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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How many integer solutions of $2 x_1 + 2 x_2 + x_3 + x_4 = 12 $?

I am wondering how many integer solutions there are of $ 2 x_1 + 2 x_2 + x_3 + x_4 = 12$ ? Should the question restrict each $x_i$ to be nonnegative? I have solved it if the $x_i$'s are nonnegative ...
Maggie McCarthy's user avatar
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Show the partition function $Q(n,k)$, the number of partitions of $n$ into $k$ distinct parts, is periodic mod m

Let $Q(n,k)$ be the number of partitions of $n$ into $k$ distinct parts. I want to show that for any $m \geq 1$ there exists $t \geq 1$ such that $$Q(n+t,i)=Q(n,i) \mod m \quad \forall n>0 \forall ...
Kinkin's user avatar
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A sum of multinomial coefficients over partitions of integer

I denote a partition of an integer $n$ by $\vec i = (i_1, i_2, \ldots)$ (with $i_1, i_2, \ldots \in \mathbb N$) and define it by $$ \sum_{p\geq1} p i_p = n. $$ I set $$ |\vec i| = \sum_{p\geq1} i_p. $$...
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A tagged partition interval subset relations problem.

The problem was taken from “Introduction to Real Analysis” by R. Bartle and D. Sherbert, Section 7.1, Exercise 4. Let $\dot{\mathcal{P}}$ be a tagged partition of $[0,3]$. (a) Show that the union $U_1$...
鈴木悠真's user avatar
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Find generating series on set of descending sequences, with weight function as taking sum of sequence

Given the set of all sequences of length k with descending (not strictly, so $3,3,2,1,0$ is allowed) terms of natural numbers (including $0$), $S_k$, and the weight function $w(x)$ as taking the sum ...
haha's user avatar
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Formula for number of monotonically decreasing sequences of non-negative integers of given length and sum?

What is a formula for number of monotonically decreasing sequences of non-negative integers of given length and sum? For instance, if length k=3 and sum n=5, then these are the 5 sequences that meet ...
JacobEgner's user avatar
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Integer partitions with summands bounded in size and number

This book says it's easy, but to me, it's not. :( As for 'at most k summands', in terms of Combinatorics, by using MSET(), $$ MSET_{\le k}(Positive Integer) = P^{1,2,3,...k}(z) = \prod_{m=1}^{k} \frac{...
David Lee's user avatar
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Generating Function for Modified Multinomial Coefficients

The multinomial coefficients can be used to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + ...} \right)^n}$ in the basis of monomial symmetric polynomials (MSP). For example, $$\...
Bear's user avatar
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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? \...
Lorenzo Vittori's user avatar
4 votes
1 answer
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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\...
Bear's user avatar
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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 ...
Random Person's user avatar
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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 ...
zg c's user avatar
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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 ...
Sanagama's user avatar
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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,...
Adam's user avatar
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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 \...
Fabius Wiesner's user avatar
5 votes
1 answer
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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{...
Knifer Plasma's user avatar
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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 ...
Seán Healy's user avatar
2 votes
1 answer
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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 $$\sum_{\lambda\vdash\mathcal{OD}}q^{\vert\lambda\vert}=\prod_{j\geq1}(1+q^{2j-1})$$ ...
T. Amdeberhan's user avatar
1 vote
0 answers
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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 $P_k^s(n)$....
user110391's user avatar
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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 \...
Michael Chu's user avatar
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1 answer
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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 ...
a boy's user avatar
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1 answer
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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 ...
Poke_Programmer's user avatar
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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 ...
Zek's user avatar
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1 answer
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Generating function of number of partitions of $n$ into parts at most $k$

I am trying to grasp the intuition behind this example. Show that $\sum_{n \geq 0} p_{\leq k}(n)x^n = \prod_{i=1}^k \frac{1}{1-x^i}$ where $p_{\leq k}(n)$ denotes the number of partitions of the ...
Zek's user avatar
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6 digit numbers no repetitions with equal part sums

I am interested in six digit numbers where the sum of the first digits numbers is equal to the last three digits but no digit repeats. Examples: $143260$ (since $1+4+3=2+6+0$) $091325$ (since $0+9+1=...
bissi's user avatar
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2 votes
1 answer
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How do I prove the integer partition problem in combinatorial mathematics?

