A Problem From The 2005 Putnam Competition
A link to the Putnam competition website appears in the section External links. The problem asks for a proof that
where the sum is over all permutations of, is the sign of, i.e. if is even and if is odd, and is the number of fixed points of .
Now the sign of is given by
where the product is over all cycles c of, as explained e.g. on the page on even and odd permutations.
Hence we consider the combinatorial class
where marks one minus the length of a contributing cycle, and marks fixed points. Translating to generating functions, we obtain
or
Now we have
and hence the desired quantity is given by
Doing the computation, we obtain
or
Extracting coefficients, we find that the coefficient of is zero. The constant is one, which does not agree with the formula (should be zero). For positive, however, we obtain
or
which is the desired result.
As an interesting aside, we observe that may be used to evaluate the following determinant of an matrix:
where . Recall the formula for the determinant:
Now the value of the product on the right for a permutation is, where f is the number of fixed points of . Hence
which yields
and finally
Read more about this topic: Random Permutation Statistics
Famous quotes containing the words problem, putnam and/or competition:
“The problem of induction is not a problem of demonstration but a problem of defining the difference between valid and invalid
predictions.”
—Nelson Goodman (1906)
“Even American women are not felt to be persons in the same sense as the male immigrants among the Hungarians, Poles, Russian Jews,not to speak of Italians, Germans, and the masters of all of usthe Irish!”
—Mary Putnam Jacobi (18421906)
“Never before has a generation of parents faced such awesome competition with the mass media for their childrens attention. While parents tout the virtues of premarital virginity, drug-free living, nonviolent resolution of social conflict, or character over physical appearance, their values are daily challenged by television soaps, rock music lyrics, tabloid headlines, and movie scenes extolling the importance of physical appearance and conformity.”
—Marianne E. Neifert (20th century)