The discrete-mathematics tag has no wiki summary.

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**1**answer

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### Bounding the entropy of a convolution

Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...

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**1**answer

286 views

### Number of blocks in a t-(v,k,l) design with empty intersection with a given set U [closed]

Question
Given a $t-(v,k,\lambda)$ design $(X,\mathcal{B})$ and a set $U\subset X$ with $|U|=u\leq t$, what is the number of blocks $B\in\mathcal{B}$ such that $B\cap U=\emptyset$?
The answer is: ...

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**1**answer

121 views

### expanding the sqare of sum

If there any way to expand the following?
$$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}}$$
and more generally, a way to expand
$$\left(\sum_{i=1}^nx_i\right)^{\frac{p}{q}}$$
where $gcd(p,q) = 1$
...

**3**

votes

**2**answers

332 views

### anyone help me with this inequality

I'm have some difficulties in bounding the following inequality:
I want to find a c as small as possible s.t.
$$\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq ...

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votes

**0**answers

83 views

### An upperbound related to inductively reducing a set by adding the two least elements

I hope this is not too trivial to be asked here, but here it goes anyway. This is just out of curiosity (regarding a problem with graphs but I have "reduced" it to the problem below:)
Let ...

**1**

vote

**1**answer

251 views

### Simple and general relation between continuant polynomials

Continued fraction $[a_0,a_1,...,a_n]$ may be expressed as quotient of two polynomials of $(a_0,a_1,...,a_n)$, named continuants (see http://en.wikipedia.org/wiki/Continuant_%28mathematics%29 )
...

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**5**answers

1k views

### Special arithmetic progressions involving perfect squares

Some time ago the following rather easy problem appeared in an online publication called "Problems in Elementary NT" by Hojoo Lee:
Prove that there are infinitely many positive integers $a$, $b$, $c$ ...

**8**

votes

**1**answer

361 views

### Die-rolling Hamiltonian cycles

Let $R$ be a rectangular region of the integer lattice $\mathbb{Z}^2$,
each of whose unit squares is labeled with a number
in $\lbrace 1, 2, 3, 4, 5, 6 \rbrace$.
Say that such a labeled $R$ is ...

**1**

vote

**0**answers

207 views

### Trigonometric semialgebraic conditions for two floors to be unequal [edited]

EDIT: Since posting my original question I have simplified the problem to finding a sufficient condition for $\lfloor \frac{a}{\phi}\rfloor \neq \lfloor \frac{b}{\phi}\rfloor$ which is trigonometric ...

**3**

votes

**2**answers

203 views

### Designing a tree to match a distribution

I want to design a tree to approximate a given
sequence of numbers, in the following sense.
Let $X=(x_1,\ldots,x_n)$ be $n$ numbers, with $0 < x_i \le 1$
and $\sum_i x_1 = 1$.
For a rooted tree ...

**0**

votes

**0**answers

180 views

### Partial Recurrence Equation

Hey people
I have the following equation, which I don't manage to solve.
The background of the problem is the clustergrowth of two chemical species, resulting in a final relation I'd like to solve:
...

**3**

votes

**0**answers

97 views

### Are there existing resources on modular-esque recurrence relations?

Does anyone know where I would be able to get information on analyzing a class of polynomial recurrence relations of a form like this?
$\begin{align*}
f_{n,k}(x) & ...

**1**

vote

**2**answers

472 views

### Randomized algorithm?

The problem is as follows. Given a set $S$ of natural numbers of size $n$ where each $x_i \in S$ is from the set $[n^2]$. Elements of $S$ are not necessarily pairwise different, i.e., there can be ...

**2**

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**3**answers

531 views

### how to cover a set in a grid with as few rectangles as possible

In calculus, when estimating a area of a set in a 2-dimensional space, we use rectangles to approximate. To get sufficient precision, how many rectangles are needed if the shape of the set is close ...

**2**

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**0**answers

301 views

### Guessing game with guess cost

This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...

**1**

vote

**1**answer

578 views

### Decomposition of a complete graph into maximal matching subgraphs

Is there a general way to decompose a complete graph $K_n$ into an union of maximal matching subgraphs such that no two subgraphs share an edge?
For example, consider $K_4$ with vertices ...

**3**

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**0**answers

130 views

### A fast way to test whether a partial function can be extended to a chirotope of rank 3 ?

Hello,
What is a fast way to test whether a partial function can be extended to a chirotope of rank 3 ? That is: I have a domain
$E = \{1,...,n\}$
and a partial function
$f: E^3 \to \{-1, 0, 1\}$
...

