All Questions
Tagged with integer-partitions set-partition
31
questions
9
votes
1
answer
363
views
Seeking a textbook proof of a formula for the number of set partitions whose parts induce a given integer partition
Let $t \geq 1$ and $\pi$ be an integer partition of $t$. Then the number of set partitions $Q$ of $\{1,2,\ldots,t\}$ for which the multiset $\{|q|:q \in Q\}=\pi$ is given by \[\frac{t!}{\prod_{i \geq ...
- 21.1k
3
votes
2
answers
241
views
Ways of distributing passengers in ships
I need help with the following combinatorial problem. There are $ K $ passengers and $ K $ ships. The passengers are denoted by $ U_1, U_2, \dots, U_K $. The objective is to find in how many ways the $...
- 778
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
3
votes
1
answer
67
views
Is this true for every partitioning?
I have two categories (category1 and category2 ) and The size of both categories is equal to each other. if we partition each categories arbibtrary .Is this proposition proven? or rejected?
$n_T \...
- 41
3
votes
2
answers
114
views
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 ...
- 31
2
votes
1
answer
415
views
Given a set of both positive and negative numbers, what is a time optimal approach to find the two numbers whose sum, plus a third number is zero
Coming from an engineering background I want to solve this question.
Question:
Given a set of positive, and negative numbers, how do I time optimally find two numbers whose sum is the mathematical ...
- 173
2
votes
2
answers
272
views
Find a bijection between the $(n-1)$ paths and the $n$-paths which have no downramps of even length.
So here is the Question :-
A Dyck $n$-path is a lattice path of n upsteps $(x,y)$ $\rightarrow$ $(x + 1,y + 1)$ and $n$
downsteps $(x,y) \rightarrow (x + 1,y-1)$ that starts at the origin and never ...
- 314
2
votes
1
answer
98
views
2-split of $n$ is $\left\{ \lfloor \frac{n}{2} \rfloor,\lceil \frac{n}{2} \rceil \right\}$. What about 3, 4, ...?
Clarification: $k$-split of $n$ is an ordered integer sequence $\left\{ a_1,\cdots,a_k \right\}\quad \text{s.t.}$
$0\le a_1\le\cdots\le a_k$
$a_1+\cdots+a_k=n$
${\left(a_k-a_1\right)}$ is minimized.
...
- 195
2
votes
1
answer
480
views
Find Integer Partition using only integers belonging to S = { 1, 2, 3 }
I spent all afternoon looking for this but I wasn't able to find anything, so...
Is there a formula to know the NUMBER of partitions with a constraint on the integer domain ?
E.g.: Find the number of ...
- 158
2
votes
1
answer
138
views
A number partition problem
I have encountered the following interesting integer partitioning problem.
Let $n,k,t \in \mathbb{N}$ be given parameters and let $S_1,\ldots, S_t$ be a partition of the numbers $1,2,\ldots,n$ such ...
- 193
1
vote
1
answer
77
views
Another formulation for Stirling numbers of the second kind
I find another formulation for Stirling numbers of the second kind:
Let $n\ge k\ge 1$. Denote by
$$\mathbb N_<^n := \{ \alpha = (\alpha_1,\cdots,\alpha_n): 0\le \alpha_1\le\cdots\le\alpha_n, \...
- 1,972
1
vote
1
answer
121
views
Why is the following not $S(n,3)$ where $S(n,k)$ is a Stirling number of the second kind? (almost solved)
In an attempt to relate the number of partitions of integers to that of partitions of distincts objects I stumbled, in the particular case of $k:=3$, on the following sum
$$\sum_{\genfrac{}{}{0pt}{1}{...
- 3,679
1
vote
0
answers
127
views
Minimizing the magnitude of the sum of a vector of complex numbers with an integer constraint
Let $h_{1}, ..., h_{N} \in \mathbb{C} $
Consider minimizing the function below:
$ \min \left| \sum\limits_{i=1}^N h_{i} x_{i} \right| $
with the constraints $x_{i}^2 = 1$
i.e., $x_{i}$ can only take ...
- 11
1
vote
0
answers
37
views
An interesting way of partitioning with inner ordered combinations
Assume $ K $ labeled blocks $ s_1, s_2, \dots, s_K $ ($ s_1 < s_2 < \dots < s_K $) that arrive sequentially and need to be accomodated as they arrive in $ N $ containers (partitions with ...
- 778
0
votes
1
answer
126
views
What is the appropriate weight ($W_k$) (for two arbitrary partitions)?
I already asked a similar question, And from the answer I received, another question came to my mind.
A positive integer can be partitioned, for example, the number 7 can be partitioned into the ...
- 41