 + 2°) = 26.

We may regard the fraction

as a redundant generating function, the enumeration of the compositions being given by the coefficient of

(t1α1)p1(t2α2)p2... (tsαs)ps.

The transformation of the pure generating function into a factorized redundant form supplies the key to the solution of a large number of questions in the theory of ordinary permutations, as will be seen later.

[The transformation of the last section involvesThe theory of permutations.a comprehensive theory of Permutations, which it is convenient to discuss shortly here.

If X1, X2, X3, ... Xnbe linear functions given by the matricular relation

that portion of the algebraic fraction,

which is a function of the products s1x1, s2x2, s3x3, ... snxnonly is

where the denominator is in a symbolic form and denotes on expansion

1 - Σ |a11|s1x1+ Σ |a11a22|s1s2x1x2- ... + (-)n|a11a22a33... ann| s1s2... snx1x2... xn,

where |a11|, |a11a22|, ... |a11a22, ... ann| denote the several co-axial minors of the determinant

|a11a22... ann|

of the matrix. (For the proof of this theorem see MacMahon, “A certain Class of Generating Functions in the Theory of Numbers,”Phil. Trans. R. S.vol. clxxxv. A, 1894). It follows that the coefficient of

xξ11xξ22... xξnn

in the product

(a11x1+ a12x2+ ... + a1nxn)ξ1(a21x1+ a22x2+ ... + a2nxn)ξ2... (an1x1+ an2x2+ ... + annxn)ξn

is equal to the coefficient of the same term in the expansion ascending-wise of the fraction

If the elements of the determinant be all of them equal to unity, we obtain the functions which enumerate the unrestricted permutations of the letters in

xξ11xξ22... xξnn,

viz.

(x1+ x2+ ... - xn)ξ1+ξ2+ ... +ξn

and

Suppose that we wish to find the generating function for the enumeration of those permutations of the letters in xξ11xξ22... xξnnwhich are such that no letter xsis in a position originally occupied by an x3for all values of s. This is a generalization of the “Problème des rencontres” or of “derangements.” We have merely to put

a11= a22= a33= ... = ann= 0

and the remaining elements equal to unity. The generating product is

(x2+ x2+ ... + xn)ξ1(x1+ x3+ ... + xn)ξ2... (x1+ x2+ ... + xn-1)ξn,

and to obtain the condensed form we have to evaluate the co-axial minors of the invertebrate determinant—

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