Qualifying Exam Problems
due December 2, 2011
You may use EC1 (first or second editions) and EC2, but no other sources including the
internet or other people. You may also use a software package such as Maple or Mathematica.
Do as many problems as possible. It is not necessary to do all the problems to pass. Some
of the problems are quite difficult.
1. Show that the number of permutations w ∈ Sn fixed by the fundamental transforma∧
tion Sn → Sn of Proposition 1.3.1 (i.e., w = w)
! is the Fibonacci number Fn+1 .
2. Prove that the number of partitions of n in which no part appears exactly once equals
the number of partitions of n into parts not congruent to ±1 (mod 6).
3. Find the number f (n) of binary sequences w = a1 a2 · · · ak (where k is arbitrary) such
that a1 = 1, ak = 0, and inv(w) = n. For instance, f (4) = 5, corresponding to the
sequences 10000, 11110, 10110, 10010, 1100. How many of these sequences have exactly
j 1’s?
4. Let P be a finite n-element poset. Simplify the two sums
"
¯
f (P ) =
e(I)e(I)
I∈J(P )
g(P ) =
" #n$
¯
e(I)e(I),
#I
I∈J(P )
where I¯ denotes the complement P − I of the order ideal I.
5. Let P be a finite poset. For t ∈ P let Vt = {s ∈ P : s ≥ t}. Simplify the sum
f (P ) =
"
t1 <···<tn
1
,
(#Vt1 − 1) · · · (#Vtn−1 − 1)
where the sum ranges over all nonempty chains t1 < · · · < tn of P for which tn is a
maximal element of P . (When n = 1, the empty product in the denominator equals
1
= 3.
1.) For instance, if P is a 3-element chain then we get 11 + 11 + 12 + 1·2
6. Let A be the hyperplane arrangement in Rn with the hyperplanes
xi = 0, 1 ≤ i ≤ n
xi + xi+1 = 0, 1 ≤ i ≤ n − 1
x1 + xn = 0.
Find the characteristic polynomial of A.
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7. Let f (n, i) denote the total number of intervals of W (Sn ) (the weak Bruhat order on
the symmetric group Sn ) that are isomorphic to the boolean algebra Bi . Show that
""
f (n, i)q i
n≥0 i≥0
xn
1
=
n!
1−x−
qx2
2
x3
x2
+ (6q + 6)
2!
3!
4
x
x5
+(6q 2 + 36q + 24) + (90q 2 + 240q + 120)
4!
5!
6
x
(90q 3 + 1080q 2 + 1800q + 720) + · · · .
6!
= 1 + x + (q + 2)
8. Let P be an Eulerian poset of rank d + 1 with d atoms, such that P −{1̂} is a simplicial
poset. Show that if d is even, then P has an even number of coatoms.
9. (a) For n ≥ 0 let
pn (u) =
"
uν(T ) ,
T
where T runs over all tilings of a 1 × n rectangle with 1 × 1 squares (monomers)
and 1 × 2 rectangles (dimers). Show that
"
pn (u)xn =
n≥0
(b) Use combinatorial reasoning to find
10. Let α ∈ C, and define for n ∈ N,
fα (n) =
%
1
.
1 − ux − x2
n
n≥0 pn (u)pn (v)x .
$
n #
"
n−k
k=0
k
αk .
%
Show that Fα (x) := n≥0 fα (n)xn is a rational function, and compute it explicitly.
Find an explicit formula for fα (n). What value of α requires special treatment?
11. (a) Let f (n) denote the number of rooted trees on the vertex set [n] whose endpoints
(leaves) are colored either red or blue. Find a functional equation (analogous to
the equation y = xey satisfied by the exponential generating function for rooted
trees) satisfied by the exponential generating function
y =
"
f (n)
n≥1
= 2x + 4
xn
n!
x3
x4
x2
+ 24 + 224 + · · · .
2!
3!
4!
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(b) Use the Lagrange inversion formula and (a) to deduce that
f (n) =
n # $
"
n
k=0
k
k n−1 .
(When n = 1 and k = 0 in the above sum, set 00 = 1.)
12. Let f (n) for n ≥ 2 be the number of Eulerian digraphs on the vertex set [n] with
no loops and with exactly one Eulerian tour (up to cyclic shift). Show that f (n) =
1
(n − 1)!Cn = (2n − 1)n−2, where Cn denotes a Catalan number.
2
%
13. Let yn = λ%n s2λ . Find the generating function
"
F (q) =
'yn , yn (q n .
n≥0
&
Express your answer in terms of the generating function P (x, t) = i≥1 (1 − txi )−1 .
Here sλ is a Schur function, and '·, ·( is the standard scalar product on symmetric
functions.
14. Fix integers 1 ≤ m ≤ n. Find simple formulas for the four sums
" " "
a(m, n) =
f µ f ν f λ cλµν
µ%m ν%n−m λ%n
b(m, n) =
" " "
f µ f ν cλµν
µ%m ν%n−m λ%n
c(m, n) =
" " "
f ν f λ cλµν
µ%m ν%n−m λ%n
d(m, n) =
" " "
f λ cλµν ,
µ%m ν%n−m λ%n
where cλµν denotes a Littlewood-Richardson coefficient and f λ is the number of standard
Young tableaux of shape λ. Some of the formulas may involve the number t(k) of
involutions in Sk for certain k.
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