Question: Show that the number of partitions of $n$ into $k$th powers $(k>1)$ in which no part appears more than $k-1$ times is always equal to 1. This is actually an exercise from the book Integer ...
Always's user avatar
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A question about partition function of integers

First I would like to mention that I am not too well versed with the properties of the partition function of an integer so I apologise if this question is too elementary or daft. Recall we define the ...
Herr Warum's user avatar
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Do we have recurrences for evaluating the Partition Function on Graphs?

Inspired by this question about defining the partition function on non integers, I was thinking about what sorts of other objects can a partition function be defined on. I noticed that if we have an ...
Sidharth Ghoshal's user avatar
19 votes
1 answer
991 views

Can we partition the reciprocals of $\mathbb{N}$ so that each sum equals 1

Let $S = \{1, 1/2,1/3,\dots\}$ Can we find a partition $P$ of $S$ so that each part sums to 1, e.g. $$P_1 = {1}$$ $$P_2 = { 1/2,1/5,1/7,1/10,1/14,1/70}$$ $$P_3 = {1/3,1/4,1/6,1/9,1/12,1/18}$$ $$P_4 = \...
AndroidBeginner's user avatar
2 votes
1 answer
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Identity involving Sum of Inverse of Product of Integer partitions [closed]

Is there a way to prove the following identity \begin{equation} \sum_{l = 1}^{k} \left( \frac{(-s)^l}{l!} \sum_{n_1 + n_2 + \ldots n_l = k} \frac{1}{n_1n_2 \ldots n_k} \right)= (-1)^k {s \choose k} \,...
alpha's user avatar
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4 answers
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Is there any mathematical operation that can be reversed to two unique numbers?

Is there any mathematical operation “op”, such that when applied to two integers a, b: a op b = n. We can use that n and reverse the operation, to get n = a op b? For example, the sum is not ...
Geronimo Castaño's user avatar
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1 answer
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Find closed form of real valued function

Let $f:\mathbb{N}\rightarrow\mathbb{R}, h:\mathbb{N}\rightarrow\mathbb{R}$ be two functions satisfying $f(0)=h(0)=1$ and: $$f(n) = \sum_{i=0}^{n}h(i)h(n-i)$$ Find a closed form for $h(n)$ in terms of $...
Duffoure's user avatar
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All the different ways to add a set of lengths - need explanation of the answer

I wish to make an integer length n of boards laid down end to end. Each board is an integer length and among boards of one length there are unique types. For example, boards of length one come in two ...
Steven Lord's user avatar
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0 answers
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Let $P(n)$ be a number of all integer partitions of n. Then, $P(1) + P(2) + ... P(n) < P(2n)$

Let $P(n)$ be a number of all integer partitions of n. Then, show that $P(1) + P(2) + ... P(n) < P(2n)$. The solution from the textbook: If $\pi$ is a partition of $i$ for $i \leq n$, then its ...
Zek's user avatar
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4 votes
0 answers
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An identity related to the series $\sum_{n\geq 0}p(5n+4)x^n$ in Ramanujan's lost notebook

While browsing through Ramanujan's original manuscript titled "The Lost Notebook" (the link is a PDF file with 379 scanned pages, so instead of a click it is preferable to download) I found ...
Paramanand Singh's user avatar
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3 votes
1 answer
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Finding the number of integer composition using only a specific pair of numbers [closed]

I want to find the number of integer compositions of 19 using numbers 2 and 3. If I wanted to find the number of integer compositions of 19 using 1 and 2, I could write it as $F(n)=F(n-1)+F(n-2)$ ...
Zek's user avatar
  • 309
6 votes
1 answer
67 views

Partition of a number $n$ whose each part is coprime with this number

I'm trying to solve the following problem: given an integer $n$, under which conditions of $n$ the following statement is true: For any $1 < k \leq n$, there is always a partition of $k$ parts of $...
Ta Thanh Dinh's user avatar
1 vote
1 answer
41 views

Finding groupings of numbers that add up to values in a target set

I am wondering if there is an efficient way to partition a set of numbers into subsets that sum to values in a set of target values. Specifically, let's assume that we are given $k$ distinct values (i....
Physics Penguin's user avatar
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On partition of a number $n$ into positive integers

