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The number of way to distribute n indistinguishable balls into k indistinguishable boxes arbitrarily
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Before even attempting at the proof, I need a clarification of a point in the question. I don't understand how the $k$ in $k geq lambda_1geq ...geq lambda_rgeq 1$ is relevant. For $n=7$ and $k=3$, for example, isn't $1,5,1$ a valid multiset of ball distribution, but $(5,1,1)$ is not a valid integer partition of 7, since $lambda_1 = 5 > 3 = k$?
Also, to prove the statement, I'm thinking of setting up a bijection between the set of ball distributions and the set of integer partitions of n. Would this be a sensible approach?
combinatorics discrete-mathematics integer-partitions
asked Jul 20 at 19:39
ensbana
279113
add a comment |Â
up vote
0
down vote
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Before even attempting at the proof, I need a clarification of a point in the question. I don't understand how the $k$ in $k geq lambda_1geq ...geq lambda_rgeq 1$ is relevant. For $n=7$ and $k=3$, for example, isn't $1,5,1$ a valid multiset of ball distribution, but $(5,1,1)$ is not a valid integer partition of 7, since $lambda_1 = 5 > 3 = k$?
Also, to prove the statement, I'm thinking of setting up a bijection between the set of ball distributions and the set of integer partitions of n. Would this be a sensible approach?
combinatorics discrete-mathematics integer-partitions
asked Jul 20 at 19:39
ensbana
279113
The number of partitions of $n$ into at most $k$ parts is equal to the number of partitions of size at most $k$. In a Ferrers diagram you can count the dots by rows, or by columns.
– saulspatz
Jul 20 at 20:11
Could you please explain the first sentence? As an example, why doesn't the tuple (5,1,1) above work?
– ensbana
Jul 20 at 21:19
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Before even attempting at the proof, I need a clarification of a point in the question. I don't understand how the $k$ in $k geq lambda_1geq ...geq lambda_rgeq 1$ is relevant. For $n=7$ and $k=3$, for example, isn't $1,5,1$ a valid multiset of ball distribution, but $(5,1,1)$ is not a valid integer partition of 7, since $lambda_1 = 5 > 3 = k$?
Also, to prove the statement, I'm thinking of setting up a bijection between the set of ball distributions and the set of integer partitions of n. Would this be a sensible approach?
combinatorics discrete-mathematics integer-partitions
asked Jul 20 at 19:39
ensbana
279113
Before even attempting at the proof, I need a clarification of a point in the question. I don't understand how the $k$ in $k geq lambda_1geq ...geq lambda_rgeq 1$ is relevant. For $n=7$ and $k=3$, for example, isn't $1,5,1$ a valid multiset of ball distribution, but $(5,1,1)$ is not a valid integer partition of 7, since $lambda_1 = 5 > 3 = k$?
Also, to prove the statement, I'm thinking of setting up a bijection between the set of ball distributions and the set of integer partitions of n. Would this be a sensible approach?
combinatorics discrete-mathematics integer-partitions
asked Jul 20 at 19:39
ensbana
279113
edited Jul 20 at 19:56
asked Jul 20 at 19:39
ensbana
279113
asked Jul 20 at 19:39
ensbana
279113
asked Jul 20 at 19:39
ensbana
279113
279113
The number of partitions of $n$ into at most $k$ parts is equal to the number of partitions of size at most $k$. In a Ferrers diagram you can count the dots by rows, or by columns.
– saulspatz
Jul 20 at 20:11
Could you please explain the first sentence? As an example, why doesn't the tuple (5,1,1) above work?
– ensbana
Jul 20 at 21:19
add a comment |Â
The number of partitions of $n$ into at most $k$ parts is equal to the number of partitions of size at most $k$. In a Ferrers diagram you can count the dots by rows, or by columns.
– saulspatz
Jul 20 at 20:11
Could you please explain the first sentence? As an example, why doesn't the tuple (5,1,1) above work?
– ensbana
Jul 20 at 21:19
The number of partitions of $n$ into at most $k$ parts is equal to the number of partitions of size at most $k$. In a Ferrers diagram you can count the dots by rows, or by columns.
