This tag is for basic questions about the study of finite or countable discrete structures — specifically how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions. ...

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37
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843 views

Why are asymptotically one half of the integer compositions gap-free?

This is a longish post about something that has been haunting me for a while about a kind of restricted composition, namely gap-free and complete compositions. First, I will define the terms that are ...
17
votes
0answers
376 views

Making Friends around a Circular Table

I have $n$ people seated around a circular table, initially in random order. At each step, I choose two people and switch their seats. What is the minimum number of steps required such that every ...
15
votes
0answers
244 views

Combinatorial Interpretation of a Certain Product of Factorials

Let $\mu$ denote the Moebius function. What is a combinatorial interpretation of the following integer, \begin{align} \prod_{d \mid n} d!^{\,\mu(n/d)}, \end{align} where the product is taken over ...
13
votes
0answers
473 views

$f(x)=\sum_{t}{x \choose t}{n-x \choose k-t}$ - even or odd?

The following function popped in my research: $$f(x)=\sum_{\array{0\le t\le k \\ t\equiv_p a}}{x \choose t}{n-x \choose k-t}$$ Where: n,k are natural numbers and $k\le n$. t is taken over all ...
11
votes
0answers
382 views

Prove this determinant identity combinatorially

This is for those of you who understand the Lindstrom-Gessel-Viennot lemma. I am looking for a proof of the following identity using paths and such: Let $A$ be an $n\times n$ matrix, and for ...
10
votes
0answers
155 views

$1^2+2^2+\cdots+24^2=70^2$ and squarily squaring the torus

The unique nontrivial solution to $1^2+2^2+\cdots+n^2=m^2$ is $(n,m)=(24,70)$. (This fact has connections to modular forms, special functions, lattices and string theory.) Martin Gardner, in the ...
10
votes
0answers
492 views

How to find a total order with constrained comparisons

There are 25 horses with different speeds. My goal is to rank all of them, by using only runs with 5 horses, and taking partial rankings. How many runs do I need, at minumum, to complete my task? As ...
9
votes
0answers
312 views

Minimal time gossip problem

The gossip problem (telephone problem) is an information dissemination problem where each of $n$ nodes of a communication network has a unique piece of information that must be transmitted to all the ...
9
votes
0answers
275 views

Generating function solution to previous question $a_{n}=a_{\lfloor n/2\rfloor}+a_{\lfloor n/3 \rfloor}+a_{\lfloor n/6\rfloor}$

In attempting to answer this question, I reduced it to a seemingly simple generating functions question, but after days of work was unable to construct a proof. Since I do not have experience trying ...
8
votes
0answers
163 views

How to estimate $Pr[vr_i=ur_i]$ in the presence of rotations

Suppose we want to compute the probability that for two different random vectors (with elements that are $0$ or $1$), denoted by $v$ and $u$, multiplying them with the rotations of a random vector $r$ ...
8
votes
0answers
113 views

To what extent is it possible to generalise a natural bijection between trees and $7$-tuples of trees, suggested by divergent series?

In the paper "Seven Trees In One" by Andreas Blass, a "very explicit" bijection is found between trees and 7-tuples of such trees. The idea to construct such a bijection stems from looking at some ...
8
votes
0answers
197 views

Symmetric polynomials

I've got a seemingly simple question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots ...
8
votes
0answers
198 views

Extracting an (almost) independent large subset from a pairwise independent set of Bernoulli variables

Let $n>1$, and let $X_1,X_2, \ldots ,X_n$ be non-constant random variables with values in $\lbrace 0,1 \rbrace$. Let us say that a subset of variables $X_{i_1},X_{i_2}, \ldots,X_{i_d}$ is complete ...
8
votes
0answers
426 views

How strong is the statement that Thompson F is amenable?

Justin Moore's proof turned out to have an error I just attended Justin Moore's talk on this today. Since I am neither a group theorist nor a combinatorist, and is not familiar with ultrafilters I ...
8
votes
0answers
176 views

Different notions of q-numbers

It seems that most of the literature dealing with q-analogs defines q-numbers according to $$[n]_q\equiv \frac{q^n-1}{q-1}.$$ Even Mathematica uses this definition: with the built-in function QGamma ...
8
votes
0answers
153 views

Reconstruction Conjecture and Partial 2-trees

Reconstruction conjecture says that graphs (with at least three vertices) are determined uniquely by their vertex deleted subgraphs. This conjecture is five decades old. Searching relevant ...
7
votes
0answers
134 views

Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
7
votes
0answers
99 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
7
votes
0answers
139 views

The right way to motivate lattice theory in a combinatorics class

I am attending a course on combinatorics. I was asked to present Möbius functions on lattices for this course. I was trying to look for a simple non-trivial problem that illustrates the need for ...
7
votes
0answers
189 views

Citation for subset complement result

Let $S = \{s_1, \ldots, s_n\} \subset \{1, \ldots, 2n\}$. Consider two operations on $S$, unfortunately both called complement in different setting: let $A(S) = \{1, \ldots, 2n\} \setminus S$ (set ...
7
votes
0answers
156 views

A variation on the Look and Say Sequence and some questions about it.

