All Questions
Tagged with integer-partitions discrete-mathematics
119
questions
5
votes
1
answer
62
views
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+....
- 609
1
vote
1
answer
62
views
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". ...
- 1,312
1
vote
1
answer
53
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 ...
- 675
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
views
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 ...
- 675
1
vote
1
answer
38
views
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
$$
\frac{q^r}{(q ; q)_r}=\sum_{\...
2
votes
2
answers
171
views
Number of ways to complete a partial Young tableau
Suppose we have a Young tableau with missing entries. Then there can be many number of ways we can complete the Young Tableau.
Is there any specific method to find the number of ways we can complete a ...
- 399
0
votes
1
answer
60
views
Self transpose of partitions and odd parts
Let a partition be called self-transpose if its Ferrers diagram is symmetric about the diagonal,
i.e. if the partition is its own transpose. For instance, there are two self-transpose partitions of
8: ...
user1154312
2
votes
1
answer
78
views
the bracket partition function ? $3 = 1 + 1 + 1 = 1 + 2 = 1 + (1 + 1) $
I want to know in how many ways we can write a positive integer by using strict positive integers, addition and brackets. The order of addition does not matter.
For instance
$$3 = 1 + 1 + 1 = 1 + 2 = ...
- 16.3k
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 ...
- 165
1
vote
1
answer
85
views
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 ...
- 333
0
votes
0
answers
75
views
Question about number of generating functions
I know that the generating function for the number of integer partitions of $n$ into distinct parts is
$$\sum_{n \ge 0} p_d(n)x^n = \prod_{i \ge 1}(1+x^i)$$
I'm trying to use this generating function ...
- 27
1
vote
1
answer
184
views
Find the number of non-negative integer solutions to $x+y+z=11$
How do I find the number of nonnegative integer solutions to $x+y+z=11$ provided that $x\leq 3, y\leq 4, z\leq 6$ using the sum rule (counting)?
I know the answer is 6, but I'm having difficulty ...
- 2,072
3
votes
1
answer
109
views
Distributing $n$ distinct objects into $m$ types of urns with $k_1,k_2...k_m$ urns of each type
I came accross this (rather complex?) combinatorial problem:
I have $18$ distinct objects, $3$ red urns, $7$ blue urns, and $11$
green urns. In how many ways can I distribute the objects into those
...
- 1,022
4
votes
1
answer
254
views
Alternative way of writing the stars and bars formula where each bar is associated with at least one star.
I was looking for a different way of writing the formula of the number of different $k$-tuples of non-negative integers whose sum is equal to $n$ and I thought of this formula followed by this ...
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