Problem set 3
Due date: 04/01/2016
Before you start working on the homework problems read the extended
syllabus1 carefully and check which problems you need to solve.
1. Practice problems
P1. Let π be a permutation, and let P (π) be the insertion tableaux obtained
from π by the RSK-algorithm. Let the shape of P (π) be λ1 ≥ λ2 ≥ · · · ≥
λm > 0.
(a) Show that the longest increasing sequence in π has length λ1 .
(b) Show that the longest decreasing sequence in π has length m.
P2. Show that
)
∞ (
∑
n+m−1
n
n=0
tn =
q
1
(1 − qt)(1 −
q 2 t) . . . (1
− q m−1 t)
.
2. Homework problems
1. Execute the RSK-algorithm to the permutation 4 8 3 1 7 11 5 6 9 2 12 10.
(Give the partial Young-tableauxs too, not just the final answer.)
2. For a permutation π ∈ Sn let I(π) = |{(i, j) | i < j and π(i) > π(j)}|.
Show that
∑
q I(π) = (n)q !.
π∈Sn
(Recall that (n)q ! = (1 + q)(1 + q + q 2 ) . . . (1 + q + q 2 + · · · + q n−1 ).)
3. Show that
( )
k
n k
(
)
q 2
t = (1 + t)(1 + qt) . . . (1 + q n−1 t).
k
q
k=0
n
∑
4. Let p(n) be the number of partitions of n, where p(0) = 1. Let sn (i) be the
number of i’s in all partitions of n.
1
The extended syllabus can be found at
http://math.mit.edu/~csikvari/extended_syllabus_algebraic_combinatorics.pdf
1
2
(a) Show that sn (1) = p(n − 1) + p(n − 2) + · · · + p(0).
(b) Find similar formulas for sn (2), sn (3), . . . .
(c) Show that
1∑
p(n) =
σ(k)p(n − k),
n k=1
n
where σ(k) is the sum of the positive divisors of k: σ(k) =
∑
d | k
d≥1
d.
5. Show that
∞
∑
k=0
2
∞
∏
1
xk
=
.
2
k
2
n
((1 − x)(1 − x ) . . . (1 − x ))
1
−
x
n=1
(In the above sum k = 0 simply means that 1 is the first term.)
6. Let x, y be variables such that yx = qxy. Show that
n ( )
∑
n
n
xk y n−k .
(x + y) =
k
q
k=0
7. Let (λ1 , . . . , λn ) be a partition of n: λ1 ≥ · · · ≥ λm > 0 and λ1 +· · ·+λm = n.
1
1
Let A = (aij ) be the m × m matrix with entries aij = (λi −i+j)!
. (Note that m!
is defined to be 0 if m < 0.) Show that f λ = n! det(A).
8. Let λ be a partition of n, and let f λ denote the number of standard Youngtableauxs of shape λ. Prove that
∑
f µ = (n + 1)f λ .
µ>λ
µ⊢n+1
(The notation µ ⊢ n + 1 means that µ is a partition of n + 1, and µ > λ means
that the Young-diagram of µ contains the Young-diagram of λ.)
9. Let m, n be positive integers. Let λ be a partition fitting into the m × n
rectangle. Let λ be the complement of λ with respect to this rectangle, so if λ =
(λ1 , . . . , λm ) such that n ≥ λ1 ≥ λ2 · · · ≥ λm ≥ 0, then λ = (n−λm , . . . , n−λ1 ).
Let f (λ) be the number of standard Young-tableauxs with shape λ. Let
∑
s(m, n) =
f (λ)f (λ),
λ
where the sum runs over all partitions fitting into the m×n rectangle including
the partition of 0 for which f (0) = 1. What is s(m, n)?
3
10. Let p(n) be the number of partitions of n. Let r(n) be the number of
partitions of n into distinct odd numbers. Show that p(n) ≡ r(n) (mod 2).
11. What is the set of eigenvalues of the bipartite graph Gn,k whose classes
consist of the k and k + 1 element subsets of the set {1, 2, . . . , n} and a set S
of size k and a set T of size k + 1 are adjacent if S ⊂ T ?
12. Let n = pα1 1 · . . . · pαr r be a positive integer with its prime factorization.
What is the maximal size of a subset D of the divisors of n if d1 does not
divide d2 whenever d1 , d2 ∈ D?