Bell number

In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s.

Starting with B0 = B1 = 1, the first few Bell numbers are:

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, ... (sequence A000110 in the OEIS).

The nth of these numbers, Bn, counts the number of different ways to partition a set that has exactly n elements, or equivalently, the number of equivalence relations on it. Outside of mathematics, the same number also counts the number of different rhyme schemes for n-line poems.[1]

As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, Bn is the nth moment of a Poisson distribution with mean 1.

Counting

Set partitions

Bell numbers subset partial order
Partitions of sets can be arranged in a partial order, showing that each partition of a set of size n "uses" one of the partitions of a set of size n-1.
Set partitions 5; circles
The 52 partitions of a set with 5 elements

In general, Bn is the number of partitions of a set of size n. A partition of a set S is defined as a set of nonempty, pairwise disjoint subsets of S whose union is S. For example, B3 = 5 because the 3-element set {abc} can be partitioned in 5 distinct ways:

{ {a}, {b}, {c} }
{ {a}, {b, c} }
{ {b}, {a, c} }
{ {c}, {a, b} }
{ {a, b, c} }.

B0 is 1 because there is exactly one partition of the empty set. Every member of the empty set is a nonempty set (that is vacuously true), and their union is the empty set. Therefore, the empty set is the only partition of itself. As suggested by the set notation above, we consider neither the order of the partitions nor the order of elements within each partition. This means that the following partitionings are all considered identical:

{ {b}, {a, c} }
{ {a, c}, {b} }
{ {b}, {c, a} }
{ {c, a}, {b} }.

If, instead, different orderings of the sets are considered to be different partitions, then the number of these ordered partitions is given by the ordered Bell numbers.

Factorizations

If a number N is a squarefree positive integer (meaning that it is the product of some number n of distinct prime numbers), then Bn gives the number of different multiplicative partitions of N. These are factorizations of N into numbers greater than one, treating two factorizations as the same if they have the same factors in a different order.[2] For instance, 30 is the product of the three primes 2, 3, and 5, and has B3 = 5 factorizations:

Rhyme schemes

The Bell numbers also count the rhyme schemes of an n-line poem or stanza. A rhyme scheme describes which lines rhyme with each other, and so may be interpreted as a partition of the set of lines into rhyming subsets. Rhyme schemes are usually written as a sequence of Roman letters, one per line, with rhyming lines given the same letter as each other, and with the first lines in each rhyming set labeled in alphabetical order. Thus, the 15 possible four-line rhyme schemes are AAAA, AAAB, AABA, AABB, AABC, ABAA, ABAB, ABAC, ABBA, ABBB, ABBC, ABCA, ABCB, ABCC, and ABCD.[1]

Permutations

The Bell numbers come up in a card shuffling problem mentioned in the addendum to Gardner (1978). If a deck of n cards is shuffled by repeatedly removing the top card and reinserting it anywhere in the deck (including its original position at the top of the deck), with exactly n repetitions of this operation, then there are nn different shuffles that can be performed. Of these, the number that return the deck to its original sorted order is exactly Bn. Thus, the probability that the deck is in its original order after shuffling it in this way is Bn/nn, which is significantly larger than the 1/n! probability that would describe a uniformly random permutation of the deck.

Related to card shuffling are several other problems of counting special kinds of permutations that are also answered by the Bell numbers. For instance, the nth Bell number equals number of permutations on n items in which no three values that are in sorted order have the last two of these three consecutive. In a notation for generalized permutation patterns where values that must be consecutive are written adjacent to each other, and values that can appear non-consecutively are separated by a dash, these permutations can be described as the permutations that avoid the pattern 1-23. The permutations that avoid the generalized patterns 12-3, 32-1, 3-21, 1-32, 3-12, 21-3, and 23-1 are also counted by the Bell numbers.[3] The permutations in which every 321 pattern (without restriction on consecutive values) can be extended to a 3241 pattern are also counted by the Bell numbers.[4] However, the Bell numbers grow too quickly to count the permutations that avoid a pattern that has not been generalized in this way: by the (now proven) Stanley–Wilf conjecture, the number of such permutations is singly exponential, and the Bell numbers have a higher asymptotic growth rate than that.

