Difference Between Generating Functions And Exponential Generating Functions [duplicate]

1

$begingroup$

This question already has an answer here:

• 
  What is the need of exponential generating functions on combinatorial problems?
  
  2 answers
  
  

What is the difference between generating functions and exponential generating functions?

I mean, we can solve a combinatorics sum by using just simple generating functions.

Or simply, for which parts in combinatorics do we use simple generating functions and in which parts we use exponential generating functions?

combinatorics

share|cite|improve this question

asked Jan 15 at 11:08

SaeeSaee

387

$endgroup$

marked as duplicate by Hans Lundmark, amWhy, darij grinberg, Math1000, Shailesh Jan 16 at 0:01

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

add a comment | 

1

$begingroup$

This question already has an answer here:

• 
  What is the need of exponential generating functions on combinatorial problems?
  
  2 answers
  
  

What is the difference between generating functions and exponential generating functions?

I mean, we can solve a combinatorics sum by using just simple generating functions.

Or simply, for which parts in combinatorics do we use simple generating functions and in which parts we use exponential generating functions?

combinatorics

share|cite|improve this question

asked Jan 15 at 11:08

SaeeSaee

387

$endgroup$

marked as duplicate by Hans Lundmark, amWhy, darij grinberg, Math1000, Shailesh Jan 16 at 0:01

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

add a comment | 

1

1

1

0

$begingroup$

This question already has an answer here:

• 
  What is the need of exponential generating functions on combinatorial problems?
  
  2 answers
  
  

What is the difference between generating functions and exponential generating functions?

I mean, we can solve a combinatorics sum by using just simple generating functions.

Or simply, for which parts in combinatorics do we use simple generating functions and in which parts we use exponential generating functions?

combinatorics

share|cite|improve this question

asked Jan 15 at 11:08

SaeeSaee

387

$endgroup$

This question already has an answer here:

• 
  What is the need of exponential generating functions on combinatorial problems?
  
  2 answers
  
  

What is the difference between generating functions and exponential generating functions?

I mean, we can solve a combinatorics sum by using just simple generating functions.

Or simply, for which parts in combinatorics do we use simple generating functions and in which parts we use exponential generating functions?

This question already has an answer here:

• 
  What is the need of exponential generating functions on combinatorial problems?
  
  2 answers
  
  

combinatorics

combinatorics

share|cite|improve this question

asked Jan 15 at 11:08

SaeeSaee

387

share|cite|improve this question

asked Jan 15 at 11:08

SaeeSaee

387

share|cite|improve this question

share|cite|improve this question

asked Jan 15 at 11:08

SaeeSaee

387

asked Jan 15 at 11:08

SaeeSaee

387

asked Jan 15 at 11:08

SaeeSaee

387

387

marked as duplicate by Hans Lundmark, amWhy, darij grinberg, Math1000, Shailesh Jan 16 at 0:01

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

marked as duplicate by Hans Lundmark, amWhy, darij grinberg, Math1000, Shailesh Jan 16 at 0:01

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

add a comment | 

add a comment | 

2 Answers
2

active

oldest

votes

1

$begingroup$

Aside from the obvious - exponential generating functions divide by a factorial, and ordinary generating functions don't - both sorts of generating functions are transform methods. One key thing transform methods do? They turn convolution into pointwise multiplication. The difference comes in there, in what sort of convolution this applies to.

For ordinary generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n a_j b_{n-j}$$
corresponding to the usual translation-invariant counting measure on the nonnegative integers. There are a great many situations in which this appears.

For exponential generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n binom{n}{j} a_j b_{n-j}$$
This form, reminiscent of the binomial theorem, is something that occasionally comes up in combinatorics - and when it does, generating functions aren't going to work well unless we make that adjustment.

share|cite|improve this answer

answered Jan 15 at 11:34

jmerryjmerry

9,4231124

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So, can you just give me an example for which you need exponential generating functions and tell me why do you need to use exponential generating functions and not simple generating functions. Thanks
  $endgroup$
  – Saee
  Jan 15 at 15:40
  
  
  

  
  
  
  

add a comment | 

0

$begingroup$

Hint: Answer moved to the MSE post Deep understanding on exponential generating function.

share|cite|improve this answer

edited Jan 16 at 12:20

answered Jan 15 at 20:50

Markus ScheuerMarkus Scheuer

61.8k457147

$endgroup$

add a comment | 

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

1

$begingroup$

Aside from the obvious - exponential generating functions divide by a factorial, and ordinary generating functions don't - both sorts of generating functions are transform methods. One key thing transform methods do? They turn convolution into pointwise multiplication. The difference comes in there, in what sort of convolution this applies to.

For ordinary generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n a_j b_{n-j}$$
corresponding to the usual translation-invariant counting measure on the nonnegative integers. There are a great many situations in which this appears.

For exponential generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n binom{n}{j} a_j b_{n-j}$$
This form, reminiscent of the binomial theorem, is something that occasionally comes up in combinatorics - and when it does, generating functions aren't going to work well unless we make that adjustment.

share|cite|improve this answer

answered Jan 15 at 11:34

jmerryjmerry

9,4231124

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So, can you just give me an example for which you need exponential generating functions and tell me why do you need to use exponential generating functions and not simple generating functions. Thanks
  $endgroup$
  – Saee
  Jan 15 at 15:40
  
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Aside from the obvious - exponential generating functions divide by a factorial, and ordinary generating functions don't - both sorts of generating functions are transform methods. One key thing transform methods do? They turn convolution into pointwise multiplication. The difference comes in there, in what sort of convolution this applies to.

