# Questions tagged [integer-partitions]

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

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### Solving a recursion formula involving products of compositions of an integer

I have the following recursive formula that I want to solve in order to find a general, non-recursive expression for arbitrary $S_N$ (real positive number). Here it is: \begin{equation*} S_{N+1} = \...
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### Optimal balanced ternary set of weights - infinite version

A fairly well known puzzle is to pick the optimal weights for use in a simple, symmetric balance. It is assumed that any combination of weights with or without the sample will fit in either pan. The ...
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### Generating all sorted positive integer sequences of given length that sum to a given total

In this thread, a bijective function is requested which, given two positive integers $n$ and $k$, maps between natural-number identifiers and sequences of $k$ positive integers that sum to $n$. Due to ...
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### Proof of a statement in OEIS A260533 about partitions

Defining the coefficient of a partition $p=(p[1]\geq p[2] \geq \dots \geq p[m])$ of $n=p[1]+\dots+p[m]$ as $c(p)=\binom{p[1]}{p[2]} \binom{p[2]}{p[3]}\dots\binom{p[m-1]}{p[m]}$ A260533 states that ...
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### Name of such combinatorial numbers. [duplicate]

Let $k,n\in\mathbb{N}$. Let $N(k,n)$ denote the size of the finite set $$\{(x_1,\cdots,x_k)\in\mathbb{N}^k:x_1+2x_2+\cdots+kx_k=n\}.$$ I feel it special and important. Do $N(k,n)$ have names? Is ...
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