At the end of the 18th century and the beginning of the 19th, combinatorics flourished in German mathematics research, under the leadership of Carl Hindenburg. Later opinion has not been kind:

"ill-advised and purposeless modification"Sir Thomas Muir 1906

"of limited scope and restricted application"Encyclopaedia Brittanica 1910

"mired in the mathematical trivia by which the School itself was plagued"Manning 1975

"The faults of his [Euler's] time found their culmination in the Combinatorial School in Germany, which has now passed into oblivion"Cajori 1991

"thousands of pages filled with esoteric symbolism that must have impressed many nonmathematicians" — Knuth 2006

Knuth goes on to call Heinrich August Rothe "Hindenberg's best student", and says that his work is "not completely trivial", but cites as evidence only an algorithm for finding the successor and predecessor of a Morse code sequence in lexicographic order.

Here are some Rothe's other contributions, from just one of his publications:

  • The first definition of the inverse of a permutation
  • A proof that the number of inversions of a permutation (the concept that Muir called "ill-advised and purposeless") is the same as for its inverse
  • A proof that the determinant of a matrix is the same as for its transpose
  • A diagram (the Rothe diagram) still used for visualization of permutations and inversions
  • The first definition of a self-inverse permutation
  • A simple recurrence formula for the number of self-inverse permutations

Not completely trivial?!

Yes, these results are easy nowadays, but in part that's because we learn about permutations from our beginning years of college, if not earlier. And where did what we learn about come from?