等價關係,整數分割

助教好,

不知道不是老師書上的適不適合在這裡問?如果不能問麻煩跟我說我再刪掉.謝謝

1.
Let A be a nite set. Prove that there is a one-
to-one correspondence between the set of equivalence relations on A and its
set of partitions.

Ans: Any equivalence relation R on A induces a natural partition of A :
{[a] : a 屬於 A}. Conversely, any partition of A gives rise to an equivalence relation on A.

我覺得這個答案好像沒在證明,但是又是出題老師寫的

請問是不是應該要像老師上課那樣證比較好呢?
另外,一個分割對應到一個等價關係請問應該怎麼證比較好?因為他說要

2.想請教這個lemma,他最後為什麼可以直接把 q# 改成 q 了?


Partition of Integers into Distinct Summands Revisited

Let q#(m, n) denote the number partitions of m ∈ Z+ into n distinct positive summands.

Lemma 81 q#(m; n) = q(m −c(n,2), n).

• x1 + x2 + · · · + xn = m, where 0 < x1 < x2 < · · · < xn,
  has q#(m, n) integer solutions.


• Adopt the following bijective transformation:
  x1 = w1;
  x2 = w2 + 1;
  x3 = w3 + 2;
  ...
  xn = wn + (n − 1):


• The equation becomes
w1 + w2 + · · · + wn = m −c(n,2)
where 0 < w1 ≤ w2 ≤ w3 ≤ · · · ≤ wn.

• This equation has q(m −c(n,2),n) integer solutions