Heinrich August Rothe

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

Life

Rothe was born on 3 September 1773 in Dresden and visited from 1785 Cross school. He matriculated in 1789 at the University of Leipzig in trade law, but soon changed to mathematics. In 1792, he earned his master's degree under the direction of Carl Friedrich Hindenburg. He was promoted to lecturer in 1796 and to associate professor in 1793. In 1804 he was a full professor at the University of Erlangen, where he took over the chair of Karl Christian von Langsdorf. In 1818 he was elected to the German Academy of Sciences Leopoldina. He went in 1823 at the age of 50 years in retirement, and died in 1842. His chair was taken over by John William Pfaff, the younger brother of Johann Friedrich Pfaff.

Research

In his dissertation in 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 contains a formula for calculating the Taylor series of the inverse of a function from the Taylor series of the function itself, which is related to the Lagrange inversion theorem.

In his work on permutations from 1800 Rothe first defined the inverse of a permutation. He also developed a technique for the visualization of permutations, which is known as Rothe diagram today. Rothe, a diagram is a square pattern having a dot in a cell when the permutation mapping the element to the element and a cross in each cell for which a later point in the same line, and another point subsequently in the same column, stands. The crosses then select the incorrect readings of the permutation. After the Rothe diagram of the inverse permutation is the transposed diagram of the starting position, so he could show that the number of faulty items does not change by the inversion. Thus he was able to show that the determinant of a transpose matrix is equal to the output matrix. If in fact the determinant developed in a polynomial, each term corresponds to a permutation, where the sign of the term corresponds to the sign of the permutation, which in turn can be determined by the number of faulty state. After each term of the determinant of the transposed matrix corresponds to a term in the output matrix with the corresponding inverse permutation, and the incorrect status number is not changed, the two determinants must be equal.

Next Rothe considered in this work for the first time self- inverse permutations, ie, permutations that are its inverse is equal to or possess equivalent to a symmetric Rothe diagram. For the number of these permutations he found the recurrence

The solution of the follow-

Is. This sequence also counts the number of possible Young tableaux and the number of matchings in a complete graph. Rothe formulated in 1811 to continue the q- binomial formula, a generalization of the binomial theorem.

Selected Publications

• Heinrich August Rothe: Formulae De Serierum Reversione demonstrations Universalis Signis Localibus Combinatorio - Analyticorum Vicariis EXHIBITA: Dissertatio Academica, Leipzig, 1793.
• Heinrich August Rothe: About permutations, in relation to the locations of its elements. Application of the derived propositions on the elimination problem. In Carl Hindenburg (ed.), Collection Combinato Risch- Analytical Essays, pp. 263-305, Bey G. Fleischer the younger, in 1800.
• Heinrich August Rothe: Handbook of pure mathematics / Systematic textbook of arithmetic, two volumes, Barth, Leipzig, 1804 and 1811.
• Heinrich August Rothe: Theory of combi -natorial integrals, bolts and Wießner, Nuremberg, 1820.