## combinatoricsThe study of the ways of choosing and arranging objects from given collections and the study of other kinds of problems to do with counting the number of ways to do something. ## Miscellaneous topics in combinatorics## Bell numberThe number of ways thatn distinguishable objects
(such as differently colored balls) can be grouped into sets (such as buckets)
if no set can be empty. For example, if there are three balls, colored red
(R), green (G), and blue (B), they can be grouped in five different ways:
(RGB), (RG)(B), (RB)(G), (BG)(R), and (R)(G)(B), so that the third Bell
number is 5. The sequence of Bell numbers, 1, 2, 5, 15, 52, 203, 877, 4140,
21147,..., can be built up in the form of a triangle, as follows. The first
row has just the number one. Each successive row begins with the last number
of the previous row and continues by adding the number just written down
to the number immediately above and to the right of it. 1 2 2 3 5 5 7 10 15 15 20 27 37 52 52 ... The Bell numbers appear down the left-hand side of the triangle. These normal Bell numbers contrast with ordered Bell numbers, which
count the number of ways of placing n distinguishable object (balls)
into one or more distinguishable sets (buckets) The ordered Bell numbers
are 1, 3, 13, 75, 541, 4683, 47293, 545835, ... Bell numbers, named after Eric Temple Bell, who was one of the first to analyze them in depth, are related to the Catalan numbers. ## binomialA polynomial expression containing two terms, joined by + or -. The binomial theorem (see below) gives the result of raising a binomial expression to a power; the expansion and the series it leads to are called thebinomial
expansion and the binomial series. A binomial
distribution is described by a formula related to the binomial
expansion. A binomial equation is a particular kind of equation that contains
two terms. ## binomial coefficientAny of the coefficients of the powers ofx or y in the expansion (see binomial
theorem, below) of (x + y)^{n}. The binomial coefficient of a typical term x^{(n - m)}y,
written ^{m}_{n}C_{m}, or (^{m}_{n}),
gives the number of ways of picking m unordered outcomes from n possibilities, and is also known as a combination.
It has the value n!/(n - m)!m! The binomial coefficients form the rows of Pascal's triangle. ## binomial theoremA method of expanding the binomial (see above) expression (x + y)^{n} into a finite or infinite series
of powers of x and y, where n is a number either
integral or fractional, positive or negative, rational or irrational; thus x + y)^{n} = x^{n} + a_{n-1}x^{n-1}y + a_{n-2}x^{n-2}y^{2} + ... + y^{n}
The coefficients a are called binomial
coefficients. _{i}The binomial theorem was discovered by Isaac Newton about 1666, and was first published in 1704 in the second appendix to Newton's Optics. That particular case of the theorem when n is
a positive integer was known to mathematicians prior to Newton (e.g., Briggs and Pascal), and Newton himself gave no demonstration
of the truth of his theorem. ## Catalan numberAny number,u, from the Catalan sequence defined by_{n}u = (2_{n}n)! / (n + 1)!n!
It begins: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, ... The values of u represent the number of
ways a polygon with _{n}n + 2 sides can
be cut into n triangles using straight
lines joining vertices and are named after
the Belgian mathematician Eugène Catalan (1814-1894). They also arise
in other counting problems, for example in determining how many ways 2n beans can be divided into two containers if one container can never have
less than the second. ## Cayley's mousetrapA permutation problem invented by Arthur Cayley. Write the numbers 1, 2, ...,n on a set of cards, and shuffle the deck. Now, start counting using the top
card. If the card chosen does not equal the count, move it to the bottom
of the deck and continue counting forward. If the card chosen does equal
the count, discard the chosen card and begin counting again at 1. The game
is won if all cards are discarded, and lost if the count reaches n + 1. The number of ways the cards can be arranged such that at least one
card is in the proper place for n = 1, 2, ..., are 1, 1, 4, 15,
76, 455, ...## change ringing
round, the order of the bells is changed slightly
in a predetermined way. With 5 bells, there are 5 × 4 × 3 ×
2 × 1, or 120, possible changes, which take about 4 minutes to ring.
With 6, 7, or 8 bells, the number of unique changes is 720, 5,040, and 40,320,
respectively. To produce pleasing variations in the sound, bells are made
to change places with adjacent bells in the row, for example:
Experienced ringers test and extend their abilities by ringing peals: 5,000 or more changes without breaks or repeating a row. Peals customarily last about three hours. The first peal was rung in England in 1715. Chiming bells (swinging them through a short arc using a rope and a lever) goes well back into the Middle Ages, but it wasn't until the 17th century that ringers developed the full wheel, which allowed enough control for orderly ringing. In 1668 Fabian Stedman published Tintinnalogia (The Art of Change
Ringing), containing all the available information on systematic ringing.
The theory of change ringing set forth by Stedman has been refined in later
years but remains essentially unchanged today. Bells for change ringing
are hung in stout frames that allow the bells to swing through 360°.
