Questions related to the different ways of expressing an integer as a sum of integers; or, questions related to the subdivision of a set into smaller disjoint sets; questions related to the subdivision of an interval into smaller intervals that intersect only at the endpoints.
0
votes
3answers
316 views
1
vote
1answer
17 views
Count the number of unique equal sized partitions of a set.
Given the integers $[1, ck]$, they will be partitioned into $c$ subsets of size $k$. I want to count the number of unique versions of each subset (where order matters).
Clearly, there are ${ck ...
10
votes
6answers
194 views
Can every infinite set be divided into pairwise disjoint subsets of size $n\in\mathbb{N}$?
Let $S$ be an infinite set and $n$ be a natural number. Does there exist partition of $S$ in which each subset has size $n$?
This is pretty easy to do for countable sets. Is it true for ...
0
votes
1answer
23 views
Make a partition that contains a set of points??
I am given a set of $M$ points in a segment (the edges are also points in this set)
I would like to partition the segment (with equidistant points), in such a way that my partition contains all these ...
1
vote
1answer
30 views
Generating functions of partition numbers
I don't understand at all why:
\begin{equation}
\sum\limits_{n=0}^\infty p_n x^n = \prod\limits_{k=1}^\infty (1-x^k)^{-1}
\end{equation}
Where $p_n$ is the number of partitions of $n$. Specifically ...
1
vote
1answer
13 views
Can a p-core of a partition be reached by repeated stripping of p-rimhooks?
in http://mathoverflow.net/questions/42562 I read : "If you strip p-rimhook after p-rimhook off of a partition, this always results in the same p-core, and the choices don't matter."
But I must be ...
3
votes
2answers
41 views
Probability distribution of product of integers
I have a scoring system based on 5 factors with integer values from 1 to 5:
Score = A * B * C * D * E
So the Score can range from 1 to 3125. Each of the factors ...
-1
votes
2answers
27 views
Question on combinatorics, partitions. [duplicate]
Let $p$ ($n|$distinct odd parts) be the number of partitions of $n$ into distinct odd parts. Prove that $p(n)$ is odd if and only if $p$($n|$distinct odd parts) is odd by using the theorem on ...
1
vote
1answer
30 views
Partition parts
Consider the partitions of $n$. For $n = 5,7,9,\ldots$, it appears as if the number of pairwise partitions $\{a,b\}$, where both $a$ and $b$ are composite, equals the total number of individual odd ...
3
votes
4answers
201 views
In how many ways i can write 12?
In how many ways i can write 12 as an ordered sum of integers where
the smallest of that integers is 2? for example 2+10 ; 10+2 ; 2+5+2+3 ; 5+2+2+3;
2+2+2+2+2+2;2+4+6; and many more
-2
votes
1answer
122 views
From a Generating Function find $R(x)$ as an infinite product of Quotients
Let $r(n)$ be the number of partitions of $n$ so that no multiple of $3$ appears as a part. For
example, $r(8) = 13$.
Let $R(x) =\sum_0^\infty r(n)x^ n $ be the generating function for $r(n)$.
Find ...
-2
votes
1answer
153 views
Find a form for $Q(x)$ as an infinite product of polynomials
Let $q(n)$ be the number of partitions of $n$ so that no part appears three or more times. For example, $q(8) = 13$
Let $Q(x) = \sum\limits_{n=0}^\infty q(n) x^n$ be the generating function for ...
2
votes
1answer
145 views
bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n
Suppose $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ is a partition of $2n$ where $n\in\mathbb N$ satisfying the following conditions:
(1) $\lambda_k=1$.
(2) $\lambda_i−\lambda_{i+1}\leq 1$ for ...
1
vote
1answer
18 views
conjugate partition definition
i would like to understand basic definition of conjugate partition,this is what is said in my book
Let $υ = (u_1, u_2, . . . , u_n)$ be a sequence of integers such that $u_1 ≥ u_2 ≥ · · · ≥ u_n ≥ ...
3
votes
2answers
155 views
The fibres of a map form a partition of the domain.
This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post.
This problem is from ...
