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| [[Image:13-Weak-Orders.svg|thumb|The 13 possible strict weak orderings on a set of three elements {''a'', ''b'', ''c''}]]
| | Claude is her title and she totally digs that name. What she loves performing is taking part in croquet and she is attempting to make it a occupation. Interviewing is what I do for a living but I strategy on altering it. Arizona is her beginning place and she will by no means transfer.<br><br>Visit my blog - car warranty - [http://Muehle-Kruskop.de/index.php?mod=users&action=view&id=22344 navigate to this site], |
| In [[number theory]] and [[enumerative combinatorics]], the '''ordered Bell numbers''' or '''Fubini numbers''' count the number of [[weak ordering]]s on a [[set (mathematics)|set]] of ''n'' elements (orderings of the elements into a sequence allowing [[Tie (draw)|ties]], such as might arise as the outcome of a [[horse race]]).<ref name="horse">{{citation|title=Those Fascinating Numbers|first=J. M.|last=de Koninck|publisher=American Mathematical Society|year=2009|isbn=9780821886311|url=http://books.google.com/books?id=qYuC1WsDKq8C&pg=PA4|page=4}}. Because of this application, de Koninck calls these numbers "horse numbers", but this name does not appear to be in widespread use.</ref> Starting from ''n'' = 0, these numbers are
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| :1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... {{OEIS|A000670}}.
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| The ordered Bell numbers may be computed via a summation formula involving [[binomial coefficient]]s, or by using a [[recurrence relation]]. Along with the weak orderings, they count several other types of combinatorial objects that have a [[Bijective proof|bijective correspondence]] to the weak orderings, such as the ordered [[multiplicative partition]]s of a [[squarefree]] number<ref name="sklar"/> or the faces of all dimensions of a [[permutohedron]]<ref>{{citation|first=Günter M.|last=Ziegler|authorlink=Günter M. Ziegler|title=Lectures on Polytopes|publisher=Springer|series=Graduate Texts in Mathematics|volume=152|year=1995|page=18}}.</ref> (e.g. the sum of faces of all dimensions in the [[truncated octahedron]] is 1 + 14 + 36 + 24 = 75<ref>1, 14, 36, 24 is the fourth row [http://oeis.org/A019538/table of this triangle] {{OEIS|A019538}}</ref>).
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| ==History==
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| [[File:Cayley ordered Bell trees.svg|thumb|300px|13 plane trees with ordered leaves and equal-length root-leaf paths, with the gaps between adjacent leaves labeled by the height above the leaves of the nearest common ancestor. These labels induce a weak ordering on the gaps, showing that the trees of this type are counted by the ordered Bell numbers.]]
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| The ordered Bell numbers appear in the work of {{harvtxt|Cayley|1859}}, who used them to count certain [[Tree (graph theory)#Plane Tree|plane trees]] with ''n'' + 1 totally ordered leaves. In the trees considered by Cayley, each root-to-leaf path has the same length, and the number of nodes at distance ''i'' from the root must be strictly smaller than the number of nodes at distance ''i'' + 1, until reaching the leaves.<ref>{{citation|first=A.|last=Cayley|authorlink=Arthur Cayley|year=1859|title=On the analytical forms called trees, second part|journal=[[Philosophical Magazine]]|series=Series IV|volume=18|issue=121|pages=374–378|doi=10.1017/CBO9780511703706.026}}. In [http://books.google.com/books?id=F3gNAQAAMAAJ&pg=PA113 ''Collected Works of Arthur Cayley'', p. 113].</ref> In such a tree, there are ''n'' pairs of adjacent leaves, that may be weakly ordered by the height of their [[lowest common ancestor]]; this weak ordering determines the tree. {{harvtxt|Mor|Fraenkel|1984}} call the trees of this type "Cayley trees", and they call the sequences that may be used to label their gaps (sequences of ''n'' positive integers that include at least one copy of each positive integer between one and the maximum value in the sequence) "Cayley permutations".<ref>{{citation
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| | last1 = Mor | first1 = M.
