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Let $f(x, y, z)$ is the number of distinct ways of representing $x$ as a sum of at most $y$ positive integers that are all smaller or equal to $z$. Moreover, If $yz < x$, then the function gives 0.
The function can be defined in one of the following equivalent ways.

  • The number $f(x,y,z)$.

  • The number of all partitions (meaning: without considering the ordering) $x = a_1 + \ldots + a_y$, where $a_i \in \mathbb{Z}_{\geq 0}$ and $a_i \leq z$ for all $i$.

  • The sum over the number of all partitions $x = a_1 + \ldots + a_l$, where $a_i \in \mathbb{Z}_{\geq 1}$ and $a_i \leq z$ for all $i$, where $l$ runs from $0$ to $k$.

  • The number of ways of throwing $x$ indistinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls.

Can we write $f$ as a function from $\mathbb Z^3 \to \mathbb Z$ ? probably recursively.? Or represent it somehow combinatorially using some partition function etc.?

PS: I am not looking for q-binomial formula because that gives " The number of ways of throwing $x$ distinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls." Instead, I am asking for "The number of ways of throwing $x$ indistinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls. "

Let $f(x, y, z)$ is the number of distinct ways of representing $x$ as a sum of at most $y$ positive integers that are all smaller or equal to $z$. Moreover, If $yz < x$, then the function gives 0.
The function can be defined in one of the following equivalent ways.

  • The number $f(x,y,z)$.

  • The number of all partitions (meaning: without considering the ordering) $x = a_1 + \ldots + a_y$, where $a_i \in \mathbb{Z}_{\geq 0}$ and $a_i \leq z$ for all $i$.

  • The sum over the number of all partitions $x = a_1 + \ldots + a_l$, where $a_i \in \mathbb{Z}_{\geq 1}$ and $a_i \leq z$ for all $i$, where $l$ runs from $0$ to $k$.

  • The number of ways of throwing $x$ indistinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls.

Can we write $f$ as a function from $\mathbb Z^3 \to \mathbb Z$ ? probably recursively.? Or represent it somehow combinatorially using some partition function etc.?

Let $f(x, y, z)$ is the number of distinct ways of representing $x$ as a sum of at most $y$ positive integers that are all smaller or equal to $z$. Moreover, If $yz < x$, then the function gives 0.
The function can be defined in one of the following equivalent ways.

  • The number $f(x,y,z)$.

  • The number of all partitions (meaning: without considering the ordering) $x = a_1 + \ldots + a_y$, where $a_i \in \mathbb{Z}_{\geq 0}$ and $a_i \leq z$ for all $i$.

  • The sum over the number of all partitions $x = a_1 + \ldots + a_l$, where $a_i \in \mathbb{Z}_{\geq 1}$ and $a_i \leq z$ for all $i$, where $l$ runs from $0$ to $k$.

  • The number of ways of throwing $x$ indistinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls.

Can we write $f$ as a function from $\mathbb Z^3 \to \mathbb Z$ ? probably recursively.? Or represent it somehow combinatorially using some partition function etc.?

PS: I am not looking for q-binomial formula because that gives " The number of ways of throwing $x$ distinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls." Instead, I am asking for "The number of ways of throwing $x$ indistinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls. "

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MIQ
MIQ
  • 83
  • 4

Combinatorial representation of function

Let $f(x, y, z)$ is the number of distinct ways of representing $x$ as a sum of at most $y$ positive integers that are all smaller or equal to $z$. Moreover, If $yz < x$, then the function gives 0.
The function can be defined in one of the following equivalent ways.

  • The number $f(x,y,z)$.

  • The number of all partitions (meaning: without considering the ordering) $x = a_1 + \ldots + a_y$, where $a_i \in \mathbb{Z}_{\geq 0}$ and $a_i \leq z$ for all $i$.

  • The sum over the number of all partitions $x = a_1 + \ldots + a_l$, where $a_i \in \mathbb{Z}_{\geq 1}$ and $a_i \leq z$ for all $i$, where $l$ runs from $0$ to $k$.

  • The number of ways of throwing $x$ indistinguishable balls into $y$ indistinguishable bins, where each bin can contain up to $z$ balls.

Can we write $f$ as a function from $\mathbb Z^3 \to \mathbb Z$ ? probably recursively.? Or represent it somehow combinatorially using some partition function etc.?