Each bell is attached to a wooden wheel with a handmade rope running around
it and takes about 2 seconds to rotate. The bells are arranged in the frame
so their ropes hang in a circle in the ringing chamber below. Into each
rope is woven a tuft of brightly colored wool (a sally),
which marks where the ringer must catch the rope while ringing. Bells are
rung from the "mouth up" position. With a pull of the rope, the bell swings
through a full circle to the up position again. With the next pull it swings
back in the other direction. The plot of Dorothy Sayers' The Nine Tailors (1934), considered one of her best works, revolves around the art of change
ringing. ## combinationA set of objects selected without reference to the order in which they are arranged. Compare with permutation (below).## combinatorial analysisThe branch of mathematics dealing with subdivisions of sets; its primary concerns are combinations and permutations and partitions. Apartition of a number n is its expression in the form of a sum of positive integers: i.e., setting
For example, the number 5 has 7 partitions: (1 + 1 + 1 + 1 + 1), (2 + 1 + 1 + 1), (2 + 2 + 1), (3 + 1 + 1), (3 + 2), (4 + 1), and (5). By extension combinatorial topology is the study of complex
forms in terms of their being built up from combinations of basic geometric
forms. ## partition numberA number that gives the number of ways of placingn indistinguishable
balls into n indistinguishable urns. For example:
1: (*)The sequence runs: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, ... If the urns are distinguishable, the number of ways is 2 n. If the
balls are distinguishable, the number of ways is given by the nth
Bell number. ## Pascal's triangleTartaglia's triangle (after Niccoló Tartaglia)
and in many parts of Asia it is referred to as Yang Hui's triangle.
Yang Hui was a minor Chinese official who wrote two books, dated 1261 and
1275, which use decimal fractions (long before they appeared in the West)
and contain one of the earliest accounts of the triangle; at about the same
time, Omar Khayyaam also wrote about it.
The Chinese triangle appears again in 1303 on the front of Chu Shi-Chieh's Ssu Yuan Yü Chien (Precious Mirror of the Four Elements), a book
in which Chu says the triangle was known in China more than two centuries
before his time.The numbers in Pascal's triangle give the number of ways of picking r unordered outcomes from n possibilities. This is equivalent to saying
that the numbers in each row are the binomial
coefficients in the expansion of (x + y)^{n}:( x + y)^{0} = 1( x + y)^{1} = 1x + 1y( x + y)^{2} = 1x^{2} + 2xy + 1y^{2}( x + y)^{3} =
1x^{3} + 3x^{2}y + 3xy^{2} + 1y^{3}( x + y)^{4} = 1x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + 1y^{4}and so on. In addition, the shallow diagonals of the triangle sum to give the numbers in the Fibonacci sequence. Pascal's triangle also has the following properties: - Each number is the sum of the two numbers standing above it to the left and right; e.g. 10 = 4 + 6.
- Each number is equal to the sum of all numbers in the left or right diagonal, beginning with the number immediately above to the left or right, and proceeding upwards, e.g. 15 = 5 + 4 + 3 + 2 + 1 and 15 = 10 + 4 + 1.
- Each diagonal is an arithmetic
sequence; e.g.
1st diagonal: 1, 1, 1, 1, 1, ... (arithmetic sequence of zero order) 2nd diagonal: 1, 2, 3, 4, 5, ... (arithmetic sequence of 1st order) 3rd diagonal: 1, 3, 6, 10, 15, ... (arithemtic sequence of 2nd order) 4th diagonal: 1, 4, 10, 20, ... (arithmetic sequence of 3rd order) and so on.
## partitionA particular ordering of a collection of objects. For example, if an athlete has won three medals, a bronze one (B), a silver one (S), and a gold one (G), there are six ways they can be permuted or lined up: BSG, BGS, SBG, SGB, GBS, and GSB. If six people want to sit on the same park bench, there are 720 ways in which they can organize themselves. In general,n things can be permuted in n × (n - 1) × (n - 2) × ... × 2 × 1 = n! ways (where "!" is the symbol
for factorial). How about if there are n distinct objects but we want to permute them in groups of k (where k n). How many ways can that be done? The first member of the group
can be picked in n ways because there are n objects to pick
from. The second member can be filled in (n - 1) ways since one of
the n elements has already been taken. The third member can be filled
in (n - 2) ways since 2 elements have already been used, and so.
This pattern continues until there are k things have been chosen.
This means that the last member can be filled in (n - k +
1) ways. Therefore a total of n (n - 1)(n - 2) ...
(n - k + 1) different permutations of k objects, taken
from a pool of n objects, exist. If we denote this number by P(n, k), we can write P(n, k) = n! / (n - k)! ## Ramsey theoryA branch of mathematics that asks questions such as: Can order always be found in what appears to be disorder? If so, how much can be found and how big a chunk of disorder is needed to find a particular amount of order in it? Ramsey theory is named after the English mathematician Frank P. Ramsey (1904–1926) who started the field in 1928 while wrestling with a problem in logic. (Frank's one-year-younger brother, Arthur, served as Archbishop of Canterbury from 1961 to 1974.) His life was cut short in 1930, at the age of 26, following a bout of jaundice. Ramsey suspected that if a system was big enough, even if it seemed to be disorderly to an arbitrary degree, it was bound to contain pockets of order from which information about the system could be gleaned.## schoolgirls' problemA problem in combinatorics posed by the Thomas Kirkman in 1847:^{1}
A school mistress has fifteen girl pupils and she wishes to take them on a daily walk. The girls are to walk in five rows of three girls each. It is required that no two girls should walk in the same row more than once per week. Can this be done?In fact, provided n is divisible by 3, we can ask the more general
question about n schoolgirls walking for (n - 1)/2 days so
that no girl walks with any other girl in the same triplet more than once.
Solutions for n = 9, 15, and 27 were given in 1850 and much work
was done on the problem thereafter. It is important in the modern theory
of combinatorics. - Kirkman, T. P. "On a Problem in Combinatorics."
*Cambridge and Dublin Math. J*., 2: 191-204 (1847).
## thirty-six officers problemArrange 36 officers in a 6 × 6 square so that one officer from each of six regiments appears in each row and one from each of six ranks appears in each column. This problem, first posed by Leonhard Euler in 1779, is equivalent to finding two mutually orthogonal Latin squares of order six. Euler correctly conjectured that there was no solution; the search for a proof led to important developments in combinatorics.## Related supercategory• MATHEMATICS | |||||||||||

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