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| | last2 = Fraenkel | first2 = A. S. | author2-link = Aviezri Fraenkel
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| | doi = 10.1016/0012-365X(84)90136-5
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| | issue = 1
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | mr = 732206
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| | pages = 101–112
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| | title = Cayley permutations
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| | volume = 48
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| | year = 1984}}.</ref>
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| {{harvtxt|Pippinger|2010}} traces the problem of counting weak orderings, which has the same sequence as its solution, to the work of {{harvtxt|Whitworth|1886}}.<ref name="pip"/><ref>{{citation|first=W. A.|last=Whitworth|authorlink=William Allen Whitworth|title=Choice and Chance|location=Deighton|publisher=Bell and Co.|year=1886|at=Proposition XXII, p. 93}}. As cited by {{harvtxt|Pippinger|2010}}.</ref>
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| These numbers were called Fubini numbers by Louis Comtet, because they count the number of different ways to rearrange the ordering of sums or integrals in [[Fubini's theorem]], which in turn is named after [[Guido Fubini]].<ref>{{citation|title=Advanced Combinatorics: The Art of Finite and Infinite Expansions|edition=revised and enlarged|first=Louis|last=Comtet|publisher=D. Reidel Publishing Co.|year=1974|url=http://www.plouffe.fr/simon/math/AdvancedComb.pdf|page=228}}.</ref> For instance, for a bivariate integral, Fubini's theorem states that
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| :<math>\int_A\left(\int_B f(x,y)\,\text{d}y\right)\,\text{d}x=\int_B\left(\int_A f(x,y)\,\text{d}x\right)\,\text{d}y=\int_{A\times B} f(x,y)\,\text{d}(x,y),</math>
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| where these three formulations correspond to the three weak orderings on two elements. In general, in a multivariate integral, the ordering in which the variables may be grouped into a sequence of nested integrals forms a weak ordering.
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| The [[Bell number]]s, named after [[Eric Temple Bell]], count the number of [[partition of a set|partitions of a set]], and the weak orderings that are counted by the ordered Bell numbers may be interpreted as a partition together with a [[total order]] on the sets in the partition.<ref name="km05">{{citation
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| | last1 = Knopfmacher | first1 = A.
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| | last2 = Mays | first2 = M. E.
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| | doi = 10.1142/S1793042105000315
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| | issue = 4
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| | journal = [[International Journal of Number Theory]]
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| | mr = 2196796
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| | pages = 563–581
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| | title = A survey of factorization counting functions
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| | volume = 1
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| | year = 2005}}.</ref>
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| ==Formulae==
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| The ''n''th ordered Bell number may be given by a [[summation]] formula involving the [[Stirling numbers of the second kind]], which count the number of partitions of an ''n''-element set into ''k'' nonempty subsets,<ref name="good">{{citation
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| | last = Good | first = I. J. | authorlink = I. J. Good
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| | journal = [[Fibonacci Quarterly]]
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| | mr = 0376367
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| | pages = 11–18
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| | title = The number of orderings of ''n'' candidates when ties are permitted
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| | url = http://www.fq.math.ca/Scanned/13-1/good.pdf
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| | volume = 13
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| | year = 1975}}.</ref><ref name="sprugnoli">{{citation
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| | last = Sprugnoli | first = Renzo
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| | doi = 10.1016/0012-365X(92)00570-H
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| | issue = 1-3
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | mr = 1297386
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| | pages = 267–290
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| | title = Riordan arrays and combinatorial sums
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| | volume = 132
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| | year = 1994}}.</ref>
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| expanded out into a double summation involving [[binomial coefficient]]s (using a formula expressing Stirling numbers as a sum of binomial coefficients), or given by an [[infinite series]]:<ref name="pip">{{citation
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| | last = Pippenger | first = Nicholas | author-link = Nick Pippenger
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| | arxiv = 0904.1757
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| | doi = 10.4169/002557010X529752
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| | issue = 5
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| | journal = [[Mathematics Magazine]]
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| | mr = 2762645
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| | pages = 331–346
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| | title = The hypercube of resistors, asymptotic expansions, and preferential arrangements
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| | volume = 83
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| | year = 2010}}.</ref><ref name="km05"/>
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| :<math>a(n)= \sum_{k=0}^n k! \left\{\begin{matrix} n \\ k \end{matrix}\right\}=\sum_{k=0}^n \sum_{j=0}^k (-1)^{k-j} \binom{k}{j}j^n=\frac12\sum_{m=0}^\infty\frac{m^n}{2^{m}}.</math>
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| An alternative summation formula expresses the ordered Bell numbers in terms of the [[Eulerian number]]s, which count the number of permutations of ''n'' items with ''k'' + 1 runs of increasing items:<ref name="vc95">{{citation
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| | last1 = Velleman | first1 = Daniel J.
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| | last2 = Call | first2 = Gregory S.
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| | doi = 10.2307/2690567
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| | issue = 4
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| | journal = Mathematics Magazine
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| | mr = 1363707
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| | pages = 243–253
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| | title = Permutations and combination locks
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| | volume = 68
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| | year = 1995}}.</ref>
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| :<math>a(n)=\sum_{k=0}^{n-1} 2^k \left\langle\begin{matrix} n \\ k \end{matrix}\right\rangle=A_n(2),</math>
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| where ''A<sub>n</sub>'' is the ''n''th Eulerian polynomial.