For ordinary generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n a_j b_{n-j}$$
corresponding to the usual translation-invariant counting measure on the nonnegative integers. There are a great many situations in which this appears.

For exponential generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n binom{n}{j} a_j b_{n-j}$$
This form, reminiscent of the binomial theorem, is something that occasionally comes up in combinatorics - and when it does, generating functions aren't going to work well unless we make that adjustment.

share|cite|improve this answer

answered Jan 15 at 11:34

jmerryjmerry

9,4231124

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So, can you just give me an example for which you need exponential generating functions and tell me why do you need to use exponential generating functions and not simple generating functions. Thanks
  $endgroup$
  – Saee
  Jan 15 at 15:40
  
  
  

  
  
  
  

add a comment | 

1

1

1

$begingroup$

Aside from the obvious - exponential generating functions divide by a factorial, and ordinary generating functions don't - both sorts of generating functions are transform methods. One key thing transform methods do? They turn convolution into pointwise multiplication. The difference comes in there, in what sort of convolution this applies to.

For ordinary generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n a_j b_{n-j}$$
corresponding to the usual translation-invariant counting measure on the nonnegative integers. There are a great many situations in which this appears.

For exponential generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n binom{n}{j} a_j b_{n-j}$$
This form, reminiscent of the binomial theorem, is something that occasionally comes up in combinatorics - and when it does, generating functions aren't going to work well unless we make that adjustment.

share|cite|improve this answer

answered Jan 15 at 11:34

jmerryjmerry

9,4231124

$endgroup$

Aside from the obvious - exponential generating functions divide by a factorial, and ordinary generating functions don't - both sorts of generating functions are transform methods. One key thing transform methods do? They turn convolution into pointwise multiplication. The difference comes in there, in what sort of convolution this applies to.

For ordinary generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n a_j b_{n-j}$$
corresponding to the usual translation-invariant counting measure on the nonnegative integers. There are a great many situations in which this appears.

For exponential generating functions, that convolution is
$$(a*b)_n = sum_{j=0}^n binom{n}{j} a_j b_{n-j}$$
This form, reminiscent of the binomial theorem, is something that occasionally comes up in combinatorics - and when it does, generating functions aren't going to work well unless we make that adjustment.

share|cite|improve this answer

answered Jan 15 at 11:34

jmerryjmerry

9,4231124

share|cite|improve this answer

share|cite|improve this answer

answered Jan 15 at 11:34

jmerryjmerry

9,4231124

answered Jan 15 at 11:34

jmerryjmerry

9,4231124

answered Jan 15 at 11:34

jmerryjmerry

9,4231124

9,4231124

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So, can you just give me an example for which you need exponential generating functions and tell me why do you need to use exponential generating functions and not simple generating functions. Thanks
  $endgroup$
  – Saee
  Jan 15 at 15:40
  
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  So, can you just give me an example for which you need exponential generating functions and tell me why do you need to use exponential generating functions and not simple generating functions. Thanks
  $endgroup$
  – Saee
  Jan 15 at 15:40
  
  
  

  
  
  
  

$begingroup$
So, can you just give me an example for which you need exponential generating functions and tell me why do you need to use exponential generating functions and not simple generating functions. Thanks
$endgroup$
– Saee
Jan 15 at 15:40

$begingroup$
So, can you just give me an example for which you need exponential generating functions and tell me why do you need to use exponential generating functions and not simple generating functions. Thanks
$endgroup$
– Saee
Jan 15 at 15:40

add a comment | 

0

$begingroup$

Hint: Answer moved to the MSE post Deep understanding on exponential generating function.

share|cite|improve this answer

edited Jan 16 at 12:20

answered Jan 15 at 20:50

Markus ScheuerMarkus Scheuer

61.8k457147

$endgroup$

add a comment | 

0

$begingroup$

Hint: Answer moved to the MSE post Deep understanding on exponential generating function.

share|cite|improve this answer

edited Jan 16 at 12:20

answered Jan 15 at 20:50

Markus ScheuerMarkus Scheuer

61.8k457147

$endgroup$

add a comment | 

0

0

0

$begingroup$

Hint: Answer moved to the MSE post Deep understanding on exponential generating function.

share|cite|improve this answer

edited Jan 16 at 12:20

answered Jan 15 at 20:50

Markus ScheuerMarkus Scheuer

61.8k457147

$endgroup$

Hint: Answer moved to the MSE post Deep understanding on exponential generating function.

share|cite|improve this answer

edited Jan 16 at 12:20

answered Jan 15 at 20:50

Markus ScheuerMarkus Scheuer

61.8k457147

share|cite|improve this answer

share|cite|improve this answer

edited Jan 16 at 12:20

edited Jan 16 at 12:20

edited Jan 16 at 12:20

answered Jan 15 at 20:50

Markus ScheuerMarkus Scheuer

61.8k457147

answered Jan 15 at 20:50

Markus ScheuerMarkus Scheuer

61.8k457147

answered Jan 15 at 20:50

Markus ScheuerMarkus Scheuer

61.8k457147

61.8k457147

add a comment | 

add a comment | 

This page is only for reference, If you need detailed information, please check here