Algebraic and enumerative combinatorics, Spring 2016
PROBLEM SET 1
To be handed in no later than March 1. You may cooperate but you must write your solutions
by yourselves. Please write full proofs! Each problem is worth 5p.
(1) Prove that the Hadamard product of two rational series S, T ∈ Krat hhXii is rational.
(2) Let X be an alphabet and L ⊆ X ∗ . Define an equivalence relation on X ∗ by w1 ∼ w2 if
and only if for all v ∈ X ∗ :
vw1 ∈ L
if and only if
vw2 ∈ L.
Prove that if ∼ has a finite number of equivalence classes, then the series
X
S=
w
w∈L
is rational.
(3) Let X = {1, 2, . . . , n} and let v = v1 v2 · · · v` ∈ X ∗ . An occurence of v in another word
w = w1 · · · wm ∈ X ∗ is a sequence 1 ≤ t1 < t2 < · · · < t` ≤ m such that for all
i, j ∈ {1, . . . , `}
vi < vj if and only if wti < wtj .
For k ∈ N, let Lk (v) ⊆ X ∗ be the set of words that has exactly k occurrences of v. Prove
that
X
Sk (v) :=
w
w∈Lk (v)
is rational. Hint. Use (2) and try k = 0 first.
(4) Prove that Kalg hhXii is a subalgebra of KhhXii. Prove also that if S ∈ Kalg hhXii is
invertible, then S −1 ∈ Kalg hhXii.
(5) Let X be a finite alphabet and x ∈ X. If w = x1 · · · xn ∈ X ∗ and 1 ≤ i ≤ n, let
(
x1 · · · xi−1 xi+1 · · · xn if xi = x,
(i)
w =
0
if xi 6= x,
and let
n
X
∂
w=
w(i) .
∂x
i=1
Extend ∂/∂x to a linear operator on KhhXii and prove that if S ∈ Kalg hhXii, then
∂S/∂x ∈ Kalg hhXii.
(6) Suppose S ∈ Calg hhXii, where X is finite. If w = x1 x2 · · · xn ∈ X ∗ , let `(w) = n denote
the length of w. Prove that there is a constant C such that
X
|hS, wi| ≤ C n ,
`(w)=n
for all n ∈ N (where the sum is over all w ∈ X ∗ of length n).