# Tagged Questions

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.

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### Formula for evaluation of character on a transposition

Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of ...
94 views

### Finite Partitions of the Unit Interval

Does the unit interval have a finite partition $P$ such that no element of $P$ contains an open interval? I would think that the answer is no, because each element of $P$ would have Lebesgue measure ...
231 views

### Integer solutions

How many positive integer solutions are there to $x_1 + x_2 + x_3 + x_4 < 100$? I haven't seen any problems with "less than", so I'm a bit thrown off. I'm not sure if my answer is correct, but ...
184 views

### Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png An equilateral triangle can be divided into 3 congruent shapes. Questions: ...
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### Is this a valid partition?

Say we have the $S$ which is the set of all compositions of n >= 0 with an odd num. parts. Define $S_1$ to be the set of all compositions of n >= 0 with an odd num. of parts where at least one part ...
137 views

### Question on lower & upper Riemann sums for piecewise.

Partition{$\frac{-\pi}{6},3,2\pi$} $f(x)= 1/2$ if x is rational $sin(x)$ if x is irrational. Im not sure if im doing it correctly: ...
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### If $f$ is quasi and left continuous on $[a,b]$ and $\alpha$ is increasing and right continuous then $f$ is Riemann-Stieltjes integrable over $[a,b]$

Question: If $f$ is quasi and left continuous on $[a,b]$ and $\alpha$ is increasing and right continuous then $f$ is Riemann-Stieltjes integrable over $[a,b]$ with integrator $\alpha$. ...
363 views

### Number of partitions contained within Young shape $\lambda$

It is well known that the number of partitions contained within an $m\times n$ rectangle is $\binom{m+n}{n}$. Furthermore, it is not difficult to calculate the number of partitions contained within a ...
1k views

### Partitions of an integer into k parts.

I am interested in knowing whether an exact formula (analogous to the Hardy-Ramanujan-Rademacher formula for $p(n)$) for the number of partitions of a positive integer into k parts is known. I tried ...
313 views

### Creating generating functions for integer partitions

Say I have a generating function $\Phi_\mathcal{A}$ for the set of partitions $\mathcal{A}$ which have no parts congruent to 2 mod 4, and I have the generating function for $\Phi_\mathcal{B}$ for the ...
184 views

### Showing two generating functions to be equal

Let $\mathcal{A}$ be the set of partitions in which each part may occur 0, 1, 4, or 5 times and let $\mathcal{B}$ be the set of partitions which have no parts congruent to 2mod4, and in which parts ...
493 views

### Integer Partitions Formulas [duplicate]

Possible Duplicate: Identity involving partitions of even and odd parts. How would I go about to show the following: Let $pe(n)$ be the number of partitions of size n with an even number of ...
622 views

### Find all ways to factor a number

An example of what I'm looking for will probably explain the question best. 24 can be written as: 12 · 2 6 · 2 · 2 3 · 2 · 2 · 2 8 · 3 4 · 2 · 3 6 · 4 I'm familiar with finding all the prime ...
112 views

### Number decomposition

Recently I encountered a problem I was not familiar with. So hope someone can help me for this. Here is the problem. Given any odd integer, how many different ways of decomposition into sum of three ...
2k views

### Median of medians algorithm

I am referring to the algorithm presented here used to find a good pivot: http://en.wikipedia.org/wiki/Selection_algorithm#Linear_general_selection_algorithm_-_Median_of_Medians_algorithm My ...
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### What is minimum Eulerian partition?

I have a homework in my graph course. It asks something about minimum Eulerian partition but it doesn't give any information about it. I googled it but couldn't come up anything useful and clear. Is ...
123 views

### Nonnegative integers with unique restricted partitions

I have come over the following problem in elementary number theory: suppose that a positive integer $k$ is given. Is it possible to find, for every such $k$, a nonnegative integer $N(k)$ and a set of ...
2k views

### Distribution of the sum of a multinomial distribution

I have distilled an error analysis problem into the following: I have a multinomial distribution, $X$, consisting of $n$ independent trials where each trial takes on the values $\{0,1,\ldots,k-1\}$ ...
185 views

### On the path diagram of a partition

A path diagram is a Ferrers diagram for a partition $\lambda$ in which the dots have been replaced with numbers computed as follows. The number in position $(i,j)$ the number of paths (going either ...
882 views

### Enumeration of partitions

The Stirling number of the second kind $S(n,k)$, where $S(n,k) = \frac{1}{k!}\sum\limits_{j=0}^k(-1)^{k-j}\left(\begin{array}{l}k\\j\end{array}\right)j^n$ Gives the number of unique unlabeled, ...
254 views