Triangle scheme for calculations

BellNumberAnimated
The triangular array whose right-hand diagonal sequence consists of Bell numbers

The Bell numbers can easily be calculated by creating the so-called Bell triangle, also called Aitken's array or the Peirce triangle after Alexander Aitken and Charles Sanders Peirce.[5]

  1. Start with the number one. Put this on a row by itself. ()
  2. Start a new row with the rightmost element from the previous row as the leftmost number ( where r is the last element of (i-1)-th row)
  3. Determine the numbers not on the left column by taking the sum of the number to the left and the number above the number to the left, that is, the number diagonally up and left of the number we are calculating
  4. Repeat step three until there is a new row with one more number than the previous row (do step 3 until )
  5. The number on the left hand side of a given row is the Bell number for that row. ()

Here are the first five rows of the triangle constructed by these rules:

 1
 1   2
 2   3   5
 5   7  10  15
15  20  27  37  52

The Bell numbers appear on both the left and right sides of the triangle.

Properties

Summation formulas

The Bell numbers satisfy a recurrence relation involving binomial coefficients:[6]

It can be explained by observing that, from an arbitrary partition of n + 1 items, removing the set containing the first item leaves a partition of a smaller set of k items for some number k that may range from 0 to n. There are choices for the k items that remain after one set is removed, and Bk choices of how to partition them.

A different summation formula represents each Bell number as a sum of Stirling numbers of the second kind

The Stirling number is the number of ways to partition a set of cardinality n into exactly k nonempty subsets. Thus, in the equation relating the Bell numbers to the Stirling numbers, each partition counted on the left hand side of the equation is counted in exactly one of the terms of the sum on the right hand side, the one for which k is the number of sets in the partition.[7]

Spivey (2008) has given a formula that combines both of these summations:

Generating function

The exponential generating function of the Bell numbers is

In this formula, the summation in the middle is the general form used to define the exponential generating function for any sequence of numbers, and the formula on the right is the result of performing the summation in the specific case of the Bell numbers.

One way to derive this result uses analytic combinatorics, a style of mathematical reasoning in which sets of mathematical objects are described by formulas explaining their construction from simpler objects, and then those formulas are manipulated to derive the combinatorial properties of the objects. In the language of analytic combinatorics, a set partition may be described as a set of nonempty urns into which elements labelled from 1 to n have been distributed, and the combinatorial class of all partitions (for all n) may be expressed by the notation

Here, is a combinatorial class with only a single member of size one, an element that can be placed into an urn. The inner operator describes a set or urn that contains one or more labelled elements, and the outer describes the overall partition as a set of these urns. The exponential generating function may then be read off from this notation by translating the operator into the exponential function and the nonemptiness constraint ≥1 into subtraction by one.[8]

An alternative method for deriving the same generating function uses the recurrence relation for the Bell numbers in terms of binomial coefficients to show that the exponential generating function satisfies the differential equation . The function itself can be found by solving this equation.[9][10][11]

Moments of probability distributions

The Bell numbers satisfy Dobinski's formula[12][9][11]

This formula can be derived by expanding the exponential generating function using the Taylor series for the exponential function, and then collecting terms with the same exponent.[8] It allows Bn to be interpreted as the nth moment of a Poisson distribution with expected value 1.

The nth Bell number is also the sum of the coefficients in the nth complete Bell polynomial, which expresses the nth moment of any probability distribution as a function of the first n cumulants.

