# Questions tagged [integer-partitions]

Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.

854 questions
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### Why are asymptotically one half of the integer compositions gap-free?

Question summary The number of gap-free compositions of $n$ can already for quite small $n$ be very well approximated by the total number of compositions of $n$ divided by $2$. This question seeks ...
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### Very curious properties of ordered partitions relating to Fibonacci numbers

I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon. We call an ordered ...
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### Making Change for a Dollar (and other number partitioning problems)

I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quoted ...
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### Feeding real or even complex numbers to the integer partition function $p(n)$?

Like most people, when I first encountered $n!$ in grade school, I graphed it, then connected the dots with a smooth curve and reasoned that there must be some meaning to $\left(\frac43\right)!$ — and,...
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### Ellipse 3-partition: same area and perimeter

Inspired by the question, "How to partition area of an ellipse into odd number of regions?," I ask for a partition an ellipse into three convex pieces, each of which has the same area and the same ...
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### Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
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### On the inequality $\frac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$

For all positive integers $n$, $p(n)$ is the number of partitions of $n$ as the sum of positive integers (the partition numbers); e.g. $p(4)=5$ since $4=1+1+1+1=1+1+2=1+3=2+2=4.$ Prove ...
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### Closed-form Expression of the Partition Function $p(n)$

I feel like I have seen news that a paper was recently published, at most a few months ago, that solved the well-known problem of finding a closed-form expression for the partition function $p(n)$ ...
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### The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts

This seems to be a common result. I've been trying to follow the bijective proof of it, which can be found easily online, but the explanations go over my head. It would be wonderful if you could give ...
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### How solutions of distinct non-negative solutions are there to $k_1+\cdots+k_n=k$?

How many distinct $n$-tuples with distinct non-negative integer elements are there that add to $k$. For example there are $6$ triples that add to $4$. Namely $(0, 1, 3)$ and its $6$ permutations. Is ...
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### Permutation induced by a partition

Let $\lambda$ be a partition of length $n$ and suppose its largest diagonal block, the Durfee square of $\lambda$, has size $r$. By this I mean that $\lambda = (\lambda_1,\ldots,\lambda_n)$ is a non-...
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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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### $a+b+c+d+e=79$ with constraints

How many non-negative integer solutions are there to $a+b+c+d+e=79$ with the constraints $a\ge7$, $b\le34$ and $3\le c\le41$? I get that for $a\ge7$ you do $79-7=72$, $\binom{72+5-1}{5-1}=\binom{76}4$...
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### Partition Generating Function

a) Let $$P(x)=\sum_{n=0}^{\infty} p_nx^n=1+x+2x^2+3x^3+5x^4+7x^5+11x^6+\cdots$$ be the partition generating function, and let $Q(x)=\sum_{n=0}^{\infty} q_nx^n$, where $q_n$ is the number of ...
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### How to prove it? (one of the Rogers-Ramanujan identities)

Prove the following identity (one of the Rogers-Ramanujan identities) on formal power series by interpreting each side as a generating function for partitions: 1+\sum_{k\geq1}\frac{z^k}{(1-z)(1-z^2)...