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'''Heinrich August Rothe''' (1773–1842) was a German mathematician, a professor of mathematics at [[University of Erlangen-Nuremberg|Erlangen]]. He was a student of [[Carl Hindenburg]] and a member of Hindenberg's school of [[combinatorics]].<ref>{{citation|title=Martin Ohm, 1792-1872: Universitäts- und Schulmathematik in der neuhumanistischen Bildungsreform|volume=4|series=Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik|first=Bernd|last=Bekemeier|publisher=Vandenhoeck & Ruprecht|year=1987|isbn=9783525403112|page=83|language=German}}.</ref><ref>{{citation|title=Mathematik und Bildung in der Humboldtschen Reform|volume=8|series=Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik|first=Hans Niels|last=Jahnke|publisher=Vandenhoeck & Ruprecht|year=1990|isbn=9783525403150|page=175|language=German}}.</ref>
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==Biography==
Rothe was born in 1773 in [[Dresden]], and in 1793 became a docent at the [[University of Leipzig]]. He became an extraordinary professor at Leipzig in 1796, and in 1804 he moved to Erlangen as a full professor, taking over the chair formerly held by [[Karl Christian von Langsdorf]]. He died in 1842, and his position at Erlangen was in turn taken by Johann Wilhelm Pfaff, the brother of the more famous mathematician [[Johann Friedrich Pfaff]].<ref>{{citation|title=Geschichte der Mathematik in Deutschland|volume=17|series=Geschichte der Wissenschaften in Deutschland: Neuere Zeit|first=Karl Immanuel|last=Gerhardt|publisher=R. Oldenbourg|year=1877|page=204|url=http://books.google.com/books?id=ExYLAAAAQAAJ&pg=PA204|language=German}}.</ref><ref>{{citation|first=David E.|last=Rowe|contribution=In search of Steiner's Ghosts : Imaginary elements in the nineteenth-century geometry|title=Le Nombre : une Hydre à ''n'' visages, Entre nombres complexes et vecteurs|editor-first=Dominique|editor-last=Flament|pages=193–208|publisher=Fondation Maison des sciences de l'homme|year=1997}}.</ref>
 
==Research==
The [[Rothe–Hagen identity]], a [[summation]] formula for [[binomial coefficient]]s, appeared in Rothe's 1793 thesis. It is named for him and for the later work of [[Johann Georg Hagen]].<ref>{{citation
| last = Gould | first = H. W.
| journal = [[The American Mathematical Monthly]]
| jstor = 2306429
| mr = 0075170
| pages = 84–91
| title = Some generalizations of Vandermonde's convolution
| volume = 63
| year = 1956}}.</ref> The same thesis also included a formula for computing the [[Taylor series]] of an [[inverse function]] from the Taylor series for the function itself, related to the [[Lagrange inversion theorem]].<ref>{{citation|title=Vita Mathematica: Historical Research and Integration With Teaching|volume=40|series=Mathematical Association of America Notes|first=Ronald|last=Calinger|publisher=Cambridge University Press|year=1996|isbn=9780883850978|pages=146–147|url=http://books.google.com/books?id=D21wKHoYGg0C&pg=PA146}}.</ref>
 
In the study of [[permutation]]s, Rothe was the first to define the inverse of a permutation, in 1800. He developed a technique for visualizing permutations now known as a Rothe diagram, a square table that has a dot in each cell (''i'',''j'') for which the permutation maps position ''i'' to position ''j'' and a cross in each cell (''i'',''j'') for which there is a dot later in row ''i'' and another dot later in column ''j''. Using Rothe diagrams, he showed that the number of [[Inversion (discrete mathematics)|inversions]] in a permutation is the same as in its inverse, for the inverse permutation has as its diagram the [[transpose]] of the original diagram, and the inversions of both permutations are marked by the crosses. Rothe used this fact to show that the [[determinant]] of a [[Matrix (mathematics)|matrix]] is the same as the determinant of the transpose: if one expands a determinant as a [[polynomial]], each term corresponds to a permutation, and the sign of the term is determined by the [[parity (mathematics)|parity]] of its number of inversions. Since each term of the determinant of the transpose corresponds to a term of the original matrix with the inverse permutation and the same number of inversions, it has the same sign, and so the two determinants are also the same.<ref>{{citation
| last = Knuth | first = Donald E. | author-link = Donald Knuth
| location = Reading, Mass.
| mr = 0445948
| pages = 14–15
| publisher = Addison-Wesley
| title = [[The Art of Computer Programming]], Volume 3: Sorting and Searching
| year = 1973}}.</ref>
 
In his 1800 work on permutations, Rothe also was the first to consider permutations that are [[Involution (mathematics)|involutions]]; that is, they are their own inverse, or equivalently they have symmetric Rothe diagrams. He found the [[recurrence relation]]
:<math>T(n) = T(n-1) + (n-1)T(n-2)</math>
for [[Enumerative combinatorics|counting]] these permutations, which also counts the number of [[Young tableau]]x, and which has as its solution the [[Telephone number (mathematics)|telephone numbers]]
:1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... {{OEIS|A000085}}.<ref>{{harvtxt|Knuth|1973}}, pp. 48 & 65.</ref>
 
Rothe was also the first to formulate the [[Basic hypergeometric series#The q-binomial theorem|''q''-binomial theorem]], a [[q-analog|''q''-analog]] of the [[binomial theorem]], in an 1811 publication.<ref>{{citation
| last = Bressoud | first = D. M.
| doi = 10.1017/S0305004100058114
| issue = 2
| journal = Mathematical Proceedings of the Cambridge Philosophical Society
| mr = 600238
| pages = 211–223
| title = Some identities for terminating ''q''-series
| volume = 89
| year = 1981}}.</ref><ref>{{citation
| last = Benaoum | first = H. B.
| arxiv = math-ph/9812011
| doi = 10.1088/0305-4470/31/46/001
| issue = 46
| journal = Journal of Physics A: Mathematical and General
  | pages = L751–L754
| title = ''h''-analogue of Newton's binomial formula
| volume = 31}}.</ref>
 
==Selected publications==
* ''[http://books.google.com/books/about/Formulae_De_Serierum_Reversione_Demonstr.html Formulae De Serierum Reversione Demonstratio Universalis Signis Localibus Combinatorio-Analyticorum Vicariis Exhibita: Dissertatio Academica]'', Leipzig, 1793.
* "[http://astronomie-rara.ethbib.ethz.ch/zut/content/pageview/1341041 Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. Anwendung der daraus abgeleiteten Satze auf das Eliminationsproblem]". In Hindenberg, Carl, ed., ''Sammlung Combinatorisch-Analytischer Abhandlungen'', pp. 263–305, Bey G. Fleischer dem jüngern, 1800.
* ''Systematisches Lehrbuch der Arithmetik'', Leipzig, 1811
 
==References==
{{reflist}}
 
{{Persondata
| NAME              = Rothe, Heinrich August
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = German mathematician
| DATE OF BIRTH    = 1773
| PLACE OF BIRTH    =
| DATE OF DEATH    = 1842
| PLACE OF DEATH    =
}}
{{DEFAULTSORT:Rothe, Heinrich August}}
[[Category:1773 births]]
[[Category:1842 deaths]]
[[Category:German mathematicians]]
[[Category:Combinatorialists]]
[[Category:University of Leipzig alumni]]
[[Category:University of Leipzig faculty]]
[[Category:University of Erlangen-Nuremberg faculty]]

Latest revision as of 23:46, 1 December 2014

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