5
votes
1answer
270 views

Number of partitions whose blocks form arithmetic progressions

As is known, the set $\{1,\ldots,n\}$ has $2^n$ many subsets and $B_n$ (the $n$th Bell number) many partitions, where clearly $B_n<2^{2^n}$ and it is actually known that $B_n<n^n$ for large $n$. ...
33
votes
1answer
1k views

Wanted: a “Coq for the working mathematician”

Sorry for a possibly off-topic question -- there are four StackExchange subs each of which could be construed as the proper place for this question, and I've just picked the one I'm most familiar ...
6
votes
1answer
122 views

Separating infinite words sharing factors by automata

Two infinite words $\xi, \eta \in X^{\omega}$ are separated by an (Büchi-)automaton if it accepts one but not the other. Denote by $F_n(\xi)$ the factors of length $n$ of an infinite word $\xi$ and ...
5
votes
1answer
73 views

Generalising the adherence operator and its closure properties with regard to regular (rational) languages

Let $X$ be an alphabet and denote by $X^{\omega}$ the set of all infinite sequences (i.e. words) in $X$. A subset $L \subseteq X^{\omega}$ is called $\omega$-regular if it is acceptable by some ...
15
votes
7answers
1k views

Two questions from combinatorics on words

Question 1. Assume that an infinite word $u\in\{0,1\}^{\mathbb Z}$ is not balanced. Is it true that there exists a finite 0-1 word $w$ such that $0w01w1$ or $1w10w0$ is a factor of $u$? Is it true ...
5
votes
1answer
114 views

What prefix and factors determine a ultimately periodic word uniquely

Let $\xi$ be an ultimately periodic sequence, i.e. there exists finite sequences $p, q \in X^*$ such that $\xi = pq^{\omega}$. Does there exists a $n > 0$ such that the prefix of length $n$ and all ...
0
votes
0answers
68 views

Proof of conjecture that permutation-free automata restrict the possible states visitable from a stringset sharing prefixes and infixes

An automaton $\mathcal A = (X, Q, \delta, q_0)$ is called permutation-free iff no word $w \in X^*$ induces a nontrivial permutation of a subset of the states of $\mathcal A$. More formally for any $R ...
5
votes
2answers
412 views

A simple language and systematic computations

The following somewhat popular simple computer language was enjoyed on sci.math, sci.math.research, pl.sci.matematyka, and perhaps before and after at several places (I wish I knew it's exact ...
0
votes
0answers
38 views

Finite window transformations--input+output (pure algebra)

This q. presents a complete approach to my previous q.: Indecomposability of image transformations ... Let $\ A\ B\ $ be finite sets of cardinality $\ > 1$.   Let $\ D:=A\times B$,   ...
1
vote
0answers
89 views

Indecomposability of image transformations (pure algebra). Open questions

W-transformations -- definitions We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
4
votes
0answers
160 views

Rough structure of the double coset space/Graph bijections up to automorphisms

I am dealing with bijective maps $\pi:\Gamma_1\to \Gamma_2$ between two graphs with the same number of vertices $N=O(10)$. The graphs have a significant automorphism group (these are disconnected ...
2
votes
0answers
53 views

Is the $d$-dimensional Arrangement of Trees still $NP$-hard?

The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...
3
votes
1answer
137 views

Question about the elementary divisors of a special matrix

I have the following question: Is there a closed formula for the elementary divisors of the Matrix $M=\lbrace (m_{ij})\rbrace_{i=1,...,n,\ j=1,...,k}$, where $m_{ij}$ is the greatest common ...
0
votes
0answers
190 views

An interesting version of the problem “balls into bins”

Consider n people, each has k identical balls. Each people choose k different bins from m bins, constrained by the condition that there are no two people choose exactly the same k bins. For instance, ...
11
votes
0answers
300 views

Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but: Question: Is there a reformulation of the Dynamic ...
18
votes
2answers
863 views

Expected edit distance

The edit or Levenshtein distance between two strings is the minimum number of single symbol insertions, deletions and substitutions to transform one string into another. For example ...
1
vote
1answer
250 views

Distance between vertices in a vertex transitive graphs. [closed]

