Questions tagged [integer-partitions]
Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.
1,430
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How to define initial values for recurrence relation
I was given the following problem : let $f(n,k)$ be the number of possible partitions of $n$ into $k$ different non-negative integers. Find a recurrence relation and initial values for $f(n,k)$.
So I ...
3
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2
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$(w_{1},w_{2},w_{3},\dots,w_{7})$ integers with $20\le w_{i} \le 22$ such that $\sum_{i=1}^{7}w_{i} = 148$
How many $(w_{1},w_{2},w_{3},\dots,w_{7})$ where each of the $w_{i}$'s are integers and $20\le w_{1},w_{2},w_{3},\dots,w_{7}\le 22$ such that they satisfy
$$w_{1}+w_{2}+w_{3}+\dots+w_{7}=148$$
ATTEMPT
...
- 189
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Finding formula for $a+b+c=n$ where $(a,b,c)$ are positive integers.
I'm currently studying a book by Paul Zeitz and currently stuck on exercise 6.2.23, below is the problem:
Find a formula for the number of different ordered triples $(a,b,c)$ of positive integers ...
- 189
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18
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dominance order of conjugate partition
Let $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n $ be two sets of non-strictly decreasing non-negative integers such that $\sum_{i=1}^n a_i = \sum_{i=1}^n b_i = m > 0 $. Let $a_i'$ and $b_i'$ ...
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1
answer
45
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Confusion between relation of stars and bars and q-binomial coefficient
Suppose we want to know the number of integer solutions to the equation $$x_1 + \cdots x_m = N$$ where $0 \leq x_i \leq t - 1$ for $1 \leq i \leq m$. One way to do this is by finding the coefficient ...
- 1,011
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The number of partitions of $\{1, \ldots, n+1\}$ into subsets of nonconsecutive integers
For each integer $n$, let $a_n$ be the number of partitions of $\{1, \ldots, n+1\}$ into subsets of nonconsecutive integers.
I found (by listing) that $ a_1, a_2, a_3, a_4$ are $1, 2, 5, 15$ ...
- 721
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Generating function and currency
We assume that we have a country's currency that contains three coins worth 1, 3, and 4. How many ways can we get an amount of $n$ using these three pieces?
In others words what is the number of ...
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On $(0,1)$-strings and counting
Consider a binary string of length $n$ that starts with a $1$ and ends in a $0$. Clearly there are $2^{n-2}$ such bit strings. I would like to condition these sequences by insisting that the number of ...
- 339
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high school math: summands
Let's say we have a question that asks you to find the amount of all possible integers adding up to a random number, lets just say 1287. However, the possible integers is restricted to explicitly 1's ...
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proving $2$ generating functions are equal [duplicate]
I have been doing a problem on generating functions and I need to prove that these are equal:
$\displaystyle\prod_{i=1}^\infty\frac{1+x^{2i-1}}{1-x^{2i}}$
and
$\displaystyle\prod_{i=1}^\infty\frac{1-x^...
- 27
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1
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Generating function of partitions of $n$ in $k$ prime parts.
I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$.
I know ...
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Conjugate of a Gel'fand pattern
Background:
A Gel'fand pattern is a set of numbers
$$
\left[\begin{array}{}
\lambda_{1,n} & & \lambda_{2,n} & & & \dots & & & \lambda_{n-1,n}...
- 248
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About the product $\prod_{k=1}^n (1-x^k)$
In this question asked by S. Huntsman, he asks about an expression for the product:
$$\prod_{k=1}^n (1-x^k)$$
Where the first answer made by Mariano Suárez-Álvarez states that given the Pentagonal ...
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1
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A variant of the partition problem or subset sum problem
Given a target list $T = (t_1, t_2, \ldots, t_N)$ and a multiset $S = \{s_1, s_2, \ldots, s_M\}$, both with non-negative integer elements, $t_k\in \mathbb{N}_>$ and $s_k\in \mathbb{N}_>$, ...
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Bell numbers - Cardinality of odd number of parts in partitions of the finite set $[n]$.
