# Tagged Questions

Questions related to the different ways of expressing an integer as a sum of integers; or, questions related to the subdivision of a set into smaller disjoint sets; questions related to the subdivision of an interval into smaller intervals that intersect only at the endpoints.

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### How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...
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### Integer partitions with distinct parts

Let $~~p(n)~~$ denote the number of all partitions of positive integer $~~n~~$ with distinct parts. I would like to find some effective algorithm for calculating $~~p(n)~~$. It seems that dynamic ...
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### How to partition $nk$ objects $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size $k$, so that no combination of $k$ is repeated.

What is an algorithm to partition $nk$ objects a total of $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size exactly $k$, so that no subset of size $k$ is ever repeated? For example, ...
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### # of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers?

How do I see that the number of partitions of the integer $n$ into at most $r$ positive integers is equal to the number of partitions of $n + r$ into exactly $r$ positive integers?
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### Verification of Rogers-Ramanujan identities

In Hardy's book 'Ramanujan', section 6.8 on the Rogers-Ramanujan identities, it states: None of these proofs can be called both "simple" and "straightforward", since the simplest are essentially ...
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### partitioning of a set with kn members into k subsets such that each subset has n members

we know that $S(n,k)$ is the number of ways we can partition a set with $n$ members to $k$ subsets ( each subset has at least one member). imagine we have a set with $k*n$ members. we want to ...
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### Partitioning of sets

Question: Consider set $A= \{ 1, 2, 3, ..., n\}$. For what values of $n$ can $A$ be partitioned into 3 subsets $A_1, A_2, A_3$, such that sum of the elements of each of them are equal? My Attempt: ...
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### Trouble understanding noncrossing partitions

I am trying to understand what a non-crossing partition means. I was reading a paper and it states A partition is noncrossing if there do not exist four distinct elements $$a < b < c < d$$ ...
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### Computing partition numbers

Today a friend and myself came up with the question of computing partitions of numbers, i.e.: given a number $n$, what is the number $p(n)$ of was of different ways writing $n$ as a sum of non-zero ...
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### Young Tableaux as Matrices

These questions are motivated only by curiosity. Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$. Is there any ...
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### Partitioning a Queue

Imaging a situation that we have n people in a queue and each people represent with number 1 and I want to partition the queue in smaller part, there are several ways to partition the queue. For ...
I happened across the following question: Determine the number of ways of putting $m$ indistinguishable balls into $n$ indistinguishable boxes with the restriction that no box is empty. The ...