# Tagged Questions

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### A variation on Bulgarian solitare

It appears that a variation on Bulgarian solitare has a fixed point regardless of the starting $n$. For example, let $n=69$, and consider this partition: $$(8,8,7,7,5,5,5,5,5,4,3,3,2,2)$$ In ...
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### Probability of k overlapping subsets in N trials

Ok, here is what I am attempting to find an answer to: I draw M uniformly random subsets of size K from the set of numbers $\Omega=\{1, \dots, N\}$ (where uniformly random means that each unique ...
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### about the structure of components of tensor product if more than one bipartite graph is taken

I was reading about tensor product of graphs. We know that if we take tensor product of n graphs and want this product to be a connected graph then at most one graph should be bipartite. In the book ...
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### Rank function and closure operator for a set system

I would like to trace the concepts "rank function" and "closure operator" back to some structures as primitive as possible. For a set system $(E,F)$ which is an independence system or a greedoid, I ...
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### Family of sets with unique subsets

I am given a set $M\subseteq\{1,\dots,n\},\,|M|=m$ and a famliy of $k$-sets ($k<m$) $\mathcal{U}=\{U_1,\dots,U_p\},\,U_{i}\subset M$. For this family, I would like to check one of the following ...
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### 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 ...
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### Cardinality of intersection of a random subset with a fixed subset

How can I simply prove the following fact: Let $A := \{1, \dots n \}$ and $B := \{1, \dots, \lfloor \frac{n}{4} \rfloor \}$. Let $d \in (0,1)$ and let $R$ be a randomly choosen (with uniform ...
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### Computing the lexicographic indices of integer partition

If we order all the partitions of a integer in a lexicographic order, how can we compute the position of each partition in this order without having to explicitly list all other partitions that ...
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### Indexing combinations with repetition

Let $\Sigma$ be a finite set of symbols with total order. Let $C_k$ be the set of all $k$-multiset (unordered collection of $k$ elements from $S$, with repetition allowed). We can order all the ...
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### Obtaining each set (hyperedge) in a set system (hypergraph) as a union of sets in a smaller set sytem

Let $V$ be a set and $E$ a set of subsets of $V$. I'd like to know the proper terminology for the following concept. Let me call it "generator". A generator is a set $F$ of subsets of $V$ such that ...
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### Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
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### Question on weights and minimal degree

Edit: question has been changed from 'lexicographic' (cf. "D K"'s answer below) to 'degree' minimality. Let $x_1,x_2,x_3$ be indeterminates. Fix an integer $k> 3$. Consider the set $M$ of all ...
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### Integer solution to special system of linear equations

This problem appear in my research, but I am unable to solve it. There should be an easy argument, but I have not yet found it. Informal version An integer $k\geq 2$ is fixed. We are given a matrix ...
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### 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 ...
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### References on techniques for solving equations with discontinuous functions such as floor and ceiling?

Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good ...
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### Citation for subset complement result

Let $S=\lbrace s_1,\ldots,s_n \rbrace \subset \lbrace1,\ldots,2n\rbrace$. Consider two operations on $S$: the complement $C(S)=\lbrace 1,\ldots,2n \rbrace \setminus S$ and a reflection* ...
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### Does erosion mix faster than a riffle shuffle?

It is a famous result of Aldous and Diaconis1 that seven shuffles are necessary and suffice to approximately randomize 52 cards.2 Here the shuffles are the standard riffle shuffle, where the ...
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### A good upper bound on the size of k-biclique in random bipartite graphs.

Let $G = (X \cup Y, E)$ be a random bipartite graph, where $X$ and $Y$ are the set of vertices and $E$ the set of edges. I want to find an upper bound of the largest biclique in which exactly $k$ ...
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### expected size of unbalanced biclique in random bipartite graph

I am discovering random graph and I am trying to prove the following result. This is a follow-up on a previous question of mine what's an upper bound on the size of the largest biclique in random ...
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### what's an upper bound on the size of the largest biclique in random bipartite graph?

I am not an expert in random graph but I need the following result and I couldn't find any reference on this. Let $G(X \cup Y,p)$ be a random bipartite graph where the set of edges is $X \cup Y$, $X$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...