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| The [[exponential generating function]] of the ordered Bell numbers is<ref name="pip"/><ref name="km05"/><ref name="sprugnoli"/><ref>{{citation
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| | last1 = Getu | first1 = Seyoum
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| | last2 = Shapiro | first2 = Louis W.
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| | last3 = Woan | first3 = Wen Jin
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| | last4 = Woodson | first4 = Leon C.
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| | doi = 10.1137/0405040
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| | issue = 4
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| | journal = [[SIAM Journal on Discrete Mathematics]]
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| | mr = 1186818
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| | pages = 497–499
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| | title = How to guess a generating function
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| | volume = 5
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| | year = 1992}}.</ref>
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| :<math>\sum_{n=0}^\infty a(n)\frac{x^n}{n!} = \frac{1}{2-e^x}.</math>
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| This can be expressed equivalently as the fact that the ordered Bell numbers are the numbers in the first column of the [[infinite matrix]] (2''I'' − ''P'')<sup>−1</sup>, where ''I'' is the [[identity matrix]] and ''P'' is an infinite matrix form of [[Pascal's triangle]].<ref>{{citation
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| | last = Lewis | first = Barry
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| | doi = 10.4169/000298910X474989
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| | issue = 1
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| | journal = [[American Mathematical Monthly]]
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| | mr = 2599467
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| | pages = 50–66
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| | title = Revisiting the Pascal matrix
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| | volume = 117
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| | year = 2010}}.</ref> Based on a [[contour integration]] of this generating function, the ordered Bell numbers may be approximated as<ref name="sklar">{{citation
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| | last = Sklar | first = Abe
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| | doi = 10.1090/S0002-9939-1952-0050620-1
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| | journal = Proceedings of the American Mathematical Society
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| | jstor = 2032169
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| | mr = 0050620
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| | pages = 701–705
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| | title = On the factorization of squarefree integers
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| | volume = 3
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| | year = 1952}}.</ref><ref name="sprugnoli"/><ref name="gross">{{citation
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| | last = Gross | first = O. A.
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| | doi = 10.2307/2312725
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| | journal = [[The American Mathematical Monthly]]
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| | mr = 0130837
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| | pages = 4–8
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| | title = Preferential arrangements
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| | volume = 69
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| | year = 1962}}.</ref><ref>{{citation
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| | last = Barthélémy | first = J.-P.
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| | doi = 10.1016/0012-365X(80)90159-4
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| | issue = 3
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | mr = 560774
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| | pages = 311–313
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| | title = An asymptotic equivalent for the number of total preorders on a finite set
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| | volume = 29
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| | year = 1980}}.</ref><ref>{{citation
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| | last = Bailey | first = Ralph W.
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| | doi = 10.1007/s003550050123
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| | issue = 4
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| | journal = Social Choice and Welfare
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| | mr = 1647055
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| | pages = 559–562
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| | title = The number of weak orderings of a finite set
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| | volume = 15
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| | year = 1998}}.</ref>
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| :<math>a(n)\approx \frac{n!}{2(\log 2)^{n+1}}.</math>
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| Because log 2 is less than one, the form of this approximation shows that the ordered Bell numbers exceed the corresponding [[factorial]]s by an exponential factor. The asymptotic convergence of this approximation may be expressed as
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| :<math>\lim_{n\to\infty} \frac{n\,a(n-1)}{a(n)}=\log 2.</math>
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| ==Recurrence and modular periodicity==
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| As well as the formulae above, the ordered Bell numbers may be calculated by the [[recurrence relation]]<ref name="pip"/><ref name="gross"/>
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| :<math>a(n) = \sum_{i=1}^{n}\binom{n}{i}a(n-i).</math>
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| The intuitive meaning of this formula is that a weak ordering on ''n'' items may be broken down into a choice of some nonempty set of ''i'' items that go into the first equivalence class of the ordering, together with a smaller weak ordering on the remaining ''n'' − ''i'' items. As a base case for the recurrence, ''a''(0) = 1 (there is one weak ordering on zero items). Based on this recurrence, these numbers can be shown to obey certain periodic patterns in [[modular arithmetic]]: for sufficiently large ''n'',
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| :<math>a(n+4) \equiv a(n) \pmod{10},</math><ref name="gross"/><ref name="mezo">{{citation|title=Periodicity of the Last Digits of Some Combinatorial Sequences|first=I.|last=Mező|journal=Journal of Integer Sequences|year=2014|issue=17|url=https://cs.uwaterloo.ca/journals/JIS/VOL17/Mezo/mezo19.html}}</ref>
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| :<math>a(n+20) \equiv a(n) \pmod{100},</math>
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| :<math>a(n+100) \equiv a(n) \pmod{1000},</math> and
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| :<math>a(n+500) \equiv a(n) \pmod{10000}.</math><ref>{{citation
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| | last = Kauffman | first = Dolores H.