– saulspatz
Jul 20 at 20:11
The number of partitions of $n$ into at most $k$ parts is equal to the number of partitions of size at most $k$. In a Ferrers diagram you can count the dots by rows, or by columns.
– saulspatz
Jul 20 at 20:11
Could you please explain the first sentence? As an example, why doesn't the tuple (5,1,1) above work?
– ensbana
Jul 20 at 21:19
Could you please explain the first sentence? As an example, why doesn't the tuple (5,1,1) above work?
– ensbana
Jul 20 at 21:19
add a comment |Â
1 Answer
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$n=lambda_1+lambda_2+ ldots+lambda_r$ corresponds to a partition of $n$ into $r$ parts. The condition $k ge lambda_i$ indicates that each part has size at most $k$.
Note that the question refers to number of partitions of $n$ into parts of size at most $k$. So $5,1,1$ is not the partition we're looking for with $n=7,k=3$ since there is one part (i.e. $5$) that has size greater than $3$. For example, with $n=7,k=3$, $3,2,1,1$ is a valid partition.
The condition $lambda_1 ge lambda_2 ge ldots ge lambda_r$ means that we don't have to care about the order of the parts when counting number of partitions. For example, $3,2,1,1,3,1,1,2$ and $3,1,2,1$ are considered to be one partition $3,2,1,1$ for $n=7,k=3$. Why do we need this condition? Try to look back the situation with indistinguishable boxes. Do we have to care about the order of the boxes?
Also, to prove the statement, I'm thinking of setting up a bijection between the set of ball distributions and the set of integer partitions of n. Would this be a sensible approach?
Yes, bijection is the way to go. Before finding the main bijection, you should "translate" these two situations into one language. In particular, since the balls and boxes are indistinguishable, the number of distributions of $n$ indistinguishable balls into $k$ indistinguishable boxes is the same as number of partitions of $n$ into $k$ parts.
answered Jul 20 at 23:26
Tengu
2,3391920
What I’m asking is, what does that condition $lambda_i leq k$ have to do with the distribution of balls into boxes? A distribution with 5 balls in 1 box, 1 ball in another box and 1 ball in yet another box is certainly a valid way to put 7 balls into 3 boxes, but why is its corresponding integers partition of 7, i.e. (5,1,1), not valid? Wouldn’t the condition $lambda_i leq k$ then exclude some cases from the set of integer partitions of n (and hence we wouldn’t have a desired bijection)?
– ensbana
Jul 20 at 23:42
Well, if you want to go with balls and boxes instead of integer partition: The condition $lambda_i le k$ indicates that each box must have at most $k$ balls. So even though $5,1,1$ is a valid partition for the first situation (where we talk about balls and boxes), it is not a valid partition for the second situation with integer partition where $n=7,k=3$ since $5>3$.
– Tengu
Jul 20 at 23:48
My current assumption is that, if we say that a bijection exists, then for each distribution of balls we would find a corresponding integers partition. Also I assume that the integers partition (5,1,1) corresponds to the distribution 5 balls - 1 ball - 1 ball, yet the way integers partition is defined in the question somehow forbids tuples like (5,1,1)? What would the tuple corresponding to the abovementioned balls distribution be?
– ensbana
Jul 20 at 23:55
To be even more explicit, if F is the bijection from the set of balls distributions to the set of integers partitions, what is F(5 balls, 1 ball, 1 ball)?
– ensbana
Jul 21 at 0:01
1
By answering this question of your, I think it will give away too much about the solution (which you can easily search on google). You should first spend some time finding a bijection. Try to take some small examples of $n,k$, list all the distributions/partitions for each situation then try to give a mapping between the two. See if you find any pattern that you can generalize for any $n,k$.
– Tengu
Jul 21 at 0:05
 |Â
show 2 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
-1
down vote
$n=lambda_1+lambda_2+ ldots+lambda_r$ corresponds to a partition of $n$ into $r$ parts. The condition $k ge lambda_i$ indicates that each part has size at most $k$.
Note that the question refers to number of partitions of $n$ into parts of size at most $k$. So $5,1,1$ is not the partition we're looking for with $n=7,k=3$ since there is one part (i.e. $5$) that has size greater than $3$. For example, with $n=7,k=3$, $3,2,1,1$ is a valid partition.
