
In Many Circles. Permutations as Products of Cycles. 135
(b) What can we say about the number of all n-permutations having
exactly one fixed point whose descent set is equal to {i}?
61. (+) Find a bijective proof of the equality SQ
2n
· (2n +1)=SQ
2n+1
.
62. Prove, without using generating functions, that SQ
2n+1
· (2n +2) ≥
SQ
2n+2
.
63. Exercise 6 of Chapter 1 provided a probabilistic interpretation for the
Eulerian numbers. Why cannot we use that interpretation to deduce by
L´evy’s theorem (Theorem 3.37) that the Eulerian polynomials have real
roots only? Note that the Eulerian polynomials have irrational roots in
general, while the interpretation of the