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| [[File:K4 matchings.svg|thumb|The [[complete graph]] ''K''<sub>4</sub> has ten matchings, corresponding to the value ''T''(4) = 10 of the fourth telephone number.]]
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| In [[mathematics]], the '''telephone numbers''' or '''involution numbers''' are a [[integer sequence|sequence of integers]] that count the number of connection patterns in a telephone system with ''n'' subscribers some of which can be linked in pairs, the number of [[matching (graph theory)|matchings]] (the [[Hosoya index]]) of a [[complete graph]] on ''n'' vertices, the number of [[permutation]]s on ''n'' elements that are [[involution (mathematics)|involution]]s, the sum of absolute values of coefficients of the [[Hermite polynomial]]s, the number of standard [[Young tableau]]x with ''n'' cells, and the sum of the degrees of the [[irreducible representation]]s of the [[symmetric group]]. Involution numbers were first studied in 1800 by [[Heinrich August Rothe]], who gave a [[recurrence equation]] by which they may be calculated,<ref name="knuth">{{citation
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| | last = Knuth | first = Donald E. | author-link = Donald Knuth
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| | location = Reading, Mass.
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| | mr = 0445948
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| | pages = 65–67
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| | publisher = Addison-Wesley
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| | title = [[The Art of Computer Programming]], Volume 3: Sorting and Searching
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| | year = 1973}}.</ref> giving the values (starting from ''n'' = 0)
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| :1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... {{OEIS|A000085}}.
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| ==Combinatorial interpretations==
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| [[John Riordan (mathematician)|John Riordan]] provides the following explanation for these numbers: suppose that a telephone service has ''n'' subscribers, any two of whom may be connected to each other by a telephone call. How many different patterns of connection are possible? For instance, with three subscribers, there are three ways of forming a single telephone call, and one additional pattern in which no calls are being made, for a total of four patterns.<ref>{{citation
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| | last = Riordan | first = John | author-link = John Riordan (mathematician)
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| | pages = 85–86
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| | publisher = Dover
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| | title = Introduction to Combinatorial Analysis
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| | year = 2002}}.</ref> For this reason, the numbers counting how many patterns are possible are sometimes called the telephone numbers.<ref>{{citation
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| | last1 = Peart | first1 = Paul
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| | last2 = Woan | first2 = Wen-Jin
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| | issue = 2
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| | journal = Journal of Integer Sequences
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| | mr = 1778992
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| | at = Article 00.2.1
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| | title = Generating functions via Hankel and Stieltjes matrices
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| | url = http://www.emis.ams.org/journals/JIS/VOL3/PEART/peart1.pdf
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| | volume = 3
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| | year = 2000}}.</ref><ref>{{citation
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| | last = Getu | first = Seyoum
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| | doi = 10.2307/2690455
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| | issue = 1
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| | journal = Mathematics Magazine
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| | mr = 1092195
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| | pages = 45–53
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| | title = Evaluating determinants via generating functions
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| | volume = 64
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| | year = 1991}}.</ref>
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|
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| Every pattern of pairwise connections between ''n'' subscribers defines an [[involution (mathematics)|involution]], a [[permutation]] of the subscribers that is its own inverse, in which two subscribers who are making a call to each other are swapped with each other and all remaining subscribers stay in place. Conversely, every possible involution has the form of a set of pairwise swaps of this type. Therefore, the telephone numbers also count involutions. The problem of counting involutions was the original [[combinatorial enumeration]] problem studied by Rothe in 1800<ref name="knuth"/> and these numbers have also been called involution numbers.<ref name="sbdhp">{{citation
