MHB Number of Partitions of n into j Parts not Exceeding m?

[poissonspot]

Remember that the number of partitions of n into parts not exceeding m is equal to the number of partitions of n into m or fewer parts? Does anyone know much about the number of partitions of n into m parts not exceeding m?

Thanks!

Edit: Or more generally, the number of partitions of n into j parts not exceeding m?
Thanks again,

 

[TheBigBadBen]

Remember that the number of partitions of n into parts not exceeding m is equal to the number of partitions of n into m or fewer parts? Does anyone know much about the number of partitions of n into m parts not exceeding m?

Thanks!

Edit: Or more generally, the number of partitions of n into j parts not exceeding m?
Thanks again,

All right, so after trying to make sense of your post, I've decided that there are the following distinct questions to be answered here:

1: In how many ways can one partition $n$ objects into groups of less than $m$
2: In how many ways can one partition $n$ objects into $m$ or fewer groups
3: In how many ways can one partition $n$ objects into $j$ parts not exceeding $m$

Please correct me if you meant something else.

 

[TheBigBadBen]

It seems that the answer is pretty complicated.

Here's a related link:
combinatorics - partitions of sets - Mathematics Stack Exchange

 

[poissonspot]

All right, so after trying to make sense of your post, I've decided that there are the following distinct questions to be answered here:

1: In how many ways can one partition $n$ objects into groups of less than $m$
2: In how many ways can one partition $n$ objects into $m$ or fewer groups
3: In how many ways can one partition $n$ objects into $j$ parts not exceeding $m$

Please correct me if you meant something else.

Hey, I'm just talking about integer partitions, but I'm sorry about the lack of clarity. I think it's well known that the number of partitions of an integer n w/ largest part m is equal to the number of partitions of n with m parts a statement equivalent to the first line "Remember".

However, I'm really interested in the question of how many integer partitions of n with j parts have parts not exceeding m.

Does that clarify things a little?

 

[TheBigBadBen]

Hey, I'm just talking about integer partitions, but I'm sorry about the lack of clarity. I think it's well known that the number of partitions of an integer n w/ largest part m is equal to the number of partitions of n with m parts a statement equivalent to the first line "Remember".

However, I'm really interested in the question of how many integer partitions of n with j parts have parts not exceeding m.

Does that clarify things a little?

I realize you mean for $m,n,j$ to be integers. It's still a tough problem.

After looking carefully, the link wasn't as related as I thought, so sorry about that.

Some things I figured out that might help: you can add up the multinomial coefficients
$$
\binom{n}{k_1,k_2,...,k_m}
$$
Over all choices of $k_j$ such that $0≤k_j≤n$, not counting repeating combinations of $k_j$, and $\sum_{j=1}^m k_j=n$, which I haven't found a way to simplify.

Or, if we're looking at the wider version of the problem where we have labeled parts $A_1,A_2,...,A_m$, then the number of ways you can distribute the $n$ objects to the $m$ sets (including potentially empty $A_j$s) is simply $m^n$.

 

[poissonspot]

I realize you mean for $m,n,j$ to be integers. It's still a tough problem.

After looking carefully, the link wasn't as related as I thought, so sorry about that.

Some things I figured out that might help: you can add up the multinomial coefficients
$$
\binom{n}{k_1,k_2,...,k_m}
$$
Over all choices of $k_j$ such that $0≤k_j≤n$, not counting repeating combinations of $k_j$, and $\sum_{j=1}^m k_j=n$, which I haven't found a way to simplify.

Or, if we're looking at the wider version of the problem where we have labeled parts $A_1,A_2,...,A_m$, then the number of ways you can distribute the $n$ objects to the $m$ sets (including potentially empty $A_j$s) is simply $m^n$.

Thanks again.
What I am hoping to find is the number of j-tuples of positive integers that are non increasing so that a₀+a₁+...+a_j=n and a_i is less than or equal to m.
Partition (number theory) - Wikipedia, the free encyclopedia