For information on the sequence mentioned in the title, see http://en.wikipedia.org/wiki/Look-and-say_sequence. This is an original problem. Suppose instead of "describing" the numbers in a string in ...
7
votes
0answers
143 views

How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?

I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be $$\small \dim \tilde{H}_t(X; {\mathbb{Z}}_2) = ...
7
votes
0answers
230 views

A sequence similar to the Catalan numbers

The $n$-th Catalan number $c_n$ has the closed form $\frac1{n+1}\binom{2n}{n}$ and follows the recursion $c_n = \sum\limits_{i = 0}^{n-1} c_{n-1-i}c_i$ I am interested in the quantity $e_n$ which ...
7
votes
0answers
220 views

Lucky chance or combinatorial cause?

Consider an $n \times 1 - $rectangle where the $n$ squares are numbered $1$ to $n$. Cover this rectangle with white squares $a$, black squares $b$ and dominoes $dd$. To each covering of the rectangle ...
7
votes
0answers
322 views

Number of possible crosswords

I was staring at my crossword this morning when I had this question: How many possible arrangement of grid squares are there in an $N\times N$ crossword, provided we impose certain constraints? I take ...
6
votes
0answers
124 views

Parity of sum of Kronecker deltas in a graph

For some fixed $n\in\mathbb N$ let $G$ be a graph on the vertex set $\{1,\dots,n\}$ with a total number of $k$ edges $e_1,\dots, e_k$. For any vertex colouring $c(i)$ of the graph, $\delta_e$ is ...
6
votes
0answers
100 views

Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
6
votes
0answers
148 views

Selecting a subset from a set such that a given quantity is minimized

Let $A$ be is a set of some $p$-dimensional points $x \in \mathbb{R}^p$. Let $d_x^A$ denote the mean Euclidean distance from the point $x$ to its $k$ nearest points in $A$ (others than $x$). Let $C ...
6
votes
0answers
79 views

Generalization of Gray codes

A friend of mine asked me if it was possible - physical difficulties aside - to generate all 32 combinations of raised/lowered fingers by changing status of a fixed number of fingers at every step. ...
6
votes
0answers
98 views

Knuth equivalence of unshuffles

For any permutation $\sigma\in S_k$ and any $i\in\left\lbrace 1,2,...,k\right\rbrace$, I will denote $\sigma\left(i\right)$ by $\sigma_i$. For any word $w$, I will denote by $\mathrm{st}w$ the ...
6
votes
0answers
93 views

Sum-Product Generating Functions

Let $A_n$ be a family of sequences $\{a_i\}_{i=1}^n$ of length $n$. I'll refer to sequence elements of $A_n$ as $a$. Then define $$G(z):=\sum_{a\in A_n}\prod_{i=1}^n(z+a_i).$$ Here's one possible ...
6
votes
0answers
143 views

Rotations of a tetrahedron

Let $P$ be a tetrahedron inside an sphere such that all of its vertices are on the surface of the sphere. Suppose that three quarters of sphere's surface is colored black. Show that there is a ...
6
votes
0answers
271 views

Finding the maximum number of subspaces of a vector space over finite field that satisfy these relations

I have a question and I am stuck. I was wondering if anyone has a thought, before I start a brute-force search. For $q$ a prime number and $n =6$, let $\mathbb {F}_{q}^{n}$ be an $n$-dimensional ...
6
votes
0answers
145 views

Can we rearrange the integers, such that the arithmetic average of any two numbers does not appear between them?

