Counting One-to-One Functions from n to m with Property f(i)<f(j)

[22990atinesh]

Homework Statement

Let F be the set of one-to-one functions from the set ${1,2,..,n}$ to the set ${1,2,...,m}$ where $m \geq n \geq 1$. Then how many functions f in F satisfy the property $f(i)<f(j)$ for some $1 \leq i \leq j \leq n$

Homework Equations

The Attempt at a Solution

$^{m-1}P_n$. Is it correct

 

[Stephen Tashi]

$^{m-1}P_n$. Is it correct

What does that notation mean?

 

[Joffan]

Two more questions:
- How many functions are there in F altogether?
- What kind of functions fail to satisfy the given property?

 

[Ray Vickson]

Homework Statement

Let F be the set of one-to-one functions from the set ${1,2,..,n}$ to the set ${1,2,...,m}$ where $m \geq n \geq 1$. Then how many functions f in F satisfy the property $f(i)<f(j)$ for some $1 \leq i \leq j \leq n$

Homework Equations

The Attempt at a Solution

$^{m-1}P_n$. Is it correct

Show your reasoning---do not just write down the answer, or what you think is the answer.

 

[22990atinesh]

Two more questions:
- How many functions are there in F altogether?
- What kind of functions fail to satisfy the given property?

That's the exact question that has been asked.

 

[22990atinesh]

What does that notation mean?

Its a Permutation

 

[Stephen Tashi]

Its a Permutation

I think should be a "number of permutations". But how is it defined? Is it the number of permutations of m-1 things taken n at a time?

 

[Joffan]

Two more questions:
- How many functions are there in F altogether?
- What kind of functions fail to satisfy the given property?

That's the exact question that has been asked.

No, those are simpler questions: the first about the whole of F, not the restricted set your question asks for, and the second about the nature of the excluded functions in your question. But they can lead to answering the original question.Incidentally...

$^{m-1}P_n$. Is it correct

No.

 

[22990atinesh]

No, those are simpler questions: the first about the whole of F, not the restricted set your question asks for, and the second about the nature of the excluded functions in your question. But they can lead to answering the original question.Incidentally...

No.

So How can we solve this

 

[Joffan]

So How can we solve this

Back to my 2 questions...

1. The total number of functions in F is relatively simple: what is that number? Obviously you could assign $m$ possible values for $f(1)$, then $m-1$ possible values for $f(2)$, etc., to count all functions

2. What functions does the condition "$f(i)<f(j)$ for some $1 \leq i \lt j \leq n$" exclude? How many ways are there to make such functions?

 

[Stephen Tashi]

So How can we solve this

Think of an example. Suppose we have the set of numbers {1,2,3,4,5}. Suppose we have the constraint "List the numbers so that at least two of them are listed in ascending order". You could count the number N of possible lists that satisfy this constraint by counting the number M that do not satisfy it and computing 5! - M = N.

The number M counts the lists satisfying the statement "It is not true that at least two of the numbers are listed in ascending order" which can be phrased as "It is not true that there exists x in the list and there exists a y in the list such that x and y are listed in ascending order.". You could approach this as an exercise in logic. How do you negate a statement that has a "there exists..." requirement? The general pattern is "It is not true that there exists..." changes to "For each... it is not true that...".

Or you might try writing an example of a list that satisfys the condition "It is not true that there are at least two numbers listed in ascending order".