Questions tagged [integer-partitions]
Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.
1,429
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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
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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 ...
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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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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 ...
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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 $1, 2, 5, 15$ ...
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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 ...
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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^...
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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
$$
\left[\begin{array}{}
\lambda_{1,n} & & \lambda_{2,n} & & & \dots & & & \lambda_{n-1,n}...
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About the product $\prod_{k=1}^n (1-x^k)$
In this question asked by S. Huntsman, he asks about an expression for the product:
$$\prod_{k=1}^n (1-x^k)$$
Where the first answer made by Mariano Suárez-Álvarez states that given the Pentagonal ...
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A variant of the partition problem or subset sum problem
Given a target list $T = (t_1, t_2, \ldots, t_N)$ and a multiset $S = \{s_1, s_2, \ldots, s_M\}$, both with non-negative integer elements, $t_k\in \mathbb{N}_>$ and $s_k\in \mathbb{N}_>$, ...
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Bell numbers - Cardinality of odd number of parts in partitions of the finite set $[n]$.
As it well known, Bell numbers denoted $B_{n}$ counts distinct partitions of the finite set $[n]$. So for example if $n=3$ there are 5 ways to the set $\left\{ a,b,c\right\}$ can be partitioned:
$$\...
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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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