A small puzzle

[Dubslow]

Here's a no doubt not-terribly-difficult problem, but it's not exactly trivial. I have little interest in actually solving it, so here it is for the forum's
enjoyment:

Given a target n and a count c, how many different ways are there for `c` integers to sum to `n`?

For instance, with n=11 and c=4, the total is eleven:

(1, 1, 1, 8)
(1, 1, 2, 7)
(1, 1, 3, 6)
(1, 1, 4, 5)
(1, 2, 2, 6)
(1, 2, 3, 5)
(1, 2, 4, 4)
(1, 3, 3, 4)
(2, 2, 2, 5)
(2, 2, 3, 4)
(2, 3, 3, 3)

[axn]

https://en.wikipedia.org/wiki/Partit...umber_of_parts

[Dubslow]

Originally Posted by axn

https://en.wikipedia.org/wiki/Partit...umber_of_parts

Ah. Well I wasn't wrong (rather, my google queries into the matter were ignorant of the extant terminology)

[retina]

Originally Posted by Dubslow

Given a target n and a count c, how many different ways are there for `c` integers to sum to `n`?

For instance, with n=11 and c=4, the total is eleven:

(1, 1, 1, 8)
(1, 1, 2, 7)
(1, 1, 3, 6)
(1, 1, 4, 5)
(1, 2, 2, 6)
(1, 2, 3, 5)
(1, 2, 4, 4)
(1, 3, 3, 4)
(2, 2, 2, 5)
(2, 2, 3, 4)
(2, 3, 3, 3)

The puzzle as stated has an infinite number of solutions.

So your puzzle in return it is discover why it has in infinite number of solutions.

[Dubslow]

Originally Posted by retina

The puzzle as stated has an infinite number of solutions.

So your puzzle in return it is discover why it has in infinite number of solutions.

Do you perhaps refer to my lack of specifying strictly positive integers?

[Uncwilly]

Originally Posted by Dubslow

Do you perhaps refer to my lack of specifying strictly positive integers?

Your original post did not specify that. BTW there is a term specifically for them "natural numbers".

[LaurV]

[deleted]
sorry, ignore this, I didn't read to the end

[davar55]

Originally Posted by LaurV

[deleted]
sorry, ignore this, I didn't read to the end

oops, me too, started to jump in too soon.
typical.

[davar55]

Originally Posted by Uncwilly

Your original post did not specify that. BTW there is a term specifically for them "natural numbers".

The naturals are 1,2,3,..., and don't usually include 0.
The axiomatic approach begins with 0 and a successor rule.
By starting with 0, instead of 1, it's easier to define the
arithmetic operations.

[Uncwilly]

Originally Posted by davar55

The naturals are 1,2,3,..., and don't usually include 0.

The way that I was taught "natural numbers" and "whole numbers" were differentiated by 0 being in one and not the other. Trying to reconfirm before posting led to some confusion. The way I recall:
[[[[[whole]natural]integers]rational]+[irrational]]=[real]
[imaginary]

[science_man_88]

Originally Posted by Dubslow

but it's not exactly trivial.

partitions(n,,[c,c])

in PARI would work