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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Probability that the maximum number of dice with the same face is k
Let say we have $N$ dice with 6 faces. I'm asking my self, what is the probability that the maximum number of dice with the same face is $k$?
In more precise terms, what is the size of this set?
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Why do Bell Polynomial coefficients show up here?
The multinomial theorem allows us to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + {x_4} + ...} \right)^n}$. I am interested in the coefficients when expanding ${\left( {\sum\...
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Prove convergence of partition sequence [duplicate]
We are given a partition of a positive integer $x$. Each step, we make a new partition of $x$ by decreasing each term in the partition by 1, removing all 0 terms, and adding a new term equal to the ...
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The Asymptotic formula of the generating function related with the partition of a positive integer
This question may be duplicate with this answer_1 and here I referred to the same paper by Hardy, G. H.; Ramanujan, S. referred to by wikipedia which is referred to in answer_1.
But here I focused on ...
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corollary of the partition congruence
I was going though the paper of Ramanujan entitled Some properties of p(n), the number of partitions of n (Proceedings of the Cambridge Philosophical Society, XIX, 1919, 207 – 210). He states he found ...
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Congrunces of partitions into distinct parts
Let $P_{d}(n)$ denote the number of partitions of n into distinct parts. The generating function of $P_{d}(n)$ s given by:
$$ \sum_{n \geq 0}P_{d}(n)q^{n}= \prod_{n \geq 0} (1+q^{n}).$$
Now let $P_{2,...
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Conjecture about integer partitions
I formulated this conjecture after reading this related question.
Let $\mathcal{P}(n) = \{P_1(n), P_2(n), \ldots \}$ be the set of all integer partitions of a positive integer $n$, and $p(n)=\vert \...
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Prove that the general formula for a sequence $a_n$ is $\frac{(-1)^n}{n!}$
Here is a sequence $a_n$ where the first five $a_n$ are:
$a_1=-\frac{1}{1!}$
$a_2=-\frac{1}{2!}+\frac{1}{1!\times1!}$
$a_3=-\frac{1}{3!}+\frac{2}{2!\times1!}-\frac{1}{1!\times1!\times1!}$
$a_4=-\frac{...
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Constrained integer partition containing particular summands
Is there a way to calculate the number of constrained integer partitions containing particular summands? By constrained, I mean, the permitted summands must be below a certain limit, such as 5. Take ...
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Counting gap sizes in a subfamily of partitions
Let $\mathcal{OD}$ be the set of all odd and distinct integer partitions. This has a generating function given by
$$\sum_{\lambda\vdash\mathcal{OD}}q^{\vert\lambda\vert}=\prod_{j\geq1}(1+q^{2j-1})$$
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Notation for $k$-partitions of $n$ containing at least one summand equal to $s$
I am looking for whether there is any notation for the $k$-partition number of $n$ where the partitions must include some summand $s$.
An example of the kind of notation I am looking for is $P_k^s(n)$....
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What is the maximum range of a convex finite additive 2-basis of cardinality k?
Conjecture:
Given any $d \in \mathbb{Z}_{\geq 2}$ and $k=2d-2$, we have \begin{align*}
\max \{ n : (\exists &f \in \{ \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0} \})\\ &[((\forall i \in \...
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Conjecture: Ramsey Number R(m,n)=(2m-1)*p(2m-6+n,m)+{1,m,m+1}, for 3<=m<=n
Today(2023-11-22), I have a conjecture on Ramsey numbers.
Fence Conjecture(栅栏猜想): Ramsey Number
R(m,n)=(2m-1)*p(2m-6+n,m)+{1,m,m+1}, for 3<=m<=n.
Here p(n,k) denotes both the number of ...
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What is the 11th unordered combination of natural numbers that add upto 6 in the partition function?
So, I was making unordered combinations of natural numbers which add upto a certain natural number. I was able to go till 6 when I got to know about the partition function. I was pleased to see that ...
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Generating function of number of partitions of $n$ into all distinct parts
I am trying to grasp this example from the book A Walk Through Combinatorics:
Show that $\sum_{n \ge 0} p_d(n)x^n = \prod_{i \ge 1}(1+x^i)$ where
$p_d$ stands for partitions of $n$ into all distinct ...
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