Concrete mathematics : a foundation for computer science

A ANSWERS TO EXERCISES 549
Incidentally, if each n-gon in Q is replaced by wzne2 we get
Q= l+z-dl-(4w+2)z+z2
211 +w)z
a formula in which the coefficient of wmzn ’ is the number of ways to divide
an n-gon into m polygons by nonintersecting diagonals.
7.51 The key first step is to observe that the square of the number of ways
is the number of cycle patterns of a certain kind, generalizing exercise 27.
These can be enumerated by evaluating the determinant of a matrix whose
eigenvalues are not difficult to determine. When m = 3 and n = 4, the fact
that cos36’ = $/2 is helpful (exercise 6.46).
7.52 The first few cases are PO(Y) = 1, pi(y) = y, pi = y2 + y,
P3(Y) = Y3 + 3Y2 + 3Y. Let p,(y) = qrn(x) where y = x(1 ~ x); we
seek a generating function that defines qln+l (x) in a convenient way. One
such function is x:, q,(x)z”/n! = 2eixZ/(eiZ + l), from which it follows that
q,,(x) = i”E, (x), where E,(x) is called an Euler polynomial. We have
t(-l)“x”bx = $(-1) ‘+’ E,(x), so Euler polynomials are analogous to Bernoulli
polynomials, and they have factors analogous to those in (6.98). By
exercise 6.23 we have nEnpl (x) = EL, (z)Bkx” k(2-2k+‘); this polynomial
has integer coefficients by exercise 6.54. Hence q2,, (x), whose coefficients
have denominators that are powers of 2, must have integer coefficients. Hence
p,(y) has integer coefficients. Finally, the relation (4y - l)pl(y) + 2pk(y) =
Ln(2n - 1 )p,-l (y) shows that
n - l
2m(2m-1) n = m(m+l) m: , + 2n(2n - 1)
I m I
I I
I m - l I ’
and it follows that the I:[’ s are positive. (A similar proof shows that the
related quantity (-l)n(2n + 2)Eln+l (x)/(2x - 1) has positive integer coefficients,
when expressed as an nth degree polynomial in y.) It can be shown
that I;‘1 is the Genocchi number (-l)np’ (22n+’ ~ 2)B2, (see exercise 6.24),
and that Inn,1 = (;), Inn21 = 2(“qf’) +3(T), etc.
7s53 It is P(l+V~~,-,+V4,,r~1/6.
PI65 = 40755.
,
Thus, for example, TJO = PJL = 210; T2s5 =
7.54 Let Ek be the operation on power series that sets all coefficients to zero
except those of zn where n mod m = k. The stated construction is equivalent
to the operation
EoSEoS(Eo+El)S . . . S(Eo+E, +...+E,,m,) /