When it comes to partitions, I know we can impose some modest restrictions (maybe even a couple) on the partitions and obtain counting formula, but I would like to impose some more serious constraints and still be able to produce a formula, generating function, or recurrence in order to count how many partitions there are that satisfy my constraints. In particular, I've come across the following scenario in some non-mathematical research.

Let $M$ and $N$ be positive integers and let $1\leq t\leq M$. I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "$s-$forcing for $t$" if the following condition holds: $$\text{$\forall A\subset \lambda,$ $\left(\sum\limits_{a\in A}a \geq t\implies |A|\geq s\right)$}$$

I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "maximally $s-$forcing for $t$" if $\lambda$ is $s-$forcing but not $(s+1)-$forcing.

In general, if we denote the set of $s-$forcing partitions by $\Lambda_s$, then we have the containment $$\Lambda_1\supseteq \Lambda_2\supseteq \cdots \supseteq \Lambda_N$$

Question 1: For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are $2-$forcing for $t$?

Answer 1: This is the number of partitions of $M$ into $N$ parts with each part having size less than $t$.

Question 1.5: For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are maximally $2-$forcing for $t$?

Question 2: For each $t$, there is a minimal positive integer $\sigma$ so that no partition is $\sigma-$forcing. How many partitions of $M$ into $N$ nonnegative parts are maximally $(\sigma-1)-$forcing?

If people have relevant references (maybe these have been studied before under a different name), I would love to do some reading. Otherwise, any mathematical help toward solutions would be greatly appreciated. Note: I am not assuming here that $M\geq N$, but that additional premise may be helpful for the time being.

When it comes to partitions, I know we can impose some modest restrictions (maybe even a couple) on the partitions and obtain counting formula, but I would like to impose some more serious constraints and still be able to produce a formula, generating function, or recurrence in order to count how many partitions there are that satisfy my constraints. In particular, I've come across the following scenario in some non-mathematical research.

Let $M$ and $N$ be positive integers and let $1\leq t\leq M$. I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "$s-$forcing for $t$" if the following condition holds: $$\text{$\forall A\subset \lambda,$ $\left(\sum\limits_{a\in A}a \geq t\implies |A|\geq s\right)$}$$

I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "maximally $s-$forcing for $t$" if $\lambda$ is $s-$forcing but not $(s+1)-$forcing.

In general, if we denote the set of $s-$forcing partitions by $\Lambda_s$, then we have the containment $$\Lambda_1\supseteq \Lambda_2\supseteq \cdots \supseteq \Lambda_N$$

Question 1: For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are $2-$forcing for $t$?

Answer 1: This is the number of partitions of $M$ into $N$ parts with each part having size less than $t$.

Question 1.5: For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are maximally $2-$forcing for $t$?

Question 2: For each $t$, there is a minimal positive integer $\sigma$ so that no partition is $\sigma-$forcing. How many partitions of $M$ into $N$ nonnegative parts are maximally $(\sigma-1)-$forcing?

If people have relevant references (maybe these have been studied before under a different name), I would love to do some reading. Otherwise, any mathematical help toward solutions would be greatly appreciated. Note: I am not assuming here that $M\geq N$, but that additional premise may be helpful for the time being.

Edit: This question is seemingly related to the question here.

When it comes to partitions, I know we can impose some modest restrictions (maybe even a couple) on the partitions and obtain counting formula, but I would like to impose some more serious constraints and still be able to produce a formula, generating function, or recurrence in order to count how many partitions there are that satisfy my constraints. In particular, I've come across the following scenario in some non-mathematical research.

Let $M$ and $N$ be positive integers and let $1\leq t\leq M$. I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "$s-$forcing for $t$" if the following condition holds: $$\text{$\forall A\subset \lambda,$ $\left(\sum\limits_{a\in A}a \geq t\implies |A|\geq s\right)$}$$

I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "maximally $s-$forcing for $t$" if $\lambda$ is $s-$forcing but not $(s+1)-$forcing.

In general, if we denote the set of $s-$forcing partitions by $\Lambda_s$, then we have the containment $$\Lambda_1\supseteq \Lambda_2\supseteq \cdots \supseteq \Lambda_N$$

$\textbf{Question 1:}$ For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are $2-$forcing for $t$?

$\textbf{Answer 1:}$ This is the number of partitions of $M$ into $N$ parts with each part having size less than $t$.

$\textbf{Question 1.5:}$ For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are maximally $2-$forcing for $t$?

$\textbf{Question 2:}$ For each $t$, there is a minimal positive integer $\sigma$ so that no partition is $\sigma-$forcing. How many partitions of $M$ into $N$ nonnegative parts are maximally $(\sigma-1)-$forcing?

If people have relevant references (maybe these have been studied before under a different name), I would love to do some reading. Otherwise, any mathematical help toward solutions would be greatly appreciated. Note: I am not assuming here that $M\geq N$, but that additional premise may be helpful for the time being.

