MATH 630–600. Enumerative Combinatorics
Assignment 6. Due on Wednesday, November 19, 2008
1. Of the 16 four-element posets, (given in page 98), exactly one of them can not be built up
from the poset 1using the operations of disjoint union and ordinal sum. Find out that poset,
and explain your reason.
2. Prove the following two lattice identities are equivalent.
x∨(y∧z)=(x∨y)∧(x∨z),(1)
x∧(y∨z)=(x∧y)∨(x∧z).(2)
3. Let Vbe a vector space, and L(V) be the lattice of subspaces of V. Not using the dimension,
prove directly that L(V) is modular. That is, for any three subspaces X, Y , Z of Vwith
X⊆Z,
X∨(Y∧Z)=(X∨Y)∧(X∨Z).
4. Let Pbe a finite partially ordered set, and let rbe the largest size of a chain. Then Pcan
be partitioned into rbut no fewer antichains.
5. Let Lbe a finite lattice. Assume in L, if two elements xand yboth cover x∧y, then x∨y
covers both xand y. Prove that Lis graded.
6. Let Bnbe the Boolean algebra of rank nm where n= 2k+ 1. Assume that Ais an anti-chain
of size n
k. Let Ambe the sub-family of Awhich contains all subsets in Aof size m. From
Sperner Theorem, we know that Am6=∅iff m=kor k+ 1.
Prove that Ak=∅, or Ak+1 =∅.
7. Textbook, Exercise 27a on page 158.
8. Textbook, Exercise 31 on page 160.
9. Textbook, Exercise 44 on page 162.
10. Textbook. Exercise 45 on page 162.
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