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| | doi = 10.2307/2312790
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| | journal = [[The American Mathematical Monthly]]
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| | mr = 0144827
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| | page = 62
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| | title = Note on preferential arrangements
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| | volume = 70
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| | year = 1963}}.</ref>
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| Several additional modular identities are given by {{harvtxt|Good|1975}} and {{harvtxt|Poonen|1988}}.<ref name="good"/><ref>{{citation
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| | last = Poonen | first = Bjorn | authorlink = Bjorn Poonen
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| | issue = 1
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| | journal = [[Fibonacci Quarterly]]
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| | mr = 931425
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| | pages = 70–76
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| | title = Periodicity of a combinatorial sequence
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| | volume = 26
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| | year = 1988}}.</ref>
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| ==Additional applications==
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| As has already been mentioned, the ordered Bell numbers count weak orderings, [[permutohedron]] faces, Cayley trees, Cayley permutations, ordered multiplicative partitions of squarefree numbers, and equivalent formulae in Fubini's theorem. Weak orderings in turn have many other applications. For instance, in [[horse racing]], [[photo finish]]es have eliminated most but not all ties, called in this context [[List of dead heat horse races|dead heats]], and the outcome of a race that may contain ties (including all the horses, not just the first three finishers) may be described using a weak ordering. For this reason, the ordered Bell numbers count the possible number of outcomes of a horse race,<ref name="horse"/> or the possible outcomes of a multi-candidate [[election]].<ref>{{citation|title=Famous Puzzles of Great Mathematicians|first=Miodrag|last=Petković|publisher=American Mathematical Society|year=2009|isbn=9780821886304|page=194|url=http://books.google.com/books?id=pmSftwkAocAC&pg=PA194}}.</ref> In contrast, when items are ordered or ranked in a way that does not allow ties (such as occurs with the ordering of cards in a deck of cards, or batting orders among [[baseball]] players), the number of orderings for ''n'' items is a [[factorial|factorial number]] ''n''!,<ref>{{citation|title=Combinatorics and Graph Theory|series=Undergraduate Texts in Mathematics|first1=John|last1=Harris|first2=Jeffry L.|last2=Hirst|first3=Michael J.|last3=Mossinghoff|edition=2nd|publisher=Springer|year=2008|isbn=9780387797106|page=132|url=http://books.google.com/books?id=CxSoZcNymacC&pg=PA132}}.</ref> which is significantly smaller than the corresponding ordered Bell number.<ref name="ek"/>
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| {{harvtxt|Kemeny|1956}} uses the ordered Bell numbers to describe the "complexity" of an [[Finitary relation|''n''-ary relation]], by which he means the number of other relations one can form from it by permuting and repeating its arguments (lowering the arity with every repetition).<ref>{{citation|title=Two measures of complexity|first=John G.|last=Kemeny|authorlink=John Kemeny (computer scientist)|journal=[[The Journal of Philosophy]]|volume=52|issue=24|pages=722–733|jstor=2022697}}.</ref> In this application, for each derived relation, the arguments of the original relation are weakly ordered by the positions of the corresponding arguments of the derived relation.
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| {{harvtxt|Velleman|Call|1995}} consider [[combination lock]]s with a numeric keypad, in which several keys may be pressed simultaneously and a combination consists of a sequence of keypresses that includes each key exactly once. As they show, the number of different combinations in such a system is given by the ordered Bell numbers.<ref name="vc95"/>
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| {{harvtxt|Ellison|Klein|2001}} point out an application of these numbers to [[optimality theory]] in [[linguistics]]. In this theory, grammars for [[natural language]]s are constructed by ranking certain constraints, and (in a phenomenon called factorial typology) the number of different grammars that can be formed in this way is limited to the number of permutations of the constraints. A paper reviewed by Ellison and Klein suggested an extension of this linguistic model in which ties between constraints are allowed, so that the ranking of constraints becomes a weak order rather than a total order. As they point out, the much larger magnitude of the ordered Bell numbers, relative to the corresponding [[factorial]]s, allows this theory to generate a much richer set of grammars.<ref name="ek">{{citation|title=Review: The Best of All Possible Words (review of ''Optimality Theory: An Overview'', Archangeli, Diana & Langendoen, D. Terence, eds., Blackwell, 1997)|first1=T. Mark|last1=Ellison|first2=Ewan|last2=Klein|journal=Journal of Linguistics|volume=37|issue=1|year=2001|pages=127–143|jstor=4176645}}.</ref>
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| ==References==
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| {{reflist|colwidth=30em}}
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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| [[Category:Enumerative combinatorics]]
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