Heinrich August Rothe

Heinrich August Rothe (born September 3, 1773 in Dresden , † 1842 in Erlangen ) was a German mathematician who dealt with combinatorics . He was a student of Carl Friedrich Hindenburg and taught as a professor at the universities in Leipzig and Erlangen . The Rothe-Hagen identity and the Rothe diagram are named after him.

Life

Rothe was born on September 3, 1773 in Dresden and attended the Kreuzschule from 1785 . He enrolled in law at the University of Leipzig in 1789 , but soon switched to mathematics. In 1792 he acquired his master's degree under the direction of Carl Friedrich Hindenburg . He was appointed lecturer there in 1793 and associate professor in 1796. In 1800 he was elected a corresponding member of the Göttingen Academy of Sciences . In 1804 he went to the University of Erlangen as a full professor , where he took over the chair from Karl Christian von Langsdorf . In 1818 he was accepted into the German Academy of Sciences Leopoldina . He retired in 1823 at the age of 50 and died in 1842. His chair was taken over by Johann Wilhelm Pfaff , the younger brother of Johann Friedrich Pfaff .

research

In his dissertation from 1793, he developed the Rothe-Hagen identity , a sum formula for binomial coefficients , which was named after him and Johann Georg Hagen . The work also includes a formula for computing the Taylor series of the inverses of a function from the Taylor series of the function itself, which is related to the Lagrangian inversion theorem .

Rothe diagram of the
permutation (2,4,1,3,5)

	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle 3}$	${\displaystyle 4}$	${\displaystyle 5}$
${\displaystyle 1}$	${\displaystyle \times }$	${\displaystyle \bullet }$
${\displaystyle 2}$	${\displaystyle \times }$		${\displaystyle \times }$	${\displaystyle \bullet }$
${\displaystyle 3}$	${\displaystyle \bullet }$
${\displaystyle 4}$			${\displaystyle \bullet }$
${\displaystyle 5}$					${\displaystyle \bullet }$

In his work on permutations from 1800, Rothe first defined the inverse of a permutation . He also developed a technique for visualizing permutations now known as the Rothe diagram . A Rothe diagram is a square scheme that has a point in a cell if the permutation maps the element to the element and a cross in each cell , for which a point later in the same row and another point later in the same column stands. The crosses then mark the deficiencies in the permutation. Since the Rothe diagram of the inverse permutation is the transposed diagram of the starting position, he was able to show that the number of deficiencies does not change due to the inversion. He was able to further show that the determinant of a transposed matrix is ​​the same as that of the output matrix . If the determinant is developed into a polynomial , each term corresponds to a permutation, the sign of the term corresponding to the sign of the permutation, which in turn can be determined via the error number. Since each term of the determinant of the transposed matrix corresponds to a term of the output matrix with the corresponding inverse permutation and the missing number does not change, the two determinants must be the same. ${\displaystyle (i,j)}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle (i,j)}$

In this work Rothe also considered self-inverse permutations for the first time, i.e. permutations that are equal to their inverses or, equivalently, have a symmetrical Rothe diagram. He found the recurrence for the number of these permutations

${\displaystyle T(n)=T(n-1)+(n-1)T(n-2)}$,

whose solution is the result

${\displaystyle 1,2,4,10,26,76,232,764,2620,9496,\ldots }$    (Follow A000085 in OEIS )

is. This sequence also counts the number of possible Young tableaus and the number of matchings in a complete graph . In 1811 Rothe continued to formulate the q -inomial formula, a generalization of the binomial theorem .

Selected publications

• Formulae De Serierum Reversione Demonstratio Universalis Signis Localibus Combinatorio-Analyticorum Vicariis Exhibita: Dissertatio Academica , Leipzig 1793.
• About permutations, in relation to the places of their elements. Application of the derived theorems to the elimination problem . In Carl Hindenburg (ed.): Collection of combinatorial-analytical treatises. Bey G. Fleischer dem youngern, 1800, pp. 263-305.
• Handbook of pure mathematics / systematic textbook of arithmetic. two volumes, Barth, Leipzig 1804 and 1811.
• Theory of combinatorial integrals. Riegel and Wießner, Nuremberg 1820.

Individual evidence

1. ↑ Bernd Bekemeier: Martin Ohm, 1792–1872: University and school mathematics in the neo-humanist educational reform (=  studies on the scientific, social and educational history of mathematics . Volume 4 ). Vandenhoeck & Ruprecht, 1987, ISBN 3-525-40311-9 , pp. 83 .
2. ↑ Hans Niels Jahnke: Mathematics and education in the Humboldtian reform (=  studies on the scientific, social and educational history of mathematics . Volume 8 ). Vandenhoeck & Ruprecht, 1990, ISBN 3-525-40315-1 , pp. 175 .
3. ↑ Holger Krahnke: The members of the Academy of Sciences in Göttingen 1751-2001 (= Treatises of the Academy of Sciences in Göttingen, Philological-Historical Class. Volume 3, Vol. 246 = Treatises of the Academy of Sciences in Göttingen, Mathematical-Physical Class. Episode 3, vol. 50). Vandenhoeck & Ruprecht, Göttingen 2001, ISBN 3-525-82516-1 , p. 206.
4. ^ Karl Immanuel Gerhardt: History of Mathematics in Germany (=  History of Sciences in Germany: Modern Times . Volume 17 ). R. Oldenbourg, 1877, p. 204 .
5. ↑ David E. Rowe: In search of Steiner's Ghosts: Imaginary elements in the nineteenth-century geometry . In: Dominique Flament (ed.): Le Nombre: une Hydre à n visages, Entre nombres complexes et vecteurs . Fondation Maison des Sciences de l'Homme , 1997, p. 193-208 .
6. ^ Moritz Cantor:  Rothe, Heinrich August . In: Allgemeine Deutsche Biographie (ADB). Volume 29, Duncker & Humblot, Leipzig 1889, p. 349 f.
7. ^ HW Gould: Some generalizations of Vandermonde's convolution . In: American Mathematical Monthly . tape 63 , 1956, pp. 84-91 .
8. ↑ Ronald Calinger: Vita Mathematica: Historical Research and Integration With Teaching (=  Mathematical Association of America notes . Band 40 ). Cambridge University Press, 1996, ISBN 0-88385-097-4 , pp. 146-147 .
9. ^ Donald E. Knuth: The Art of Computer Programming , Volume 3: Sorting and Searching . Addison-Wesley, Reading, Mass. 1973, p. 14-15 .
10. ^ Donald E. Knuth: The Art of Computer Programming , Volume 3: Sorting and Searching . Addison-Wesley, Reading, Mass. 1973, p. 48, 56 .
11. ↑ DM Bressoud: Some identities for terminating q -series . In: Mathematical Proceedings of the Cambridge Philosophical Society . tape 89 , no. 2 , 1981, p. 211-223 .
12. ↑ HB Benaoum: h -analogue of Newton's binomial formula . In: Journal of Physics A: Mathematical and General . tape 31 , no. 46 , p. L751-L754 .