As it well known, Bell numbers denoted $B_{n}$ counts distinct partitions of the finite set $[n]$. So for example if $n=3$ there are 5 ways to the set $\left\{ a,b,c\right\}$ can be partitioned:
$$\...
- 119
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Prove $\sum_{j=0}^{n} q^{j^{2}}\binom{n}{j}_{q^{2}}$ generates the self-conjugate partitions with part at most $n$.
Prove $\sum_{j=0}^{n} q^{j^{2}}\binom{n}{j}_{q^{2}}$ generates the self-conjugate partitions with part at most $n$, and that it equals $(1+q)(1+q^{3})\cdot\cdot\cdot(1+q^{2n-1})$.
For the first part, ...
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Need help with part of a proof that $p(5n+4)\equiv 0$ mod $5$
Some definitions:
$p(n)$ denotes the number of partitions of $n$.
Let $f(q)$ and $h(q)$ be polynomials in $q$, so $f(q)=\sum_0^\infty a_n q^n$ and $h(q)=\sum_0^\infty b_n q^n$. Then, we say that $f(q)\...
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How to prove the following partition related identity?
So I want to show that the following is true, but Iam kidna stuck...
$$
\sum_{q_{1}=1}^{\infty}\sum_{q_{2}=q_{1}}^{\infty}\sum_{q_{3}=q_{1}}^{q_{2}}...\sum_{q_{k+1}=q_{1}}^{q_{k}}x^{q_{1}+q_{2}+...+q_{...
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Is there a closed form method of expressing the *content* of integer partitions of $n$?
I know that the question of a closed form for the number of partitions of $n$, often written $p(n)$, is an open one (perhaps answered by the paper referred to in this question's answer, although I'm ...
2
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1
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MacMahon partition function and prime detection (ref arXiv:2405.06451)
In the recent paper arXiv:2405.06451 the authors provide infinitely many characterizations of the primes using MacMahon partition functions: for $a>0$ the functions $M_a(n):=\sum\limits_{0<s_1&...
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There are allways $q_1,q_2,...,q_n$ such that $f'(q_1)+f'(q_2)+...+f'(q_n)=n$ for every natural n [duplicate]
Let $f$ be a differentiable between $(0,1)$ and take $f(1)=1, f(0)=0$. Prove that then there exist $q_1,q_2,...,q_n$ distinct points such that $f'(q_1)+f'(q_2)+...+f'(q_n)=n$ for every natural n. By ...
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Is this connection of (increasingly exclusive) integer partitions to the the Euler-Mascheroni constant useful?
$\mathbf{SETUP}$
From this previous question, I quote Cauchy's formula for the number of all possible cycle types
\begin{align}
N_{\lambda} =
\frac{n!}
{1^{\alpha_1} 2^{\alpha_2} ... n^{\...
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answer
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Why does this connection of increasingly exclusive partitions $P_{n,k}$ to the harmonic series $H_k$ exist?
$\mathbf{SETUP}$
In this previous question, I show how the sum of all cycles of type defined by non-unity partitions of $n$ relates to the derangement numbers / subfactorial $!n$ (number of ...
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How to explain arithmetic form of surprising equality that connects derangement numbers to non-unity partitions?
$\mathbf{SETUP}$
By rephrasing the question of counting derangements from
"how many permutations are there with no fixed points?"
to
"how many permutations have cycle types that are non-...
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Congruences of partition function
I'm trying to understand Ken Ono's results showing Erdös' conjecture for the primes $\ge5$. He first shows the following: let $m\ge5$ be prime and let $k>0$. A positive proportion of the primes $\...
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Proving an Identity on Partitions with Durfee Squares Using $q$-Binomial Coefficients and Generating Functions
Using the Durfee square, prove that
$$
\sum_{j=0}^n\left[\begin{array}{l}
n \\
j
\end{array}\right] \frac{t^j q^{j^2}}{(1-t q) \cdots\left(1-t q^j\right)}=\prod_{i=1}^n \frac{1}{1-t q^i} .
$$
My ...
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Recursion regarding number-partitions
I am learning about partitions of numbers at the moment.
