MATH 630–600. Enumerative Combinatorics
Assignment 5. Due on Wednesday, November 5, 2008
1. Let m1≥m2≥ · · · ≥ mnbe a sequence of positive integers, and
M= (mi,j) be an n×nmatrix whose ijth entry is
mij =1
(mi−i+j)!.
(Assume that all mi−i+jare nonnegative.) Prove that the determi-
nant of Mis
det(M) = Q1≤i<j≤n(mi−mj+j−i)
(m1+n−1)!(m2+n−2)! · · · mn!.
2. Find the rank-generating function F(P, q) for the following posets.
(a) DN, the set of all positive integral divisors of n, where i≤j
if iis a divisor of j. Assume that the prime factorization of n
is pα1
1pα2
2· · · pαk
k, where p1, . . . , pkare distinct primes, and αiare
positive integers.
(b) The set Πnof all partitions of [n], ordered by refinement.
(c) The set Ln(q) that consists of all subspaces of an n-dimensional
vector space Vn(q) over the q-element field Fq, ordered by inclu-
sion.
3. If posets Pand Qare graded with rank generating functions F(P, q)
and F(Q, q), then prove
F(P×Q, q) = F(P, q )F(Q, q),
and
F(P⊗Q, q) = F(P, q r+1)F(Q, q).
4. Check the following rules of cardinal arithmetic:
RP+Q=RP×RQ,
(RQ)P=RQ×P.