I need to prove this inequality, but I do not have a good background in algebra, if you can guide me: We have: $$ p_1+p_2+\ldots+p_k = q_1+q_2+\ldots+q_k+\ldots+q_t = n $$ and $$ p_1 + 2p_2 + \ldots +...
BADJARA Mohamed el Amine's user avatar
1 vote
0 answers
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Partition of n into k parts with at most m

I ran into a problem in evaluating a sum over kronecker delta. I want to evaluate $$\sum_{\ell_1,...,\ell_{2m}=1}^s\delta_{\ell_1+\ell_3+...+\ell_{2m-1},\ell_2+\ell_4+...+\ell_{2m}}$$ My approach was ...
Qant123's user avatar
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0 answers
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Construct bijection between partitions of $[n-1]$ with $k-1$ parts, and partitions of $[n]$ with $k$ parts with no parts contains consecutive integers

So... the title is just I want to ask for. This is exercise #104 of Chapter 1, in Lessons In Enumerative Combinatorics, GTM 290. With previous exercise, I proved that number of $k-1$-part paritions of ...
MH.Lee's user avatar
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2 answers
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Number of positive integral solution of $\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10$

I want to find the number of positive integer solutions of the equations given by $$\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10.$$ I know the case that, for any pair of ...
abcdmath's user avatar
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0 votes
1 answer
108 views

How to fill a set of container by partition of set?

Let $\{A,B,C,D\}$ be a table with $4$ containers of sizes respectively $5,5,3,2$. Let $\pi=\{B_1,B_2,\cdots B_k\}$ a partition of a set, where $k\in \mathbb{N}$. I wonder how to enumerate the ...
Josaphat Baolahy's user avatar
1 vote
1 answer
47 views

The product of components in the partition of a number.

Let $f(n,k)$ be the sum of expressions of the form $x_1 \cdot x_2 \cdot \ldots \cdot x_k$, where the sum counts over all solutions of the equation $x_1 + \ldots + x_k = n$ in natural numbers. Find ...
Michał's user avatar
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0 votes
1 answer
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Sum of square of parts, and sum of binomials over integer partition

Let $n$ be positive integer. Consider its integer partitions denoting as $(m_1,\cdots,m_k)$, where $m_1+\cdots+m_k=n$ and the order does not matter. I am interested in the following two quantity (1) $$...
happyle's user avatar
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0 answers
37 views

Upper bound a sum of over partitions of integer

Let $n,L$ are positive integers. $(m_1,\cdots,m_k)$ is partition of $n$, i.e. $m_1+\cdots+m_k=n$ or $(m_1,\cdots,m_k) \vdash n$. Note that a partition of n is a representation of n as a sum of ...
happyle's user avatar
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Whether a sum over all partitions of n decays exponentially with n

Given $L$ as an absolute constant (does not depend on $n$), and $\beta$ is a function of $n$. Consider $$\sum_{m_1+\cdots+m_k=n\\ 1\leq k\leq L\\ \text{all }m_i \text{ are positive integers}}\binom{L}{...
happyle's user avatar
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1 answer
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The same number of distinct elements and ones in the partitions of the number.

For a given partition $\pi$ of the number $n$, let $A(\pi)$ denote the number of ones in $\pi$, and $B(\pi)$ the number of distinct parts in $\boldsymbol{\pi}$. EXAMPLE: for the partition $\pi: 1+1+2+...
Michał's user avatar
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0 votes
1 answer
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Skew diagram horizontal $m$-strip defintion and board strip question

In Symmetric Functions and Hall Polynomials by Manin, Manin claims that for a skew diagram $\theta = \lambda - \mu$ to be a horizontal $m$-strip, "the sequences $\lambda$ and $\mu$ are interlaced,...
Henry Yan's user avatar
3 votes
2 answers
78 views

Homogeneous and inhomogeneous recurrences for a sequence

Some integer sequences have both homogeneous and inhomogeneous recurrence relations. If you know one, are there techniques for figuring out the other? (Or seeing if the other exists?) Below is an ...
Brian Hopkins's user avatar

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