The condition $lambda_1 ge lambda_2 ge ldots ge lambda_r$ means that we don't have to care about the order of the parts when counting number of partitions. For example, $3,2,1,1,3,1,1,2$ and $3,1,2,1$ are considered to be one partition $3,2,1,1$ for $n=7,k=3$. Why do we need this condition? Try to look back the situation with indistinguishable boxes. Do we have to care about the order of the boxes?
Also, to prove the statement, I'm thinking of setting up a bijection between the set of ball distributions and the set of integer partitions of n. Would this be a sensible approach?
Yes, bijection is the way to go. Before finding the main bijection, you should "translate" these two situations into one language. In particular, since the balls and boxes are indistinguishable, the number of distributions of $n$ indistinguishable balls into $k$ indistinguishable boxes is the same as number of partitions of $n$ into $k$ parts.
answered Jul 20 at 23:26
Tengu
2,3391920
What I’m asking is, what does that condition $lambda_i leq k$ have to do with the distribution of balls into boxes? A distribution with 5 balls in 1 box, 1 ball in another box and 1 ball in yet another box is certainly a valid way to put 7 balls into 3 boxes, but why is its corresponding integers partition of 7, i.e. (5,1,1), not valid? Wouldn’t the condition $lambda_i leq k$ then exclude some cases from the set of integer partitions of n (and hence we wouldn’t have a desired bijection)?
– ensbana
Jul 20 at 23:42
Well, if you want to go with balls and boxes instead of integer partition: The condition $lambda_i le k$ indicates that each box must have at most $k$ balls. So even though $5,1,1$ is a valid partition for the first situation (where we talk about balls and boxes), it is not a valid partition for the second situation with integer partition where $n=7,k=3$ since $5>3$.
– Tengu
Jul 20 at 23:48
My current assumption is that, if we say that a bijection exists, then for each distribution of balls we would find a corresponding integers partition. Also I assume that the integers partition (5,1,1) corresponds to the distribution 5 balls - 1 ball - 1 ball, yet the way integers partition is defined in the question somehow forbids tuples like (5,1,1)? What would the tuple corresponding to the abovementioned balls distribution be?
– ensbana
Jul 20 at 23:55
To be even more explicit, if F is the bijection from the set of balls distributions to the set of integers partitions, what is F(5 balls, 1 ball, 1 ball)?
– ensbana
Jul 21 at 0:01
1
By answering this question of your, I think it will give away too much about the solution (which you can easily search on google). You should first spend some time finding a bijection. Try to take some small examples of $n,k$, list all the distributions/partitions for each situation then try to give a mapping between the two. See if you find any pattern that you can generalize for any $n,k$.
– Tengu
Jul 21 at 0:05
 |Â
show 2 more comments
up vote
-1
down vote
$n=lambda_1+lambda_2+ ldots+lambda_r$ corresponds to a partition of $n$ into $r$ parts. The condition $k ge lambda_i$ indicates that each part has size at most $k$.
Note that the question refers to number of partitions of $n$ into parts of size at most $k$. So $5,1,1$ is not the partition we're looking for with $n=7,k=3$ since there is one part (i.e. $5$) that has size greater than $3$. For example, with $n=7,k=3$, $3,2,1,1$ is a valid partition.
The condition $lambda_1 ge lambda_2 ge ldots ge lambda_r$ means that we don't have to care about the order of the parts when counting number of partitions. For example, $3,2,1,1,3,1,1,2$ and $3,1,2,1$ are considered to be one partition $3,2,1,1$ for $n=7,k=3$. Why do we need this condition? Try to look back the situation with indistinguishable boxes. Do we have to care about the order of the boxes?
Also, to prove the statement, I'm thinking of setting up a bijection between the set of ball distributions and the set of integer partitions of n. Would this be a sensible approach?
