All Questions
Tagged with integer-partitions discrete-mathematics
120
questions
-1
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0
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Column/Digit blind solution for the "Number of possible combinations of x numbers that sum to y"
What formula will give me "the total number of possible combinations of x digits that sum to y".
This is a branch question from the original question entitled
Number of possible ...
- 275k
5
votes
1
answer
63
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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+....
- 47.4k
1
vote
1
answer
62
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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". ...
- 5,239
7
votes
7
answers
2k
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$a+b+c+d+e=79$ with constraints
How many non-negative integer solutions are there to $a+b+c+d+e=79$ with the constraints $a\ge7$, $b\le34$ and $3\le c\le41$?
I get that for $a\ge7$ you do $79-7=72$, $\binom{72+5-1}{5-1}=\binom{76}4$...
- 42.8k
1
vote
1
answer
53
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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 ...
- 319
0
votes
1
answer
39
views
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+...
- 675
2
votes
1
answer
118
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Number of partitions of $n$ into distinct even parts.
Question from my last exam:
Let $r_n$ denote the number of partitions of $n$ into distinct parts. Prove that
$$
\sum_{i=0}^n(-1)^i r_i r_{n-i}
$$
is the number of partitions of $n$ into distinct even ...
- 109k
1
vote
1
answer
38
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Equality between q shifted factorials and sum with partitions
I am looking at WilliamY.C.Chen,Qing-HuHou,and Alain Lascoux proof for the Gauss Identity in q shifted factorials. At some point they claim that it is easy to see that
$$
\frac{q^r}{(q ; q)_r}=\sum_{\...
- 21.9k
2
votes
2
answers
174
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Number of ways to complete a partial Young tableau
Suppose we have a Young tableau with missing entries. Then there can be many number of ways we can complete the Young Tableau.
Is there any specific method to find the number of ways we can complete a ...
- 83.9k
0
votes
1
answer
60
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Self transpose of partitions and odd parts
Let a partition be called self-transpose if its Ferrers diagram is symmetric about the diagonal,
i.e. if the partition is its own transpose. For instance, there are two self-transpose partitions of
8: ...
CommunityBot
- 1
2
votes
1
answer
78
views
the bracket partition function ? $3 = 1 + 1 + 1 = 1 + 2 = 1 + (1 + 1) $
I want to know in how many ways we can write a positive integer by using strict positive integers, addition and brackets. The order of addition does not matter.
For instance
$$3 = 1 + 1 + 1 = 1 + 2 = ...
- 78.3k
2
votes
2
answers
168
views
Finding the generating function.
Find the generating function for the number of solutions for the equation $x_1+x_2+x_3+x_4 = n$, where $x_1,x_2,x_3,x_4\geq1$, and $x_1 < x_2$.
My attempt so far: I have tried putting a $y$ value ...
- 47.4k
1
vote
2
answers
139
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Find a recurrence for the number of integer compositions of n which only have 1s and 2s as parts
Find a recurrence for $$i_n$$ the number of integer compositions of $n$ which only have $1$s and $2$s as parts.
How do you approach this problem?
- 1,404
3
votes
4
answers
1k
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Two hard number partition problems
For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$. Also define $p(0)=1.$
Problem 1.
Prove that $p(n)-p(n-1)...
CommunityBot
- 1
1
vote
1
answer
86
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Number of partitions with distinct even parts/parts with multiplicity $\leq 3$
I am supposed to solve a problem regarding partitions of $n \in \mathbb{N}$ into:
distinct even parts
parts with multiplicity $\leq 3$
I am supposed to prove that 1. and 2. are equal.
So I tried ...
- 104k