### How many circular distinct compositions of $n$ into $k$ parts at most $g$

how many circular distinct sequences (e.g. $(4124)\equiv(4412$)) are there that sum up to $n$, have $k$ elements may be non-negative integers at most $g$? In other words: we're looking for the number ...
1k views

### Partition of a set of size n into subsets of size 1 and 2.

Before I ask the question, I must admit that combinatorics has never been my forte. I am given a set X of size $n$, we may assume assume $X=\{1,2,...,n\}$. I want to count the partition of this set ...
127 views

### Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6$ [closed]

Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6$ such that condition 1: $1 \leq a_i \leq 7$ condition 2: $1 \leq b_i \leq 4$ condition 3: $\sum_{i=1}^{2} {a_i} = 8$ ...
2k views

### Equivalence Relations and Partitions

My university "textbook" for discrete math is Schaum's Outline. In this outline he goes over Equivalence Relations and Partitions, and I got confused at a particular theorem. From the book: Theorem ...
885 views

### Ellipse 3-partition: same area and perimeter

Inspired by the question, "How to partition area of an ellipse into odd number of regions?," I ask for a partition an ellipse into three convex pieces, each of which has the same area and the same ...
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### Combinatorics problem based on Ferrers graph

Need help with this proof using Ferrers' graph or otherwise. Show that the number of partitions of $r+k$ into $k$ parts is equal to The number of partitions of $r + {k+1 \choose 2}$ into $k$ ...
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### cosets aren't forming a partition?

If you have a group $G$ and $H \leq G$, the cosets of $H$ should partition $G$. Suppose $G=\mathbb{Z}_2\times\mathbb{Z}_4$ and $H=\langle (0,1)\rangle = \{(0,0), (0,1),(0,2),(0,3)\}$. Then both ...
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### Asymptotics for partitions of $n$ with largest part at most $k$ (or into at most $k$ parts)

Let $\bar p_k(n)$ be the number of partitions of $n$ with largest part at most $k$ (equivalently, into at most $k$ parts). Is there an elementary formula for the asymptotic behavior of $\bar p_k(n)$ ...
5k views

### How many ways you can make change for an amount [duplicate]

I am looking for a formula or at least something to use when trying to compute how many ways I can make change for an amount. Example: there are $3$ ways to give change for $4$ if you have coins ...
509 views

### Number of partitions of $2n$ with no element greater than $n$

The number of partitions of $2n$ into partitions with no element greater than $n$ (copied and slightly adapted from http://mathworld.wolfram.com/PartitionFunctionq.html), so I'm looking for a nice ...
517 views

### Finite or infinite set?

Due to my not-so-advanced math skills, this question may take a few attempts to state clearly: Consider the unordered pair (2-tuple) partitions of n (e.g. with n=4, we have {{4,0},{3,1},{2,2}}). ...
64 views

### Ordered integer partition [duplicate]

Possible Duplicate: Proving an equality involving compositions of an integer A sequence of natural numbers $\langle a_1,\ldots,a_k \rangle$ is an ordered partition of $n$ if ...
119 views

### Partitions of binary numbers into binary numbers with fixed digits?

If we are to have (two, for example) binary numbers, such that their sum is $100111010_2$, and given that the first number has 5 ones, and the second number has 3 ones, can I find the numbers that ...
320 views

### Two number partition problems

Let $p_k(n)$ be a number of ways to express $n$ as a sum of $k$ positive integers. For example $p_2(3)=1$. Problem 1. Prove that following recurrences are correct: ...
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### Partitioning a sequence of numbers based on a percentage

I have a sequence $s$ of rational numbers where $|s|=n$. I am given a percentage $p$ to partition the sequence $s$ into $\frac{1}{p}$ parts, such that each part contains $p\%$ number of elements of ...
435 views

### Average Number of Blocks in a Partition Containing a Large Block

Notation Let $[n]$ denote the set of integers from 1 up to $n$, inclusive. Let $A_n$ denote the average number of blocks over all partitions of $[n]$. Let $B_n$ denote the number of partitions of ...
537 views

### Number of partitions of a set with size constraints

One of my students made some experiments on partitions of sets. He found some results and I asked him, if he can prove some statements. After two weeks he had no result, so maybe one of you can help. ...
Possible Duplicate: Same number of partitions of a certain type? Prove that the number of partitions of $n$ without elements divisible by $4$, is equal to the number of partitions of $n$ ...
Question: Joe is in his hunting blind when he locates $20$ geese, $25$ ducks, $40$ eagles, $10$ cranes, and $5$ flamingos. Joe randomly selects six birds to target, what is the probability that at ...