Yes, bijection is the way to go. Before finding the main bijection, you should "translate" these two situations into one language. In particular, since the balls and boxes are indistinguishable, the number of distributions of $n$ indistinguishable balls into $k$ indistinguishable boxes is the same as number of partitions of $n$ into $k$ parts.
answered Jul 20 at 23:26
Tengu
2,3391920
What I’m asking is, what does that condition $lambda_i leq k$ have to do with the distribution of balls into boxes? A distribution with 5 balls in 1 box, 1 ball in another box and 1 ball in yet another box is certainly a valid way to put 7 balls into 3 boxes, but why is its corresponding integers partition of 7, i.e. (5,1,1), not valid? Wouldn’t the condition $lambda_i leq k$ then exclude some cases from the set of integer partitions of n (and hence we wouldn’t have a desired bijection)?
– ensbana
Jul 20 at 23:42
Well, if you want to go with balls and boxes instead of integer partition: The condition $lambda_i le k$ indicates that each box must have at most $k$ balls. So even though $5,1,1$ is a valid partition for the first situation (where we talk about balls and boxes), it is not a valid partition for the second situation with integer partition where $n=7,k=3$ since $5>3$.
– Tengu
Jul 20 at 23:48
My current assumption is that, if we say that a bijection exists, then for each distribution of balls we would find a corresponding integers partition. Also I assume that the integers partition (5,1,1) corresponds to the distribution 5 balls - 1 ball - 1 ball, yet the way integers partition is defined in the question somehow forbids tuples like (5,1,1)? What would the tuple corresponding to the abovementioned balls distribution be?
– ensbana
Jul 20 at 23:55
To be even more explicit, if F is the bijection from the set of balls distributions to the set of integers partitions, what is F(5 balls, 1 ball, 1 ball)?
– ensbana
Jul 21 at 0:01
1
By answering this question of your, I think it will give away too much about the solution (which you can easily search on google). You should first spend some time finding a bijection. Try to take some small examples of $n,k$, list all the distributions/partitions for each situation then try to give a mapping between the two. See if you find any pattern that you can generalize for any $n,k$.
– Tengu
Jul 21 at 0:05
 |Â
show 2 more comments
up vote
-1
down vote
up vote
-1
down vote
$n=lambda_1+lambda_2+ ldots+lambda_r$ corresponds to a partition of $n$ into $r$ parts. The condition $k ge lambda_i$ indicates that each part has size at most $k$.
Note that the question refers to number of partitions of $n$ into parts of size at most $k$. So $5,1,1$ is not the partition we're looking for with $n=7,k=3$ since there is one part (i.e. $5$) that has size greater than $3$. For example, with $n=7,k=3$, $3,2,1,1$ is a valid partition.
The condition $lambda_1 ge lambda_2 ge ldots ge lambda_r$ means that we don't have to care about the order of the parts when counting number of partitions. For example, $3,2,1,1,3,1,1,2$ and $3,1,2,1$ are considered to be one partition $3,2,1,1$ for $n=7,k=3$. Why do we need this condition? Try to look back the situation with indistinguishable boxes. Do we have to care about the order of the boxes?
Also, to prove the statement, I'm thinking of setting up a bijection between the set of ball distributions and the set of integer partitions of n. Would this be a sensible approach?
Yes, bijection is the way to go. Before finding the main bijection, you should "translate" these two situations into one language. In particular, since the balls and boxes are indistinguishable, the number of distributions of $n$ indistinguishable balls into $k$ indistinguishable boxes is the same as number of partitions of $n$ into $k$ parts.
answered Jul 20 at 23:26
Tengu
2,3391920
$n=lambda_1+lambda_2+ ldots+lambda_r$ corresponds to a partition of $n$ into $r$ parts. The condition $k ge lambda_i$ indicates that each part has size at most $k$.
Note that the question refers to number of partitions of $n$ into parts of size at most $k$. So $5,1,1$ is not the partition we're looking for with $n=7,k=3$ since there is one part (i.e. $5$) that has size greater than $3$. For example, with $n=7,k=3$, $3,2,1,1$ is a valid partition.
The condition $lambda_1 ge lambda_2 ge ldots ge lambda_r$ means that we don't have to care about the order of the parts when counting number of partitions. For example, $3,2,1,1,3,1,1,2$ and $3,1,2,1$ are considered to be one partition $3,2,1,1$ for $n=7,k=3$. Why do we need this condition? Try to look back the situation with indistinguishable boxes. Do we have to care about the order of the boxes?