Definition:
Let $n \in \mathbb{N}$. A $k$-partition of $n$ is a representation of $n$ as the sum of $k$ numbers greater than $0$, (i.e.
$n=a_1+....
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Find generating function for the number of partitions which are not divisible by $3$. [duplicate]
I'm trying to find the generating function for the number of partitions into parts, which are not divisible by $3,$ weighted by the sum of the parts. My idea is that we get the following generating ...
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Estimate the order of restricted number partitions
There are $k$ integers $m_l,1\leq l\leq k
$(maybe negetive), satisfiying $|m_l|\leq M$ and $\sum_l m_l=s$.
I want to get an order estimate of the number of solutions for $k, M$ when fixing $s$.
I came ...
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Computing integer partitions subject to certain constraints
Context:
I am programming a videogame.
Background:
Let $I$ be a set of named items such that each is assigned a difficulty score, and each is tagged either as "food" or "obstacle". ...
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A generalized subset sum problem
I'm looking at a problem I believe is combinatorial.
Find all possible solutions $\mathbf{x}$ to:
$$\mathbf{a} = [a_1, a_2, ..., a_n], a_k \in \mathbb{N^+}$$
$$\mathbf{l} = [l_1, l_2, ..., l_n], l_k \...
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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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Partial Orders on Integer Partitions
My question is the following: An integer partition $\lambda$ can be represented as an integer sequence $(f_1,f_2,f_3, \cdots)$ where $f_i$ is the number of parts used in $\lambda$. For instance, $4 + ...
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Has anyone found a closed-form expression for the strict partition function?
More precisely, if anyone has found it, could they provide links, please? I have been trying to find such a solution and have not seen one.
For context, when I try to solve the strict partition ...
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Graded ring generated by finitely many homogeneous elements of positive degree has Veronese subring finitely generated in degree one
Let $S=\bigoplus_{k\ge 0}S_n$ be a graded ring which is generated over $S_0$ by some homogeneous elements $f_1,\dotsc, f_r$ of degrees $d_1,\dotsc, d_r\ge 1$, respectively. I want to show that there ...
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Congruences of partitions and Legendre symbol.
Let $(\frac{a}{p})$ denote the Legendre symbol, and let $\psi(q)=\sum_{n\geq 0} q^{n(n+1)/2}$.
We define Ramanujan's general theta function $f(a,b)$ for $\mid ab \mid <1$ as
$$
f(a,b)=\sum_{n=-\...
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"Factorization" of the solutions set of a system of linear diophantine equations over non-negative integers
Suppose we have a system of linear diophantine equations over non-negative integers:
$$
\left\lbrace\begin{aligned}
&Ax=b\\
&x\in \mathbb{Z}^n_{\geq0}
\end{aligned}\right.
$$
where $A$ is a ...
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irregularities in partition function modulo n
It is an open problem whether the partition function is even half the time. Inspired by this, I wrote some Sage/Python code to check how many times $p(n)$ hits each residue class:
...
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Show the partition function $Q(n,k)$, the number of partitions of $n$ into $k$ distinct parts, is periodic mod m
Let $Q(n,k)$ be the number of partitions of $n$ into $k$ distinct parts.
I want to show that for any $m \geq 1$ there exists $t \geq 1$ such that
$$Q(n+t,i)=Q(n,i) \mod m \quad \forall n>0 \forall ...
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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
$$
|\vec i| = \sum_{p\geq1} i_p.
$$...
- 1,657
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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 $U_1$...
user1055322
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48
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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 ...
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54
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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 ...
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1
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80
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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(),
$$ MSET_{\le k}(Positive Integer) = P^{1,2,3,...k}(z) = \prod_{m=1}^{k} \frac{...
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Generating Function for Modified Multinomial Coefficients
The multinomial coefficients can be used to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + ...} \right)^n}$ in the basis of monomial symmetric polynomials (MSP). For example,
$$\...
- 51
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0
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45
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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?
\...
4
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1
answer
339
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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 ...
0
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1
answer
44
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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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1
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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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