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0
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0answers
85 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
1answer
330 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 ) ...
12
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5answers
2k 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
1answer
372 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
0answers
210 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
2answers
218 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
0answers
207 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
0answers
120 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
2answers
484 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
votes
3answers
745 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
votes
0answers
406 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. ...
3
votes
1answer
749 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
votes
0answers
134 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
0answers
168 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
2answers
335 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
4answers
457 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 ...
3
votes
1answer
5k 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
votes
0answers
725 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
1answer
235 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
1answer
163 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, ...
0
votes
2answers
443 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" ...
1
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0answers
169 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 ...
17
votes
2answers
759 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 ...
4
votes
0answers
218 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 ...
14
votes
3answers
804 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 ...
8
votes
2answers
1k 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 ...
7
votes
9answers
455 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 ...
1
vote
0answers
429 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)]} ...
8
votes
3answers
1k 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 ...