Modular arithmetic

The Bell numbers obey Touchard's congruence: If p is any prime number then[13]

or, generalizing[14]

Because of Touchard's congruence, the Bell numbers are periodic modulo p, for every prime number p; for instance, for p = 2, the Bell numbers repeat the pattern odd-odd-even with period three. The period of this repetition, for an arbitrary prime number p, must be a divisor of

and for all prime p ≤ 101 and p = 113, 163, 167, or 173 it is exactly this number (sequence A001039 in the OEIS).[15][16]

The period of the Bell numbers to modulo n are

1, 3, 13, 12, 781, 39, 137257, 24, 39, 2343, 28531167061, 156, 25239592216021, 411771, 10153, 48, 51702516367896047761, 39, 109912203092239643840221, 9372, 1784341, 85593501183, 949112181811268728834319677753, 312, 3905, 75718776648063, 117, 1647084, 91703076898614683377208150526107718802981, 30459, 568972471024107865287021434301977158534824481, 96, 370905171793, 155107549103688143283, 107197717, 156, ... (sequence A054767 in the OEIS)

Integral representation

An application of Cauchy's integral formula to the exponential generating function yields the complex integral representation

Some asymptotic representations can then be derived by a standard application of the method of steepest descent.[17]

Log-concavity

The Bell numbers form a logarithmically convex sequence. Dividing them by the factorials, Bn/n!, gives a logarithmically concave sequence.[18][19][20]

Growth rate

Several asymptotic formulas for the Bell numbers are known. In Berend & Tassa (2010) the following bounds were established:

for all positive integers ;

moreover, if then for all ,

where and The Bell numbers can also be approximated using the Lambert W function, a function with the same growth rate as the logarithm, as [21]

Moser & Wyman (1955) established the expansion

uniformly for as , where and each and are known expressions in .[22]

The asymptotic expression

was established by de Bruijn (1981).

Bell primes

Gardner (1978) raised the question of whether infinitely many Bell numbers are also prime numbers. The first few Bell numbers that are prime are:

2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 (sequence A051131 in the OEIS)

corresponding to the indices 2, 3, 7, 13, 42 and 55 (sequence A051130 in the OEIS).

The next Bell prime is B2841, which is approximately 9.30740105 × 106538.[23] As of 2018, it is the largest known prime Bell number. Phil Carmody showed it was a probable prime in 2002. After 17 months of computation with Marcel Martin's ECPP program Primo, Ignacio Larrosa Cañestro proved it to be prime in 2004. He ruled out any other possible primes below B6000, later extended to B30447 by Eric Weisstein.[24]

History

The Bell numbers are named after Eric Temple Bell, who wrote about them in 1938, following up a 1934 paper in which he studied the Bell polynomials.[25][26] Bell did not claim to have discovered these numbers; in his 1938 paper, he wrote that the Bell numbers "have been frequently investigated" and "have been rediscovered many times". Bell cites several earlier publications on these numbers, beginning with Dobiński (1877) which gives Dobinski's formula for the Bell numbers. Bell called these numbers "exponential numbers"; the name "Bell numbers" and the notation Bn for these numbers was given to them by Becker & Riordan (1948).[27]

The first exhaustive enumeration of set partitions appears to have occurred in medieval Japan, where (inspired by the popularity of the book The Tale of Genji) a parlor game called genji-ko sprang up, in which guests were given five packets of incense to smell and were asked to guess which ones were the same as each other and which were different. The 52 possible solutions, counted by the Bell number B5, were recorded by 52 different diagrams, which were printed above the chapter headings in some editions of The Tale of Genji.[28][29]

In Srinivasa Ramanujan's second notebook, he investigated both Bell polynomials and Bell numbers.[30] Early references for the Bell triangle, which has the Bell numbers on both of its sides, include Peirce (1880) and Aitken (1933).

