# Tagged Questions

23 views

### No simple closed form for Bell numbers

The Bell number $B_n$ is the number of partitions of $[n]$. Unlike other basic combinatorial quantities, $B_n$ has no simple finite closed form. This seems surprising to me. Can anyone explain why ...
12 views

### The number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit ...
20 views

### Ehrhart Polynomials Modulo Prime Integers

Are there any results known about computing Ehrhart Polynomials modulo prime integers?
28 views

### Hall's marriage problem, graph theoretic version

Let $G = ( V,E)$ be a bipartite graph, where $V = V_1 \cup V_2$ , $|V_1| \leq |V_2|$ . For every $A \subseteq V_1$ let $\phi(A)$ be the set of vertices in $V_2$ adjacent to vertices in $A$. By ...
53 views

### Combinatorics of a game

Suppose there are $n$ people sitting in a circle, with $n$ odd. The game is played in rounds until one player is left. Each round the remaining players point either to the person on their right or ...
35 views

### proof of Konig's Theorem for bipartite graphs from Menger's Theorem

Could someone provide me with a good reference for a proof of Konig's Theorem for bipartite graphs from Menger's Theorem? Konig's Theorem is as follows: For a bipartite graph $G$, the maximum size ...
34 views

### Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
17 views

### Embed one Coxeter System into another

What is a good reference that explains all the braid relations and diagrams for Coxeter systems concisely? In particular, how do I embed $H_3$ inside $D_6$, or $H_4$ inside $E_8$? Any hints?
179 views

### What is this myth/legend and origin of related ideas?

There is a story I recently heard but the story teller (who read about it someone on the Internet) have forgotten the majority of the story, so there is little I can work on: my search attempts went ...
81 views

### Theoretical computer science text for mathematician

I am a high school student, I know some basic programming in java,python and visual basic. I love combinatorics and I have encountered various cases in which I have found some problems are really ...
133 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, ...
54 views

### Conjectured Value of Ramsey Number $R(3,10)$

It is known that the value of the Ramsey number $R(3,10)$ is either 40, 41, or 42. Have any experts in the field offered a conjecture as to which it might be?
24 views

### Maximal chain in (strong) Bruhat order

Consider the (strong) Bruhat order, $\leq_{B}$, on the symmetric group $S_n$. Suppose there are permutations $\pi,\sigma \in S_n$ such that $\pi \geq_{B} \sigma$. Suppose further that they satisfy the ...
86 views

### Very elementary number theory and combinatorics books.

I know the basics of logic, sets, relations and the like, so studying intros to abstract algebra and real analysis is not that hard. That said, I have a deficiency when it comes to elementary number ...
42 views

### Directory for known bound of Ramsey numbers?

I must admit I'm not a google connoisseur, but I have not been able to find a place where I can find known lower bounds for many Ramsey numbers, something ideal would be if I could insert (3,44) and ...
27 views

### Number of 3-way contingency tables

Denote by $f(n)$ the number of $2\times 2 \times 2$ contingency tables with all 2-dim marginals equal to $n$. What is $f(n)$? It is easy to see that $f(n)$ is polynomial of degree 4, since it ...
53 views

### Recurrence relation for product of binomial coefficients

We all know the standard recurrence relation for binomial coefficients: $$\binom{n}{k-1} + \binom{n}{k} = \binom{n+1}{k}$$ Is there any finite-step recurrence relation one can write down for a ...
74 views

### Math competitions resource at university level

I want some problems especially in Algebra field for math competitions at undergraduate math students level. Does anybody here know book, website,... that I can use?!
87 views

### A basic Combinatorics Book

So, this is my problem...I have completed my boards and among all others, I have a great weakness in combinatorics. So this means I can utilize my free time now to address this problem. I think it is ...
437 views

### simple games with cute winning strategies?

Im thinking of games of two players ($A$ goes first and $B$ second) like the following: There are 35 chips in a table, during each turn a player can remove 1,2,3 or 4 chips. Prove player $B$ can ...
219 views

### Does $\sum_{k=0}^{k=n} {n \choose k} k!$ have a closed form for integers $k,n$?

While doing research in computer system, I came across the following summation: $$S_n = \sum_{k=0}^{n} {n \choose k} k! = \sum_{k=0}^{n} \frac{n!}{(n-k)!}$$ where both $n$ and $k$ are integers. $S_n$ ...
193 views

### Asymmetric roles that are symmetric in every instance

This is similar to something else I posted, but this time we'll pretend we've never heard of infinite sets or infinite series. \begin{align} & \sin(\alpha+\beta+\gamma+\delta+\varepsilon+\zeta) ...
230 views

### A conjecture relating Multiple Zeta Values and the Polya Enumeration Theorem

Let me state my motivation. I believe that the Polya Enumeration Theorem and Multiple Zeta Values (the classic being the Basel problem and the values of the Riemann zeta function at the even ...
26 views

### Generalized permutahedron and random polytopes

The Birkhoff polytope $B_n$ is defined as the convex hull of the set of permutation matrices, which gives us the set of doubly stochastic matrices. A concept which is intimately related is that of the ...
448 views

### Book on combinatorial identities

Do you know any good book that deals extensively with identities obtained using combinatorial and/or probabilistic arguments (e.g., by solving the same combinatorial or probability problem in two ...
112 views

### Characterizing a certain set of matrices arising from binary trees

Suppose I have a binary tree, like v1 v4 \ / -------- / \ v2 v3 I can write a matrix for this tree whose $(i,j)$th ...
25 views