Can anybody help me in finding out the distances between vertices in a vertex transitive graphs. Is there any specific formula to calculate distance between vertices in this graph. Thanks for your ...
0
votes
1answer
191 views

hypergraph cartesian join operation (over same vertex set)

consider two hypergraphs $H_1 = (V, \mathscr{E}_1), H_2 = (V, \mathscr{E}_2)$ over the same vertex set $V$. am interested in what could be called a "cartesian join" operation building a new hypergraph ...
0
votes
0answers
344 views

Examples of Hamiltonian Cycle Problem / Traveling Salesman Problem in general grid graph form

I understand that there is a polynomial algorithm to solve TSPs that are in solid grid graph form (grid graphs without holes). I am particularly interested in the non-solid grid graph form of the ...
2
votes
1answer
454 views

#P version of SUBSET SUM

The decision version of the SUBSET SUM problem asks the following: Given a set of integers $S =$ {$a_1, ..., a_n$}, is there a subset $S'$ of $S$ such that the sum of the elements in $S'$ is equal to ...
5
votes
1answer
387 views

Arrangement of integers 1..k^2 in k*k grid to minimize energy function

Question arises from considering cache oblivious algorithms. What is the optimal way arrange the numbers $1$ to $k^2$ in a grid, to minimize to average difference between any two neighbouring ...
3
votes
1answer
631 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 ...
1
vote
1answer
188 views

Recoving an unknown tree graph with knowledge of root node to leaf node distances

Imagine I have an unknown (undirected) tree graph, $G$, with some unknown number of nodes $||V||$. However, I know the edge-length between nodes is of fixed size, $L_{edge} = 1$, and I have access to ...
1
vote
3answers
346 views

Operator probability in a RPN string

Consider the set $S_n$ of all strings of length $n$ ($n$ integer, $n \geq 3$) representing an expression in RPN ( http://en.wikipedia.org/wiki/Reverse_Polish_notation. ) Assumptions (to simplify): ...
5
votes
6answers
3k views

Fast evaluation of polynomials

Hello everybody ! I was reading a book on geometry which taught me that one could compute the volume of a simplex through the determinant of a matrix, and I thought (I'm becoming a worse computer ...
3
votes
2answers
452 views

Partition a square into sub-rectangles with restrictions

Is there an algorithm to generate all partitions of given square by using $n$ vertical and $n$ horizontal lines into sub-rectangles under the following restrictions: 1- No vertical line crosses any ...
5
votes
1answer
384 views

What is the pathwidth of the 3D-grid (mesh or lattice) with sidelength k?

This question is now also on http://cstheory.stackexchange.com/questions/4081/what-is-the-pathwidth-of-the-3d-grid-mesh-or-lattice-with-sidelength-k, where a discussion started, and one reference ...
4
votes
2answers
574 views

Coloring edges on a graph s.t. the set of edges for any two vertices have no more than 'k' colors in common

Please imagine the case where one has a planar graph, $G$, with a set of $|V|$ vertices, $(v_1, ..., v_{|V|}) \in V$, and $|E|$ edges, $(e_1, ..., e_{|E|}) \in E$. Now, provided a total of $N$ ...
2
votes
3answers
317 views

Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?

I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...
5
votes
1answer
331 views

Drawing graphs on circles

Please consider the following problem: Given: a simple graph (without self-loops and without multiple edges) $G$ on $n$ vertices. Task: place equidistantly the vertices of $G$ on a circle of unit ...
27
votes
1answer
3k views

An edge partitioning problem on cubic graphs

Hello everyone, I already asked this question on the TCS Stack Exchange, but it has not been resolved yet. Maybe readers of this forum will have other ideas or information, although I suspect that ...
0
votes
3answers
293 views

boolean functions and averaging / counting

Hey guys, I have a slightly imprecise question. I would like say something about a whole set of binary strings evaluated by a binary function by just looking at some type of average. The easiest ...
2
votes
4answers
732 views

Enumerative algorithm through inclusion-exclusion

Hello everybody ! I wondered, without really knowing where to search, whether there was a "smart" way to enumerate/iterate over all the elements of a set which can be counted by inclusion-exclusion. ...
8
votes
1answer
843 views