**2**

votes

**0**answers

142 views

### Number of breakpoints in parametric maximum flow problems

The parametric maximum flow problem can be formulated as
$$f(\lambda) = \min_{x\in\{0,1\}^n} \left( \sum_{i}(a_i + b_i\lambda)x_i + \sum_{i,j}c_{ij}x_ix_j \right),
$$
where all $c_{ij}<0$ (so that ...

**4**

votes

**2**answers

317 views

### Computer platforms for combinatorial search problems/mathematical music theory?

I'm finding programming various combinatorial searches
(connected to mathematical music theory) in a general purpose computer
language tedious, so I'd like pointers to computer platforms/environment ...

**2**

votes

**4**answers

428 views

### Statistical computation in matrix. Rows before columns? riddle..

First I'll phrase the question as a riddle, and than as a general math problem.
We have 12 lettered vases $(A,B,...,L)$, in each vase there are 30 numbered balls (1-30). In each ball there is some ...

**1**

vote

**1**answer

3k views

### How to compute KL-divergence when PMF contains 0s?

From the Wikipedia page on Kullback-Leibler divergence, the way to compute this metric is to utilize the following formula:
The way I understand this is to compute the PMFs of two given sample sets ...

**3**

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**0**answers

657 views

### Method for variable substitution in multiple summation

I want to ask: is there any general method for variable substitution in multiple summation?
For example in the following equation a new variable $\lambda=n+m-2\mu$ is introduced to transform the LHS ...

**5**

votes

**1**answer

231 views

### Reference sought for Conways observation on stable matchings.

Looking for a reference on the observation that the set of stable matchings form a distributive lattice. This is attributed to Conway by Knuth in "Marriages Stables" but I would like an explicit ...

**0**

votes

**1**answer

151 views

### the maximal length of a special dicksonian sequence

First, we define a sequence $t_{1},t_{2},\cdots,t_{k}$ of n-tuples dicksonian, if $\forall 1\leq i < j\leq k,$ there does not exist a non-negative n-tuple t such that
$t_{i}+t=t_{j}.$ For example, ...

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**2**answers

434 views

### A non-associative three-valued logic

There are three elements: x, y, z and a relation C:
x C y, y C z, z C x, x C x, y C y, z C z.
Let us introduce two binary operations with respect to the C: "the leftmost" (L) and "the rightmost" ...

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**0**answers

165 views

### recursion formula for odd holonomic function

suppose we have a map
$f:\mathbb{Z}\longrightarrow\mathbb{C}[t^{\pm}]$
with property that $f(i)=-f(-i)$.
The algebra
$\mathcal{T}=\mathbb{C}[t^{\pm}][L^{\pm},M^{\pm}]/[LM=tML]$ acts on $f$ by
...

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votes

**2**answers

708 views

### Is there a discrete Cerf theory?

Towards the end of the 1990's, Robin Forman developed a discrete version of Morse theory, which concerns maps from a simplicial complex to $\mathbb{R}$ satisfying a combinatorial analogue to the ...

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208 views

### Domination in Nice Lattices

Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions.
Call a lattice ...

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**3**answers

775 views

### Partitioning the integers $1$ through $n$ so that the product of the elements in one set is equal to the sum of the elements in the other

I asked this question at math.SE a couple of months ago and only got a partial answer, so I thought I would try here.
It is known that, for $n \geq 5$, it is possible to partition the integers ...

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**2**answers

975 views

### palindromic subsequences

I'd like any insight or references to the following two conjectures (see the glossary below for definitions):
Conjecture 1: For any string $x$, there exists a longest common subsequence of $x$ and ...

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**0**answers

395 views

### Covering a set of intervals

EDIT: I posted this question to http://cstheory.stackexchange.com/questions/4358/covering-a-set-of-intervals , which looks like a more appropriate venue for such a question.
Hello,
I'm trying to ...

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**9**answers

427 views

### What are some early examples of creation of lists / catalogues of (particularly) combinatorial objects?

A lot of effort in discrete maths / combinatorics is expended in the construction of lists, catalogues or census [sic] of combinatorial objects such as groups, graphs, designs etc. These catalogues ...

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420 views

### Zeroes of a tricky function.

I am attempting to show that there does not exist an N past which every open unit interval (k, k+1) -where k is an integer- contains a zero of the following function:
$h(x)=\sum_{n=2}^{[\sqrt(x)]} ...

**7**

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**3**answers

936 views

### How to characterize a Self-avoiding path.

I cannot find any answer to that apparently simple problem :
On a square lattice, a path is given by a sequence of relative moves in {"move forward", "turn right" and "turn left"}.
Is there a rule ...