Homework 3 due Mon 9/19 (tex).
⊞
Solutions
#1: [HHM] p. 158, 160. #2. [HHM] §2.3 p 144. #3. See Stanley §1.9 Eqn 1.99 ff. #4. [HHM] §2.8.2 p 234.
#5. Franklin's transformation clearly works if Lλ and Dλ are disjoint. They share the lower right dot when |Dλ| = m, the number of non-zero parts of λ.
If |Lλ| < m or |Lλ| > m+1, the transformation also works. The remaining bad cases are: |Lλ| = |Dλ| = m, λ = (2m, 2m−1,..., m+1), |λ| = ½m(3m+1); and |Lλ| − 1 = |Dλ| = m, λ = (2m−1, 2m−2,..., m), |λ| = ½m(3m−1).