Expected number of steps for a discrete random walk to visit every point on an N-dimensional rectangular lattice

Please imagine a discrete random walk on an N-dimensional rectangular lattice with dimensional lengths $(l_1, ..., l_N) \in L$ and total lattice points $P = \prod{l_i}$, for $i = 1, ..., N$. At each ...
2
votes
1answer
290 views

Approximating a recursively-defined function

Let $$f(k) := \frac{2k-1}{k}\bigl(1-\sum\limits_{i\lt k}\frac{i\ f(i)}{k+i-1}\bigr)$$ for $k\in\mathbb{N}^{+}$. So $f(1) = 1$, $f(2) = 3/4$, $f(3) = 35/72$, etc. (This function arises when ...
1
vote
0answers
360 views

Minimizing quadratic form over permutations

Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem: $\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$, where $S_n$ ...
6
votes
2answers
880 views

Bijective proof of weak form of Stirling's approximation

There are short and sweet proofs of various forms of Stirling's approximation. But even the sweetest among them don't instill the same conviction in the reader as a direct bijective proof. Computer ...
2
votes
2answers
186 views

Indexing schemes of binary sequences

I am looking for "low-complexity" indexing methods to enumerate binary sequences of a given length and a given weight. Formally, let $T_k^n = \{x_1^n \in \{0,1\}^n: \sum_{i=1}^n x_i = k\}$. How to ...
7
votes
1answer
259 views

How long are the certificates produced by the Zeilberger and WZ methods for solving combinatorial sums (A=B)?

In the book "A = B" by Petkovesk, Wilf, and Zeilberger, (downloadable here), the authors provide several algorithmic methods for finding closed forms or recurrences for sums involving e.g. binomial ...
10
votes
1answer
461 views

I am searching for the name of a partition (if it already exists)

I derived this definition by searching for a representation of a family of sets. I am quite sure that someone should have thought to this before, because it seems to be quite straightforward given a ...
9
votes
0answers
667 views

Finding a set with the maximum number of finite alphabet strings within a fixed Levenshtein distance of one-another

Please consider the set of all possible strings of some finite size $M$ alphabet $\Sigma$, $\alpha$ $= a_1, a_2, ..., a_k, ..., a_n$, of length $|\alpha| = L$. The Levenshtein distance (or 'edit ...
5
votes
1answer
537 views

Counting Eulerian Orientation in a 4-regular undirected graph

We would like to know how hard it is to count Eulerian orientation in an undirected 4-regular graph. For a given edge orientation to be Eulerian, we mean that every vertex has 2 in-edges and 2 ...
10
votes
1answer
641 views

Counting colored rook configurations in the cube - when is it even?

Informal Statement In the $n\times n \times n$ grid, we can places rooks (those from chess) such that no two rooks can attack each other. One way to achieve this is to place a rook in position ...
13
votes
1answer
433 views

Bipartite Nim-Geography

Two players are playing a game on a bipartite graph where all of the edges are nim-heaps of various sizes. A token starts on one of the vertices, and on your turn you must move the token over an edge ...
4
votes
6answers
624 views

Reconstructing an ordering of a multiset from its consecutive submultisets

We have a multiset $S$ of size $t$ with $r$ distinct elements, where $t$ is much larger than $r$. We want to reconstruct an ordering $s_1, s_2, ... s_t$ of the elements of $S$ given the values of $t$ ...
10
votes
1answer
616 views

Characterization of Boolean-valued functions on the discrete cube based on its Fourier coefficients.

Consider functions on the discrete cube $\{-1,1\}^n$. We consider the Discrete Fourier Transform of such functions. Suppose we denote the parity function on a subset $S \subseteq [n]$ of ...
12
votes
4answers
500 views

Is there a natural family of languages whose generating functions are holonomic (i.e. D-finite)?

Let $L$ be a language on a finite alphabet and let $L_n$ be the number of words of length $n$. Let $f_L(x) = \sum_{n \ge 0} L_n x^n$. The following are well-known: If $L$ is regular, then $f_L$ is ...