### Non-Intersecting up-right lattice paths and standard Young Tableaux

Consider the Lattice $\mathbb{Z}^2$ and an initial set of points with coordinates $(0,u_1)$, $(0,u_2)$, $\cdots$ $(0,u_n)$, final set of points $(m,v_1),(m,v_2),\cdots,(m,v_n)$, where $v_i,u_i$ are ...
42 views

### Alternative reference for number of restricted partitions

I am looking for the number of partitions of some number $n$ into $k$ parts. Following the Wikipedia article on partitions, I ended up with Andrew's book [1]. Judging by Google's preview Chapter 3 ...
30 views

### Number of non-increasing boolean functions of $n$ booleans, up to permutations.

How many non-increasing boolean functions of $n$ boolean variables are there? I don't want to count functions that ignore some of their inputs. If two or more functions differ only by permuting their ...
72 views

### Gowers' proof of Szemerdi's theorem

Are there any good books or other resources (expository notes) which explains Gowers' proof of Szemerdi's theorem in detail?
30 views

### A game of repeatedly taking the union of sets

Suppose that I have the set family $\mathcal{A}_1 = \{A_1, \dots, A_k\}$. We play the following game, which consists of a sequence of iterations. In the first iteration, we choose an arbitrary ...
70 views

### Describing the sequence A224239.

I've been trying to describe mathematically the $n$th term $a_n$ of the sequence A224239. We get $a_n$ by counting the distinct ways to fill an $n\times n$ grid with squares of smaller integer size, ...
61 views

### Monoid on ordered partitions of a natural number

Fix a natural number $n$, and let $O_n$ be the set of ordered partitions of $n$. For example $O_3=\{1+1+1,1+2,2+1,3\}$ which can also be written as $\{1|2|3,1|23,12|3,123\}$. We can define two ...
50 views

### References for chromatic symmetric functions of hypergraphs

Define a hypergraph to be a pair $H = (V,E)$ where $V$ is a set of vertices and $E$ is any set of subsets of $V$ called edges. Thus if every edge $U \in E$ has only two elements, then the hypergraph ...
25 views

### looking for hypergraph decompositions

there are many thms for/types of graph decompositions. in contrast, am looking for various types of hypergraph decompositions...? also esp interested in graph analogs that translate somehow eg ...
176 views

### Book on discrete mathematics for self study

I am searching for book on discrete mathematics which is suitable for self study. This mean I want it to have exercises with answers (It would be ideal if it had solutions). I have already read ...
122 views

### What can we say about the kernel of $\phi: F_n \rightarrow S_k$

Let $F_n$ denote the free group on $n$ generators and let $S_k$ denote the symmetric group on the integers $\{1,\dots, k\}$, and the action of homomorphism $\phi$ (as given in the title) on the ...
102 views

### Articles on matchstick puzzles

There are many ingenious puzzles involving matchsticks that are arranged as squares, rectangles or triangles, and can be moved under some restrictions (for a lot of examples see ...
73 views

### Finding the super-mean (NOT the mean) of a set of numbers.

the super-mean is found by grouping pairs of numbers and finding the average successively until there is just one number. For example, (1-2-3-4-5) \to ((1+2)/2,(2+3)/2,(3+4)/2,(4+5)/2) \\ ...
30 views

### Variants of sudoku that require very little give aways

Consider the following generalization of sudoku (it might already exist under a different name, any reference is welcome): Let $n$ be a natural number. By a generalized sudoku I understand a ...
190 views

### An extrasensory perception strategy :-)

Inspired by classical Joseph Banks Rhine experiments demonstrating an extrasensory perception (see, for instance, the beginning of the respective chapter of Jeffrey Mishlove book “The Roots of ...
151 views

### Maximum edges in a square free graph

Square free graph : Graphs with minimum cycle length greater than 4. Question : What is the maximum number of edges possible for a square free graph $G(V,E)$ given that $|V|$ = n. Is it of the order ...
63 views

### Ham Sandwich theorem used in combinatorics problem involving beads on a necklace

Ok, so according to a friend of mine you can use the ham sandwich to prove the following theorem: Suppose there is a necklace with $m$ types of beads and $2n_1,2n_2...2n_m$ beads of each colors. So ...
16 views

### Online Encyclopedia of Error-Correcting Codes

Is there some kind person on the internet who is making an exhaustive collection of error-correcting codes? I'm looking for something analogous to the OEIS. I want to ask questions like "what is this ...
121 views

### Are there mathematical (combinatorial) objects to represent such set systems?

Background: This is a problem arising from my study on computer science. In its appearance, it involves set systems. A set system $\mathcal{S} = \{S_1, S_2, \ldots, S_m \}$ is a collection of ...
214 views

### Symmetries of combinatorial structures.

Studying the automorphism groups of graphs/finite geometries/designs has been quite useful and important for both group theory and combinatorics. I know of the following books which cover the ideas ...
43 views

### A Survey on Rota's Conjecture

I am looking for a survey on Rota's Conjecture (1970). I am more interested in geometric aspects of the conjecture and any geometric content related to the conjecture. Any reference would be helpful. ...
54 views

### How many Hamiltonian loop are there in a big rectangle?

Suppose I have some big rectangle made of $n \times m$ squares, and I want to place tiles on it in a manner that makes a picture of a hamiltonian loop. I can transform this problem into a problem ...
### Maximum bin load for $\alpha n$ balls into $n$ bins
In a paper I am reading the author writes: A standard result concerning balls and bins shows that if we throw at least $\alpha n$ balls into at most $n$ bins, then the maximum bin load is ...
An exercise in Stanley's Enumerative Combinatorics (Chapter 3, ex. 8) asked for an example of a finite self-dual poset, (i.e. there is a bijection $f: P\to P$ such that \$s\le t \Longleftrightarrow ...