Articles of partitions

Decomposition by subtraction

In how many ways one can decompose an integer $n$ to smaller integers at least 3? for example 13 has the following decompositions: \begin{gather*} 13\\ 3,10\\ 4,9\\ 5,8\\ 6,7\\ 3,3,7\\ 3,4,6\\ 3,5,5\\ 3,3,3,4\\ 4,4,5\\ \end{gather*} Points and hints are welcome.

Existence of uncountable set of uncountable disjoint subsets of uncountable set

“Can you to any uncountably infinite set $M$ find an uncountably infinite family $F$ consisting of pairwise disjoint uncountably infinite subsets of $M$?” Intuitively, I feel like it should be possible for the real numbers at least: you simply split the real numbers into two intervals, and since there are uncountably many points where you […]

Jordan Measure, Semi-closed sets and Partitions

From earlier question here. Consider $$\mu \left( (0,1) \cap \mathbb Q \right) = 0$$ where $$(0,1) = \left(0,\frac{1}{3}\right)\cup \left[\frac{1}{3}, \frac{2}{3}\right] \cup \left(\frac{2}{3}, 1\right)$$ so $$ \mu ((0,1) \cap \mathbb Q ) = \mu \left(\left(0,\frac{1}{3}\right) \cap \mathbb Q \right) + \mu\left( \left[\frac{1}{3}, \frac{2}{3}\right] \cap \mathbb Q\right) + \mu\left(\left(\frac{2}{3}, 1\right) \cap \mathbb Q\right) = 0 + \frac{1}{3} […]

Recurrence relation / recursive formula / closed formula for partition numbers (1,2,3,5,7,11,15,22,30,42,…)

I have already read this: Number partition – prove recursive formula But the formula from the above link requires a parameter k which is the required number of partitions, but I would like to partition it as far as it could. What I am finding is the partition number of a positive integer n, where […]

Counting Set Partitions with Constraints

I’m trying to count set partitions under the following constraints: The number of partitions on $n$ elements where the largest cell(s) have exactly $k$ elements All cells have at least two elements Starting here, I’m able to get the number of partitions where the maximum size cell size is exactly $k$ for values of $k […]

What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.

Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4×3 and 5×4 rectangles, What is the minimum square partition of an almost-square rectangle?. Let: An almost-square-rectangle be a rectangle that has a width $w$ and height $h=w-1$. […]

Number of possible combinations of x numbers that sum to y

I want to find out the number of possible combinations of $x$ numbers that sum to $y$. For example, I want to calculate all combination of 5 numbers, which their sum equals to 10. An asymptotic approixmation is also useful. This question seems to be very close to number partitioning, with the difference that a […]

Exponential generating function for restricted compositions

I wanted to know if it is possible to use exponential generating functions to evaluate composition of N using K distinct numbers (where the supply of numbers is infinite)? For e.g if N=10 and a1=2,a2=3,a3=5 then number of solutions would be (2,3,5),(2,5,3),(3,2,5),(3,5,2),(5,2,3),(5,3,2),(2,2,2,2,2),(3,3,2,2),(3,2,2,3),(2,2,3,3),(2,3,2,3),(3,2,3,2),(2,3,3,2) I tried with writing generating functions for 2,3 and 5 but was not […]

Proving an Inequality Involving Integer Partitions

I am having a bit of trouble beginning the following: Prove that for all positive integers $n$, the inequality $p(n)^2<p(n^2+2n)$ holds, where $p(n)$ is defined as the number of all partitions of $n$. I initially considered weak induction on n, but am not sure if that is the correct way to begin. Is there an […]

Permutation Partition Counting

Consider the number $n!$ for some integer $n$ In how many ways can $n!$ be expressed as $$a_1!a_2!\cdots a_n!$$ for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as $\Omega(n)$ Consider the value of $10!$ $$10! = 7!6!$$ $$10! = 7!5!3!$$ Thus we know that $\Omega(10)\ge 3$ We note that […]