| | Many people rent Videos to view with the buddies and family daily. Generally speaking this is performed at youtube videos local rental place. These days there is often a style of renting videos from a vending instrument often in order to Redbox.<br><br>I believe Twitter is seen as a pre-amazing selling machine. Every tweet can send out has the actual to create red hot pre-sold prospects attracted of your service, brand, [http://Www.Answers.com/topic/product product] or business.<br><br>Next, for a world's best sales agent, you the prospects interest within your product or presentation. Letting them know you are usually busy, is packed with dynamite. Your briefcase is no longer considered as overnight luggage, and it portrays explore fighting much more than a sale until they say no eight hours. Keep control by asking to set at a table an individual want you show them a must see option that only agreed to be introduced. Asking them questions if they mind invest off your tie.<br><br>Don't make your mistake in thinking you can begin making money without ANY investment. Neglect the idea income for little or nothing. Here is the bare minimum you should expect to shell out. If you aren't willing to spend this much money, you must go find another joint partnership. $8.95 per year for a domain name.$6.00 per month for hosting your internet.$50 to $100 for a good web editor like DreamWeaver.$400 for a very good computer (check Dell's Refurbished site for the most effective deals).and an investment in products if you have to be selling physical products online.<br><br>To sell on Amazon you check out their homepage where could certainly select; "selling on amazon". Then choose on "Sell a little" along with the best with this option usually that you do not pay any listing fees like on eBay, only paying if your item actually sells.<br><br>If you are unfamiliar although variables that are used to index your website by integrated search engines then a search engine optimization company is placed in order. Obtain the site advice from engineered so came up at helpful tips of research for SEO's, after all they got themselves on the websites for now it's the perfect time to watch them help for you. For more on [https://www.youtube.com/watch?v=yZ363L_28io selling videos] look at the web site. The advice a person receive when implemented will show results surprisingly quickly.<br><br>Advertising Start tracking your competition's advertising. If they've kept liquids ad posted on Facebook up for months, the content you produce that ad is is simply because. As you track your competition's ads up over time, you can able to model their successes. That's all I got for today, do let me know people though about post their comments division!!!!!!! |
| | last1 = Solomon | first1 = A. I.
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| | last2 = Blasiak | first2 = P.
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| | last3 = Duchamp | first3 = G.
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| | last4 = Horzela | first4 = A.
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| | last5 = Penson | first5 = K.A.
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| | editor1-last = Gruber | editor1-first = Bruno J.
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| | editor2-last = Marmo | editor2-first = Giuseppe
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| | editor3-last = Yoshinaga | editor3-first = Naotaka
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| | arxiv = quant-ph/0310174
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| | contribution = Combinatorial physics, normal order and model Feynman graphs
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| | doi = 10.1007/1-4020-2634-X_25
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| | pages = 527–536
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| | publisher = Kluwer Academic Publishers
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| | title = Symmetries in Science XI
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| | year = 2005}}.</ref><ref>{{citation
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| | last1 = Blasiak | first1 = P.
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| | last2 = Dattoli | first2 = G.
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| | last3 = Horzela | first3 = A.
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| | last4 = Penson | first4 = K. A.
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| | last5 = Zhukovsky | first5 = K.
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| | issue = 1
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| | journal = Journal of Integer Sequences
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| | mr = 2377567
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| | at = Article 08.1.1
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| | title = Motzkin numbers, central trinomial coefficients and hybrid polynomials
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| | url = http://www.cs.uwaterloo.ca/journals/JIS/VOL11/Penson/penson131.html
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| | volume = 11
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| | year = 2008}}.</ref>
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| | |
| In [[graph theory]], a subset of the edges of a graph that touches each vertex at most once is called a [[matching (graph theory)|matching]]. The number of different matchings of a given graph is important in [[chemical graph theory]], where the graphs model molecules and the number of matchings is known as the [[Hosoya index]]. The largest possible Hosoya index of an ''n''-vertex graph is given by the [[complete graph]]s, for which any pattern of pairwise connections is possible; thus, the Hosoya index of a complete graph on ''n'' vertices is the same as the ''n''th telephone number.<ref>{{citation
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| | last1 = Tichy | first1 = Robert F.
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| | last2 = Wagner | first2 = Stephan
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| | doi = 10.1089/cmb.2005.12.1004
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| | issue = 7
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| | journal = [[Journal of Computational Biology]]
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| | pages = 1004–1013
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| | title = Extremal problems for topological indices in combinatorial chemistry
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| | url = http://www.math.tugraz.at/fosp/pdfs/tugraz_main_0052.pdf
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| | volume = 12
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| | year = 2005}}.</ref>
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| [[Image:Young tableaux for 541 partition.svg|thumb|A standard Young tableau]]
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| A [[Ferrers diagram]] is a geometric shape formed by a collection of ''n'' squares in the plane, grouped into a [[polyomino]] with a horizontal top edge, a vertical left edge, and a single monotonic chain of horizontal and vertical bottom and right edges. A standard [[Young tableau]] is formed by placing the numbers from 1 to ''n'' into these squares in such a way that the numbers increase from left to right and from top to bottom throughout the tableau.
