Math Problems
(1) W3 Let n be a non negative integer. Let An be the set of subsets of {1,.. . , n} that do not contain any consecutive pair of numbers. For example, A3 = {O, {1}, {2}, {3}, {1, 3}}.
- Compute A0,A1,A2,A3, A4, A5.
- Make a conjecture about |An| for all n ≥ 1.
- Prove your conjecture.
(2) W3 Find a bijective proof for the identity 6S(n, 3) + 6S(n, 2) + 3S(n, 1) = 3n.
(3) W3 Find a bijective proof for the identity Bn = ~n−1 (n−1 )Bk. (Recall Bn is the number of set
k=0 k partitions of [n] into nonempty subsets.)
(4) W3
- Let n ≥ 2. Prove that the number of partitions of n in which the two largest parts are equal (e.g. 5 + 5 + 3 + 1) is equal top(n) – p(n – 1).
- Find/prove a formula, along the same lines, for the number of partitions of n ≥ 3 in which the three largest parts are equal.
- Prove that the sequence p(n) – p(n – 1) (for n ≥ 2) is nondecreasing. (That is, show that (p(n) – p(n – 1)) – (p(n – 1) – p(n – 2) ≥ 0 holds.)