Heinrich August Rothe

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Heinrich August Rothe (1773–1842) was a German mathematician, a professor of mathematics at Erlangen. Heinrich August Rothe_sentence_0

He was a student of Carl Hindenburg and a member of Hindenberg's school of combinatorics. Heinrich August Rothe_sentence_1

Biography Heinrich August Rothe_section_0

Rothe was born in 1773 in Dresden, and in 1793 became a docent at the University of Leipzig. Heinrich August Rothe_sentence_2

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. Heinrich August Rothe_sentence_3

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. Heinrich August Rothe_sentence_4

Research Heinrich August Rothe_section_1

The Rothe–Hagen identity, a summation formula for binomial coefficients, appeared in Rothe's 1793 thesis. Heinrich August Rothe_sentence_5

It is named for him and for the later work of Johann Georg Hagen. Heinrich August Rothe_sentence_6

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. Heinrich August Rothe_sentence_7

In the study of permutations, Rothe was the first to define the inverse of a permutation, in 1800. Heinrich August Rothe_sentence_8

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. Heinrich August Rothe_sentence_9

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. Heinrich August Rothe_sentence_10

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. Heinrich August Rothe_sentence_11

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. Heinrich August Rothe_sentence_12

He found the recurrence relation Heinrich August Rothe_sentence_13

for counting these permutations, which also counts the number of Young tableaux, and which has as its solution the telephone numbers Heinrich August Rothe_sentence_14

Heinrich August Rothe_description_list_0

  • 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence in the OEIS).Heinrich August Rothe_item_0_0

Rothe was also the first to formulate the q-binomial theorem, a q-analog of the binomial theorem, in an 1811 publication. Heinrich August Rothe_sentence_15

Selected publications Heinrich August Rothe_section_2

Heinrich August Rothe_unordered_list_1

  • , Leipzig, 1793.Heinrich August Rothe_item_1_1
  • "". In Hindenburg, Carl, ed., Sammlung Combinatorisch-Analytischer Abhandlungen, pp. 263–305, Bey G. Fleischer dem jüngern, 1800.Heinrich August Rothe_item_1_2
  • Systematisches Lehrbuch der Arithmetik, Leipzig, 1811Heinrich August Rothe_item_1_3


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Heinrich August Rothe.