Heinrich August Rothe
Heinrich August Rothe (1773–1842) was a German mathematician, a professor of mathematics at Erlangen.
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.
It is named for him and for the later work of Johann Georg Hagen.
In the study of permutations, 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 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 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 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.
In his 1800 work on permutations, Rothe also was the first to consider permutations that are involutions; that is, they are their own inverse, or equivalently they have symmetric Rothe diagrams.
He found the recurrence relation
- 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence in the OEIS).
- , Leipzig, 1793.
- "". In Hindenburg, Carl, ed., Sammlung Combinatorisch-Analytischer Abhandlungen, pp. 263–305, Bey G. Fleischer dem jüngern, 1800.
- Systematisches Lehrbuch der Arithmetik, Leipzig, 1811
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Heinrich August Rothe.