Counting Cover Relations in the Lattice of Partitions $L(m,n)$
Taken from: [1, Exercise 6.2(a)].
Exercise 6.2(a)
Show that the number $c(m, n)$ of cover relations in $L(m, n)$, i.e., the number of pairs $(\lambda, \mu)$ of partitions in $L(m, n)$ for which $\mu$ covers $\lambda$, is given by
\[c(m, n)=\frac{(m+n-1)!}{(m-1)!(n-1)!}.\]
Proof
Let $(\lambda,\mu)$ be a pair of partitions counted by $c(m,n)$...
Why the Product of Symmetric Unimodal Polynomials is Also Symmetric Unimodal
Taken from: [1, Exercise 4.3].
Exercise 4.3
Let $ F(q) $ and $ G(q) $ be symmetric unimodal polynomials with nonnegative real coefficients. Show that $ F(q)G(q) $ is also symmetric (easy) and unimodal (harder).
Proof
We prove first the result for the product $P(q)=(1+q+\cdots+q^u)(1+q+\cdots+q^v)=1+\cdots+q^{u+v}$ with $u, v$ nonnegative int...
Eigenvalue Pairing in Bipartite Graphs
Taken from: [1, Exercise 1.3].
Exercise 1.3
A bipartite graph $G$ with vertex bipartition $(A,B)$ is a graph whose vertex set is the disjoint union $A \mathbin{\unicode{x228D}} B$ of $A$ and $B$, such that every edge of $G$ is incident to one vertex in $A$ and one vertex in $B$. Show by a walk-counting argument that the nonzero eigenvalues of $...