Also, to prove the statement, I'm thinking of setting up a bijection between the set of ball distributions and the set of integer partitions of n. Would this be a sensible approach?
Yes, bijection is the way to go. Before finding the main bijection, you should "translate" these two situations into one language. In particular, since the balls and boxes are indistinguishable, the number of distributions of $n$ indistinguishable balls into $k$ indistinguishable boxes is the same as number of partitions of $n$ into $k$ parts.
answered Jul 20 at 23:26
Tengu
2,3391920
edited Jul 20 at 23:33
answered Jul 20 at 23:26
Tengu
2,3391920
answered Jul 20 at 23:26
Tengu
2,3391920
answered Jul 20 at 23:26
Tengu
2,3391920
2,3391920
What I’m asking is, what does that condition $lambda_i leq k$ have to do with the distribution of balls into boxes? A distribution with 5 balls in 1 box, 1 ball in another box and 1 ball in yet another box is certainly a valid way to put 7 balls into 3 boxes, but why is its corresponding integers partition of 7, i.e. (5,1,1), not valid? Wouldn’t the condition $lambda_i leq k$ then exclude some cases from the set of integer partitions of n (and hence we wouldn’t have a desired bijection)?
– ensbana
Jul 20 at 23:42
Well, if you want to go with balls and boxes instead of integer partition: The condition $lambda_i le k$ indicates that each box must have at most $k$ balls. So even though $5,1,1$ is a valid partition for the first situation (where we talk about balls and boxes), it is not a valid partition for the second situation with integer partition where $n=7,k=3$ since $5>3$.
– Tengu
Jul 20 at 23:48
My current assumption is that, if we say that a bijection exists, then for each distribution of balls we would find a corresponding integers partition. Also I assume that the integers partition (5,1,1) corresponds to the distribution 5 balls - 1 ball - 1 ball, yet the way integers partition is defined in the question somehow forbids tuples like (5,1,1)? What would the tuple corresponding to the abovementioned balls distribution be?
– ensbana
Jul 20 at 23:55
To be even more explicit, if F is the bijection from the set of balls distributions to the set of integers partitions, what is F(5 balls, 1 ball, 1 ball)?
– ensbana
Jul 21 at 0:01
1
By answering this question of your, I think it will give away too much about the solution (which you can easily search on google). You should first spend some time finding a bijection. Try to take some small examples of $n,k$, list all the distributions/partitions for each situation then try to give a mapping between the two. See if you find any pattern that you can generalize for any $n,k$.
– Tengu
Jul 21 at 0:05
 |Â
show 2 more comments
What I’m asking is, what does that condition $lambda_i leq k$ have to do with the distribution of balls into boxes? A distribution with 5 balls in 1 box, 1 ball in another box and 1 ball in yet another box is certainly a valid way to put 7 balls into 3 boxes, but why is its corresponding integers partition of 7, i.e. (5,1,1), not valid? Wouldn’t the condition $lambda_i leq k$ then exclude some cases from the set of integer partitions of n (and hence we wouldn’t have a desired bijection)?
– ensbana
Jul 20 at 23:42
Well, if you want to go with balls and boxes instead of integer partition: The condition $lambda_i le k$ indicates that each box must have at most $k$ balls. So even though $5,1,1$ is a valid partition for the first situation (where we talk about balls and boxes), it is not a valid partition for the second situation with integer partition where $n=7,k=3$ since $5>3$.
– Tengu
Jul 20 at 23:48
My current assumption is that, if we say that a bijection exists, then for each distribution of balls we would find a corresponding integers partition. Also I assume that the integers partition (5,1,1) corresponds to the distribution 5 balls - 1 ball - 1 ball, yet the way integers partition is defined in the question somehow forbids tuples like (5,1,1)? What would the tuple corresponding to the abovementioned balls distribution be?
– ensbana
Jul 20 at 23:55
To be even more explicit, if F is the bijection from the set of balls distributions to the set of integers partitions, what is F(5 balls, 1 ball, 1 ball)?
– ensbana
Jul 21 at 0:01
1
By answering this question of your, I think it will give away too much about the solution (which you can easily search on google). You should first spend some time finding a bijection. Try to take some small examples of $n,k$, list all the distributions/partitions for each situation then try to give a mapping between the two. See if you find any pattern that you can generalize for any $n,k$.
