组合数学 (Fall 2015)/Problem Set 1

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每道题目的解答都要有完整的解题过程。中英文不限。

Problem 1

  1. [math]\displaystyle{ k }[/math]种不同的明信片,每种明信片有无限多张,寄给[math]\displaystyle{ n }[/math]个人,每人一张,有多少种方法?
  2. [math]\displaystyle{ k }[/math]种不同的明信片,每种明信片有无限多张,寄给[math]\displaystyle{ n }[/math]个人,每人一张,每个人必须收到不同种类的明信片,有多少种方法?
  3. [math]\displaystyle{ k }[/math]种不同的明信片,每种明信片有无限多张,寄给[math]\displaystyle{ n }[/math]个人,每人收到[math]\displaystyle{ r }[/math]张不同的明信片(但不同的人可以收到相同的明信片),有多少种方法?
  4. 只有一种明信片,共有[math]\displaystyle{ m }[/math]张,寄给[math]\displaystyle{ n }[/math]个人,全部寄完,每个人可以收多张明信片或者不收明信片,有多少种方法?
  5. [math]\displaystyle{ k }[/math]种不同的明信片,其中第[math]\displaystyle{ i }[/math]种明信片有[math]\displaystyle{ m_i }[/math]张,寄给[math]\displaystyle{ n }[/math]个人,全部寄完,每个人可以收多张明信片或者不收明信片,有多少种方法?

Problem 2

  • Give a combinatorial proof for the following identity:
[math]\displaystyle{ \sum_{k=1}^n k{n\choose k}=n\cdot 2^{n-1}. }[/math]
  • Give a closed form for [math]\displaystyle{ \sum_{k=1}^n k^2{n\choose k} }[/math] and prove it.

Problem 3

A [math]\displaystyle{ 2\times n }[/math] rectangle is to be paved with [math]\displaystyle{ 1\times 2 }[/math] identical blocks and [math]\displaystyle{ 2\times 2 }[/math] identical blocks. Let [math]\displaystyle{ f(n) }[/math] denote the number of ways that can be done. Find a recurrence relation for [math]\displaystyle{ f(n) }[/math], solve the recurrence relation.