See also

Notes

  1. ^ a b Gardner 1978.
  2. ^ Williams 1945 credits this observation to Silvio Minetola's Principii di Analisi Combinatoria (1909).
  3. ^ Claesson (2001).
  4. ^ Callan (2006).
  5. ^ Sloane, N. J. A. (ed.). "Sequence A011971 (Aitken's array)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ Wilf 1994, p. 23.
  7. ^ Conway & Guy (1996).
  8. ^ a b Flajolet & Sedgewick 2009.
  9. ^ a b Rota 1964.
  10. ^ Wilf 1994, pp. 20-23.
  11. ^ a b Bender & Williamson 2006.
  12. ^ Dobiński 1877.
  13. ^ Becker & Riordan (1948).
  14. ^ Hurst & Schultz (2009).
  15. ^ Williams 1945.
  16. ^ Wagstaff 1996.
  17. ^ Simon, Barry (2010). "Example 15.4.6 (Asymptotics of Bell Numbers)". Complex Analysis (PDF). pp. 772–774.
  18. ^ Engel 1994.
  19. ^ Canfield 1995.
  20. ^ Asai, Kubo & Kuo 2000.
  21. ^ Lovász (1993).
  22. ^ Canfield, Rod (July 1994). "The Moser-Wyman expansion of the Bell numbers" (PDF). Retrieved 2013-10-24.
  23. ^ "93074010508593618333...83885253703080601131". 5000 Largest Known Primes, The Prime Database. Retrieved 2013-10-24.
  24. ^ Weisstein, Eric W. "Integer Sequence Primes". MathWorld.
  25. ^ Bell 1934.
  26. ^ Bell 1938.
  27. ^ Rota 1964. However, Rota gives an incorrect date, 1934, for Becker & Riordan 1948.
  28. ^ Knuth 2013.
  29. ^ Gardner 1978 and Berndt 2011 also mention the connection between Bell numbers and The Tale of Genji, in less detail.
  30. ^ Berndt 2011.

References

External links

1,000,000

1,000,000 (one million), or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione (milione in modern Italian), from mille, "thousand", plus the augmentative suffix -one. It is commonly abbreviated as m (not to be confused with the metric prefix for 1×10−3) or M; further MM ("thousand thousands", from Latin "Mille"; not to be confused with the Roman numeral MM = 2,000), mm, or mn in financial contexts.In scientific notation, it is written as 1×106 or 106. Physical quantities can also be expressed using the SI prefix mega (M), when dealing with SI units; for example, 1 megawatt (1 MW) equals 1,000,000 watts.

The meaning of the word "million" is common to the short scale and long scale numbering systems, unlike the larger numbers, which have different names in the two systems.

The million is sometimes used in the English language as a metaphor for a very large number, as in "Not in a million years" and "You're one in a million", or a hyperbole, as in "I've walked a million miles" and "You've asked the million-dollar question".

1,000,000,000

1,000,000,000 (one billion, short scale; one thousand million or milliard, yard, long scale) is the natural number following 999,999,999 and preceding 1,000,000,001. One billion can also be written as b or bn.In scientific notation, it is written as 1 × 109. The metric prefix giga indicates 1,000,000,000 times the base unit. Its symbol is G.

One billion years may be called eon/aeon in astronomy or geology.

Previously in British English (but not in American English), the word "billion" referred exclusively to a million millions (1,000,000,000,000). However, this is no longer common, and the word has been used to mean one thousand million (1,000,000,000) for several decades.The term milliard can also be used to refer to 1,000,000,000; whereas "milliard" is seldom used in English, variations on this name often appear in other languages.

In the South Asian numbering system, it is known as 100 crore or 1 arab.

10,000,000

10,000,000 (ten million) is the natural number following 9,999,999 and preceding 10,000,001.

In scientific notation, it is written as 107.

In South Asia, it is known as the crore.

In Cyrillic numerals, it is known as the vran (вран - raven).

100,000

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.

13 (number)

13 (thirteen) is the natural number following 12 and preceding 14.