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| According to the [[Robinson–Schensted correspondence]], permutations correspond one-for-one with ordered pairs of standard [[Young tableau]]x. Inverting a permutation corresponds to swapping the two tableaux, and so the self-inverse permutations correspond to single tableaux, paired with themselves.<ref name="b">A direct bijection between involutions and tableaux, inspired by the recurrence relation for the telephone numbers, is given by {{citation
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| | last = Beissinger | first = Janet Simpson
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| | doi = 10.1016/0012-365X(87)90024-0
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| | issue = 2
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | mr = 913181
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| | pages = 149–163
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| | title = Similar constructions for Young tableaux and involutions, and their application to shiftable tableaux
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| | volume = 67
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| | year = 1987}}.</ref> Thus, the telephone numbers also count the number of Young tableaux with ''n'' squares.<ref name="knuth"/> In [[representation theory]], the Ferrers diagrams correspond to the [[irreducible representation]]s of the [[symmetric group]] of permutations, and the Young tableaux with a given shape form a basis of the irreducible representation with that shape. Therefore, the telephone numbers give the sum of the degrees of the irreducible representations.
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| {{Chess diagram|=
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| | tright
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| |=
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| 8 |__|__|__|rl|__|__|__|__|=
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| 7 |__|__|__|__|__|__|rl|__|=
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| 6 |__|__|rl|__|__|__|__|__|=
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| 5 |rl|__|__|__|__|__|__|__|=
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| 4 |__|__|__|__|rl|__|__|__|=
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| 3 |__|__|__|__|__|__|__|rl|=
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| 2 |__|rl|__|__|__|__|__|__|=
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| 1 |__|__|__|__|__|rl|__|__|=
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| |A diagonally symmetric non-attacking placement of eight rooks on a chessboard
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| }}
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| In the [[Mathematical chess problem|mathematics of chess]], the telephone numbers count the number of ways to place ''n'' rooks on an ''n'' × ''n'' chessboard in such a way that no two rooks attack each other (the so-called [[Eight queens puzzle|eight rooks puzzle]]), and in such a way that the configuration of the rooks is symmetric under a diagonal reflection of the board. Via the [[Pólya enumeration theorem]], these numbers form one of the key components of a formula for the overall number of "essentially different" configurations of ''n'' mutually non-attacking rooks, where two configurations are counted as essentially different if there is no symmetry of the board that takes one into the other.<ref>{{citation
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| | last = Holt | first = D. F.
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| | issue = 404
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| | journal = The Mathematical Gazette
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| | jstor = 3617799
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| | pages = 131–134
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| | title = Rooks inviolate
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| | volume = 58
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| | year = 1974}}.</ref>
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| ==Recurrence==
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| The telephone numbers satisfy the [[recurrence relation]]
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| :<math>T(n) = T(n-1) + (n-1)T(n-2),</math>
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| first published in 1800 by [[Heinrich August Rothe]], by which they may easily be calculated.<ref name="knuth"/>
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| One way to explain this recurrence is to partition the ''T''(''n'') connection patterns of the ''n'' subscribers to a telephone system into the patterns in which the first subscriber is not calling anyone else, and the patterns in which the first subscriber is making a call. There are ''T''(''n'' − 1) connection patterns in which the first subscriber is disconnected, explaining the first term of the recurrence. If the first subscriber is connected to someone else, there are ''n'' − 1 choices for which other subscriber he or she is connected to, and ''T''(''n'' − 2) patterns of connection for the remaining ''n'' − 2 subscribers, explaining the second term of the recurrence.<ref name="chm"/>
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| ==Summation formula and approximation==
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| The telephone numbers may be expressed exactly as a [[summation]]
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| :<math>T(n) = \sum_{k=0}^{\lfloor n/2\rfloor}\binom{n}{2k}(2k-1)!! = \sum_{k=0}^{\lfloor n/2\rfloor}\frac{n!}{2^k (n-2k)! k!}.</math>
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| In each term of this sum, <math>k</math> gives the number of matched pairs, the [[binomial coefficient]] <math>\binom{n}{2k}</math> counts the number of ways of choosing the <math>2k</math> elements to be matched, and the [[double factorial]] <math>(2k-1)!! = (2k)!/(2^k k!)</math> is the product of the odd integers up to its argument and counts the number of ways of completely matching the <math>2k</math> selected elements.<ref name="knuth"/><ref name="chm"/> It follows from the summation formula and [[Stirling's approximation]] that, [[Asymptotic analysis|asymptotically]], ''T''(''n'') is approximately
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| :<math>T(n) = \left(\frac{n}{e}\right)^{n/2} \frac{e^{\sqrt{n}}}{(4e)^{1/4}}\bigl(1+o(1)\bigr).</math><ref name="knuth"/><ref name="chm">{{citation
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| | last1 = Chowla | first1 = S. | author1-link = Sarvadaman Chowla | |
| | last2 = Herstein | first2 = I. N. | author2-link = Israel Nathan Herstein
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| | last3 = Moore | first3 = W. K.
