How is it called this function ?

[zetafunction]

how is it called this function ??

given a number 'n' how is it called this function $$P(n,k)$$

so P(n,k) is the number of ways in which an integer 'n' can be expressed as the sum (with repeated summands) of k integers different or not

then for any integers n and k how could i compute $$P(n,k)$$ ? is there a recurrence equation ??

 

[Petek]

Would it be useful for you to restrict n and k to be positive integers? If so, then you're dealing with the theory of partitions. (If you allow negative integers, then you get infinitely many solutions. Say n = 5 and k = 3. In addition to solutions such as 5 = 3 + 1 + 1, you also get 5 = 5 + 1 + (-1) = 5 + 2 + (-2) = 5 + 3 + (-3), and so on.)

Assuming that you're interested in the case where n, k are positive, we have the following

Theorem: The number of partitions of n into k parts is equal to the number of partitions of n into parts the largest of which is k.

The proof is to consider the Ferrers diagram (see the Wikipedia article for the definition) of such partitions.

It's now easy to simplify your function P(n,k) (By the way, the Wiki article uses P(n,k) for a different function). Let P(n) be the number of unrestricted partitions of n. Then

P(n,k) = P(n-k)

The proof is simple: For each partition of n in which the largest part is k, we have a corresponding partition of n - k. In other words, if

$$n = k + k_1 + k_2 + \cdots + k_m$$

then

$$n - k = k_1 + k_2 + \cdotd + k_m$$

Thus, you can translate P(n,k) into P(n-k) and use all of the techniques and such from partition theory, only a small part of which is described in the Wiki article.

HTH

 

[zetafunction]

ok thanks ... :)

 

[FroChro]

...Let P(n) be the number of unrestricted partitions of n. Then

P(n,k) = P(n-k)

The proof is simple: For each partition of n in which the largest part is k, we have a corresponding partition of n - k. In other words, if

$$n = k + k_1 + k_2 + \cdots + k_m$$

then

$$n - k = k_1 + k_2 + \cdotd + k_m$$

Thus, you can translate P(n,k) into P(n-k) and use all of the techniques and such from partition theory, only a small part of which is described in the Wiki article.

HTH

But parts on the right hand side of $$n - k = k_1 + k_2 + \cdotd + k_m$$ still need to be smaller or equal to k. So your statement holds only for $$n \leq 2k$$. Or am I missing something?

 

[Petek]

But parts on the right hand side of $$n - k = k_1 + k_2 + \cdotd + k_m$$ still need to be smaller or equal to k. So your statement holds only for $$n \leq 2k$$. Or am I missing something?

No, you're right. Have to think some more. Thanks for pointing that out!