Strikingly folkloric aspects of the number 13 have been noted in various cultures around the world: one theory is that this is due to the cultures employing lunar-solar calendars (there are approximately 12.41 lunations per solar year, and hence 12 "true months" plus a smaller, and often portentous, thirteenth month). This can be witnessed, for example, in the "Twelve Days of Christmas" of Western European tradition.

15 (number)

15 (fifteen) is a number, numeral, and glyph. It is the natural number following 14 and preceding 16.

203 (number)

203 (two hundred [and] three) is the natural number following 202 and preceding 204.

52 (number)

52 (fifty-two) is the natural number following 51 and preceding 53.

75 (number)

75 (seventy-five) is the natural number following 74 and preceding 76.

Bell X-22

The Bell X-22 was an American V/STOL X-plane with four tilting ducted fans. Takeoff was to selectively occur either with the propellers tilted vertically upwards, or on a short runway with the nacelles tilted forward at approximately 45°. Additionally, the X-22 was to provide more insight into the tactical application of vertical takeoff troop transporters such as the preceding Hiller X-18 and the X-22 successor, the Bell XV-15. Another program requirement was a true airspeed in level flight of at least 525 km/h (326 mph; 283 knots).

Dobiński's formula

In combinatorial mathematics, Dobiński’s formula states that the n-th Bell number Bn (i.e., the number of partitions of a set of size n) equals

The formula is named after G. Dobiński, who published it in 1877.

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:

a = a (reflexive property),

if a = b then b = a (symmetric property), and

if a = b and b = c then a = c (transitive property).As a consequence of the reflexive, symmetric, and transitive properties, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.

Euler number

In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion

,

where cosh t is the hyperbolic cosine. The Euler numbers are related to a special value of the Euler polynomials, namely:

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.

Hartlip

Hartlip is a village and civil parish in the borough of Swale, in the county of Kent, England.

The population estimate was 680 in 1991, and in 2001 there were 566 registered voters. At the 2011 Census the population was 746. The village covers 1422.547 acres (5.8 km²) and is in an agricultural region of high quality fruit farming, hops and grain.

Narayana number

In combinatorics, the Narayana numbers N(n, k), n = 1, 2, 3 ..., 1 ≤ k ≤ n, form a triangular array of natural numbers, called Narayana triangle, that occur in various counting problems. They are named after Canadian mathematician T. V. Narayana (1930–1987).

Ordered Bell number

In number theory and enumerative combinatorics, the ordered Bell numbers or Fubini numbers count the number of weak orderings on a set of n elements (orderings of the elements into a sequence allowing ties, such as might arise as the outcome of a horse race). Starting from n = 0, these numbers are

1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... (sequence A000670 in the OEIS).The ordered Bell numbers may be computed via a summation formula involving binomial coefficients, or by using a recurrence relation. Along with the weak orderings, they count several other types of combinatorial objects that have a bijective correspondence to the weak orderings, such as the ordered multiplicative partitions of a squarefree number or the faces of all dimensions of a permutohedron (e.g. the sum of faces of all dimensions in the truncated octahedron is 1 + 14 + 36 + 24 = 75).

Partition of a set

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory.

Quatrain

A quatrain is a type of stanza, or a complete poem, consisting of four lines.

Existing in a variety of forms, the quatrain appears in poems from the poetic traditions of various ancient civilizations including Ancient India, Ancient Greece, Ancient Rome, and China, and continues into the 21st century, where it is seen in works published in many languages. During Europe's Dark Ages, in the Middle East and especially Iran, polymath poets such as Omar Khayyam continued to popularize this form of poetry, also known as Ruba'i, well beyond their borders and time. Michel de Nostredame (Nostradamus) used the quatrain form to deliver his famous prophecies in the 16th century.

There are fifteen possible rhyme schemes, but the most traditional and common are: AAAA, ABAB, and ABBA.

Singleton (mathematics)

In mathematics, a singleton, also known as a unit set, is a set with exactly one element. For example, the set {0} is a singleton.

The term is also used for a 1-tuple (a sequence with one member).

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