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| | doi = 10.4153/CJM-1951-038-3
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| | journal = [[Canadian Journal of Mathematics]]
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| | mr = 0041849
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| | pages = 328–334
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| | title = On recursions connected with symmetric groups. I
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| | volume = 3
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| | year = 1951}}.</ref><ref>{{citation
| |
| | last1 = Moser | first1 = Leo | author1-link = Leo Moser
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| | last2 = Wyman | first2 = Max
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| | doi = 10.4153/CJM-1955-021-8
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| | journal = [[Canadian Journal of Mathematics]]
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| | mr = 0068564
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| | pages = 159–168
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| | title = On solutions of ''x<sup>d</sup>'' = 1 in symmetric groups
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| | volume = 7
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| | year = 1955}}.</ref>
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| ==Generating function== | |
| The [[exponential generating function]] of the telephone numbers is
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| :<math>\sum_{n=0}^{\infty}\frac{T(n)x^n}{n!}=\exp\left(\frac{x^2}{2}+x\right).</math><ref name="chm"/><ref name="gfgt">{{citation
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| | last1 = Banderier | first1 = Cyril
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| | last2 = Bousquet-Mélou | first2 = Mireille
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| | last3 = Denise | first3 = Alain
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| | last4 = Flajolet | first4 = Philippe | author4-link = Philippe Flajolet
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| | last5 = Gardy | first5 = Danièle
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| | last6 = Gouyou-Beauchamps | first6 = Dominique
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| | arxiv = math/0411250
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| | doi = 10.1016/S0012-365X(01)00250-3
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| | issue = 1-3
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | mr = 1884885
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| | pages = 29–55
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| | title = Generating functions for generating trees
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| | volume = 246
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| | year = 2002}}.</ref>
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| In other words, the telephone numbers may be read off as the coefficients of the [[Taylor series]] of {{nowrap|exp(''x''<sup>2</sup>/2 + ''x'')}}, and the ''n''th telephone number is the value at zero of the ''n''th derivative of this function.
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| This function is closely related to the exponential generating function of the [[Hermite polynomial]]s, which are the [[matching polynomial]]s of the complete graphs.<ref name="gfgt"/>
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| The sum of absolute values of the coefficients of the ''n''th (probabilist) Hermite polynomial is the ''n''th telephone number, and the telephone numbers can also be realized as certain special values of the Hermite polynomials:<ref name="sbdhp"/><ref name="gfgt"/>
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| :<math>T(n)=\frac{\mathop{He}_n(i)}{i^n}.</math>
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| ==Prime factors==
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| For large values of ''n'', the ''n''th telephone number is divisible by a large [[power of two]], <math>2^{n/4+O(1)}</math>.
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| More precisely, the [[p-adic order|2-adic order]] (the number of factors of two in the [[prime factorization]]) of <math>T(4k)</math> or <math>T(4k+1)</math> is <math>k</math>; for <math>T(4k+2)</math> it is <math>k+1</math>, and for <math>T(4k+3)</math> it is <math>k+2</math>.<ref>{{citation
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| | last1 = Kim | first1 = Dongsu
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| | last2 = Kim | first2 = Jang Soo
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| | doi = 10.1016/j.jcta.2009.08.002
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| | issue = 8
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| | journal = [[Journal of Combinatorial Theory]] | series = Series A
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| | mr = 2677675
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| | pages = 1082–1094
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| | title = A combinatorial approach to the power of 2 in the number of involutions
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| | volume = 117
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| | year = 2010}}.</ref>
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| ==References==
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| {{reflist|colwidth=30em}}
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| [[Category:Integer sequences]]
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| [[Category:Enumerative combinatorics]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Matching]]
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| [[Category:Permutations]]
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