– Tengu
Jul 21 at 0:05
What I’m asking is, what does that condition $lambda_i leq k$ have to do with the distribution of balls into boxes? A distribution with 5 balls in 1 box, 1 ball in another box and 1 ball in yet another box is certainly a valid way to put 7 balls into 3 boxes, but why is its corresponding integers partition of 7, i.e. (5,1,1), not valid? Wouldn’t the condition $lambda_i leq k$ then exclude some cases from the set of integer partitions of n (and hence we wouldn’t have a desired bijection)?
– ensbana
Jul 20 at 23:42
What I’m asking is, what does that condition $lambda_i leq k$ have to do with the distribution of balls into boxes? A distribution with 5 balls in 1 box, 1 ball in another box and 1 ball in yet another box is certainly a valid way to put 7 balls into 3 boxes, but why is its corresponding integers partition of 7, i.e. (5,1,1), not valid? Wouldn’t the condition $lambda_i leq k$ then exclude some cases from the set of integer partitions of n (and hence we wouldn’t have a desired bijection)?
– ensbana
Jul 20 at 23:42
Well, if you want to go with balls and boxes instead of integer partition: The condition $lambda_i le k$ indicates that each box must have at most $k$ balls. So even though $5,1,1$ is a valid partition for the first situation (where we talk about balls and boxes), it is not a valid partition for the second situation with integer partition where $n=7,k=3$ since $5>3$.
– Tengu
Jul 20 at 23:48
Well, if you want to go with balls and boxes instead of integer partition: The condition $lambda_i le k$ indicates that each box must have at most $k$ balls. So even though $5,1,1$ is a valid partition for the first situation (where we talk about balls and boxes), it is not a valid partition for the second situation with integer partition where $n=7,k=3$ since $5>3$.
– Tengu
Jul 20 at 23:48
My current assumption is that, if we say that a bijection exists, then for each distribution of balls we would find a corresponding integers partition. Also I assume that the integers partition (5,1,1) corresponds to the distribution 5 balls - 1 ball - 1 ball, yet the way integers partition is defined in the question somehow forbids tuples like (5,1,1)? What would the tuple corresponding to the abovementioned balls distribution be?
– ensbana
Jul 20 at 23:55
My current assumption is that, if we say that a bijection exists, then for each distribution of balls we would find a corresponding integers partition. Also I assume that the integers partition (5,1,1) corresponds to the distribution 5 balls - 1 ball - 1 ball, yet the way integers partition is defined in the question somehow forbids tuples like (5,1,1)? What would the tuple corresponding to the abovementioned balls distribution be?
– ensbana
Jul 20 at 23:55
To be even more explicit, if F is the bijection from the set of balls distributions to the set of integers partitions, what is F(5 balls, 1 ball, 1 ball)?
– ensbana
Jul 21 at 0:01
To be even more explicit, if F is the bijection from the set of balls distributions to the set of integers partitions, what is F(5 balls, 1 ball, 1 ball)?
– ensbana
Jul 21 at 0:01
1
1
By answering this question of your, I think it will give away too much about the solution (which you can easily search on google). You should first spend some time finding a bijection. Try to take some small examples of $n,k$, list all the distributions/partitions for each situation then try to give a mapping between the two. See if you find any pattern that you can generalize for any $n,k$.
– Tengu
Jul 21 at 0:05
By answering this question of your, I think it will give away too much about the solution (which you can easily search on google). You should first spend some time finding a bijection. Try to take some small examples of $n,k$, list all the distributions/partitions for each situation then try to give a mapping between the two. See if you find any pattern that you can generalize for any $n,k$.
– Tengu
Jul 21 at 0:05
 |Â
show 2 more comments
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The number of partitions of $n$ into at most $k$ parts is equal to the number of partitions of size at most $k$. In a Ferrers diagram you can count the dots by rows, or by columns.
– saulspatz
Jul 20 at 20:11
Could you please explain the first sentence? As an example, why doesn't the tuple (5,1,1) above work?
– ensbana
Jul 20 at 21:19