When it comes to partitions, I know we can impose some modest restrictions (maybe even a couple) on the partitions and obtain counting formula, but I would like to impose some more serious constraints and still be able to produce a formula, generating function, or recurrence in order to count how many partitions there are that satisfy my constraints. In particular, I've come across the following scenario in some non-mathematical research.

Let $M$ and $N$ be positive integers and let $1\leq t\leq M$. I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "$s-$forcing for $t$" if the following condition holds: $$\text{$\forall A\subset \lambda,$ $\left(\sum\limits_{a\in A}a \geq t\implies |A|\geq s\right)$}$$

I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "maximally $s-$forcing for $t$" if $\lambda$ is $s-$forcing but not $(s+1)-$forcing.

In general, if we denote the set of $s-$forcing partitions by $\Lambda_s$, then we have the containment $$\Lambda_1\supseteq \Lambda_2\supseteq \cdots \supseteq \Lambda_N$$

Question 1: For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are $2-$forcing for $t$?

Answer 1: This is the number of partitions of $M$ into $N$ parts with each part having size less than $t$.

Question 1.5: For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are maximally $2-$forcing for $t$?

Question 2: For each $t$, there is a minimal positive integer $\sigma$ so that no partition is $\sigma-$forcing. How many partitions of $M$ into $N$ nonnegative parts are maximally $(\sigma-1)-$forcing?

If people have relevant references (maybe these have been studied before under a different name), I would love to do some reading. Otherwise, any mathematical help toward solutions would be greatly appreciated. Note: I am not assuming here that $M\geq N$, but that additional premise may be helpful for the time being.

When it comes to partitions, I know we can impose some modest restrictions (maybe even a couple) on the partitions and obtain counting formula, but I would like to impose some more serious constraints and still be able to produce a formula, generating function, or recurrence in order to count how many partitions there are that satisfy my constraints. In particular, I've come across the following scenario in some non-mathematical research.

Let $M$ and $N$ be positive integers and let $1\leq t\leq M$. I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "$s-$forcing for $t$" if the following condition holds: $$\text{$\forall A\subset \lambda,$ $\left(\sum\limits_{a\in A}a \geq t\implies |A|\geq s\right)$}$$

I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "maximally $s-$forcing for $t$" if $\lambda$ is $s-$forcing but not $(s+1)-$forcing.

In general, if we denote the set of $s-$forcing partitions by $\Lambda_s$, then we have the containment $$\Lambda_1\supseteq \Lambda_2\supseteq \cdots \supseteq \Lambda_N$$

$\textbf{Question 1:}$ For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are $2-$forcing for $t$?

$\textbf{Answer 1:}$ This is the number of partitions of $M$ into $N$ parts with each part having size less than $t$.

$\textbf{Question 1.5:}$ For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are maximally $2-$focring for $t$?

$\textbf{Question 2:}$ For each $t$, there is a minimal positive integer $\sigma$ so that no partition is $\sigma-$forcing. How many partitions of $M$ into $N$ nonnegative parts are maximally $(\sigma-1)-$forcing?

If people have relevant references (maybe these have been studied before under a different name), I would love to do some reading. Otherwise, any mathematical help toward solutions would be greatly appreciated. Note: I am not assuming here that $M\geq N$, but that additional premise may be helpful for the time being.

When it comes to partitions, I know we can impose some modest restrictions (maybe even a couple) on the partitions and obtain counting formula, but I would like to impose some more serious constraints and still be able to produce a formula, generating function, or recurrence in order to count how many partitions there are that satisfy my constraints. In particular, I've come across the following scenario in some non-mathematical research.

Let $M$ and $N$ be positive integers and let $1\leq t\leq M$. I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "$s-$forcing for $t$" if the following condition holds: $$\text{$\forall A\subset \lambda,$ $\left(\sum\limits_{a\in A}a \geq t\implies |A|\geq s\right)$}$$

I will call a partition $\lambda$ of $M$ into $N$ nonnegative parts "maximally $s-$forcing for $t$" if $\lambda$ is $s-$forcing but not $(s+1)-$forcing.

In general, if we denote the set of $s-$forcing partitions by $\Lambda_s$, then we have the containment $$\Lambda_1\supseteq \Lambda_2\supseteq \cdots \supseteq \Lambda_N$$

$\textbf{Question 1:}$ For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are $2-$forcing for $t$?

$\textbf{Answer 1:}$ This is the number of partitions of $M$ into $N$ parts with each part having size less than $t$.

$\textbf{Question 1.5:}$ For a given $t$, how many partitions of $M$ into $N$ nonnegative parts are maximally $2-$forcing for $t$?

$\textbf{Question 2:}$ For each $t$, there is a minimal positive integer $\sigma$ so that no partition is $\sigma-$forcing. How many partitions of $M$ into $N$ nonnegative parts are maximally $(\sigma-1)-$forcing?

If people have relevant references (maybe these have been studied before under a different name), I would love to do some reading. Otherwise, any mathematical help toward solutions would be greatly appreciated. Note: I am not assuming here that $M\geq N$, but that additional premise may be helpful for the time being.