Can we rearrange the integers, such that the arithmetic average of any two numbers does not appear between them? In other words: Can we have a sequence $\{a_n\}_{n\in \mathbb{Z}}$, where all integers ...
6
votes
0answers
154 views

Dimension of the space of algebraic Riemann curvature tensors

Given $n\in \mathbb N$, consider the vector space $\mathbb R^{n^4}$ whose elements I will denote by $(R_{abcd})$ with indices $a,b,c,d \in \{1, \dots, n\}$. This vector space is $n^4$-dimensional. The ...
6
votes
0answers
129 views

Characterization of Sequences with Integral Binomial Coefficients

For any sequence of positive integers $\{ a_i \}_{i \ge 1}$ we can define the generalized binomial coefficients $\binom{n}{k}_{a}$ as follows: $$m!_a = a_1 a_2 \cdots a_m, \binom{n}{k}_a = ...
6
votes
0answers
172 views

Combinatorial Proofs of Real Analysis Identity

In this question,a proof using real analysis is given of the following identity: $$ \sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^2}$$ Is there a ...
6
votes
0answers
356 views

Is there a Steiner system S(6,9,45)?

Background: the Belgian Lottery switched its main game this year to a draw of 6 balls out of a pool of 45, plus a bonus number which doesn't matter for the sake of this question. Assuming someone ...
6
votes
0answers
252 views

Young Tableaux as Matrices

These questions are motivated only by curiosity. Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1\geq\lambda_2\geq\ldots\geq \lambda_n)$. Is there any physical ...
6
votes
0answers
152 views

Calculating $\sum_{y=0}^x \Pr[Y= y] \Pr[Z\leq k-y]^2$ when Y,Z are binomially distributed?

Remark: I recently rewrote this post, hoping to get answers! I am analyzing the following experiment: Pick an $x \in \{0,\ldots,2k\}$ uniformly at random Pick $(2k+1)$-bit bitstring $b_1=(u,v_1)$ ...
6
votes
0answers
96 views

Optimal lower bound of $k$-sums for the integers $\{1,2,\ldots,n\}$ arranged around a circle in arbitrary order.

This question is motivated by the following two questions and is a slightly generalized version of them. Some three consecutive numbers sum to at least $32$ Integers $1, 2, \ldots, 10$ are ...
6
votes
0answers
88 views

Does the Hirsch conjecture hold for $n < 2d$?

The Hirsch conjecture states that the graph (i.e. $1$-skeleton) of a $d$-dimensional polytope with $n$ facets has diameter at most $n - d$. It was known for a long time that it sufficed to prove it ...
6
votes
0answers
219 views

Counting ordered tuples with an additional condition

Let $a_1 \le a_2 \le ... \le a_n$ be positive integers. I'm looking for a closed formula of the number $f(a_1,...,a_n)$ of all (ordered) tuples $(x_1,...,x_n)$ of positive integers statisfying $$x_1 ...
6
votes
0answers
202 views

Uses of Chevalley-Warning

In the recent IMC 2011, the last problem of the 1st day (no. 5, the hardest of that day) was as follows: We have $4n-1$ vectors in $F_2^{2n-1}$: $\{v_i\}_{i=1}^{4n-1}$. The problem asks : Prove the ...
6
votes
0answers
115 views

Generalizing Quadratic Reciprocity Law with Dilates

Eisenstein's proof of the Quadratic Reciprocity (QR) (and its Jacobi symbol generalization) both rely on counting lattice points in two congruent triangles. If we take $t$-dilates of these triangles, ...
5
votes
0answers
59 views

The number of regions created by $k$ hyperplanes

Consider $n$ dimensional Euclidean space. I am interested in the following question. For a $n$ dimensional space, what is the maximum number of $n$ dimensional regions you can get if the space ...
5
votes
0answers
67 views

Double factorial as a sum

I believe the following equality to hold for all integer $l\geq 1$ $$(2l+1)!2^l\sum_{k=0}^l\frac{(-1)^k(l-k)!}{k!(2l-2k+1)!4^k}=(-1)^l(2l-1)!!$$ (it's correct for at least $l=1,2,3,4$), but cannot ...
5
votes
0answers
81 views

Traveling salesman problem: can a terrible strategy beat a good one?

Until yesterday, I was under the naive impression that constructing a weighted graph where the nearest-neighbour algorithm gives the worst possible route, would have the property that any other ...
5
votes
0answers
118 views

Probability that at least one of four hands missing at least one suit

Deal each of four players a 13-card hand at random. What is the probability that at least one of the four hands is missing at least one suit? Let $A_i$ mean that player $i$ is missing at least one ...
5
votes
0answers
40 views

For the exponential operator $e^{f(x)\frac d{dx}}= \sum_{i=0}^\infty F_i(x) \frac{d^i}{dx^i}$, is there a formula for the $F_i$ in terms of $f$?

Consider the operator $$ e^{ f(x) \frac{d}{dx} } = \sum_{i = 0}^\infty \frac{1}{i!} \left(f \frac{d}{dx} \right)^i $$ If one commutes the derivatives with the powers of $ f $, then there are functions ...