# Tagged Questions

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### 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 ...
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### Ehrhart Polynomials Modulo Prime Integers

Are there any results known about computing Ehrhart Polynomials modulo prime integers?
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### Lattice Path Spaces.

It is well known that the number of paths from $(0,0)$ to $(n,k)$ in $\mathbb{N^2}$ with the set of steps $\{(1,0),(0,1)\}$ is ${n+k \choose k}$. This is the minimum number of steps needed to get to ...
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### Finding integer vectors in the column space of a matrix

Consider a given set $S \subset Z$. $S$ is a finite set. Matrix $A \in S^{N \times M}$ is also given. Does there exist an algorithm to find all the vectors belonging to the space Col$(A)\cap S^N$ ...
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### Counting sum of lattice points

Assume a set $S$ with $|S|$ entries. Indeed, $S$ is the set of lattice points inside a $k$-sphere. Assume $V=S\oplus S$ where $\oplus$ is the Minkowski sum of two sets. Do you know any lower bound on ...
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### number of lattice points in an n-ball

I have faced a problem in my work and I will appreciate any hint/reference as I am not much into the lattice problems. Assume an n-dimensional lattice $\Lambda_n$ with generator matrix $G$. Note that ...
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### Show transformation of lattice paths is a bijection

How do I show that the transformation (switching head and tail of a path) is indeed a bijection from lattice paths with excess k to lattice paths with excess k-1. I need to describe the inverse ...
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### Show $\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$ by block-walking interpretation of Pascal's triangle

A combinatorial proof for the identity $$\binom{n}{k}\binom{k}{a} = \binom{n}{a}\binom{n-a}{k-a}$$ is the following "committee" argument, which seems the most common. There are $\binom{n}{k}$ ...
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### Bounding the density of finite coprime sets

I am currently running into a problem related to coprime numbers. Consider a set of $d$-dimensional integer vectors, $z \subset \mathbb{Z}^d$ such that each component $z_i$ is bounded by another ...
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### What's the maximum number of sums of $ak_1,..,ak_m, b\ell_1,…,b\ell_m$ needed to solve for $a$ or $b$?

Given a set of integer multiples of $a$ and $b$, $ak_1,..,ak_m, b\ell_1,...,b\ell_m$, what is the maximum number of finite sums of the multiples you can create such that no sum of all multiples of $a$ ...
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### Lattice Paths from $(1, 1) \to (x, y)$

Moderator Note: This is a current contest question on Brilliant.org. Let $S$ be the set of $\{(1,1), (1,−1), (−1,1), (1,0), (0,1)\}$-lattice paths which begin at $(1,1),$ do not use the same ...
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### Monotonic Lattice Paths and Catalan numbers

Can someone give me a cleaner and better explained proof that the number of monotonic paths in an $n\times n$ lattice is given by ${2n\choose n} - {2n\choose n+1}$ than Wikipedia I do not understand ...
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### When does a simplex have an interior lattice point?

Given $r$ vectors $v_1, \dots, v_r$ in $\mathbb{Z}^n$, is there an easy way (in terms of the entries of the $v_i$) to determine if there is a point of $\mathbb{Z}^n$ in the interior of the simplex ...
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### Counting lattice points interior to a polygon

If I define an integer lattice $\Lambda \subseteq \mathbb{Z}^2$ with a basis given by $$\omega_{1} = a \hat{i} + b\hat{j}, \;\;\; \omega_{2} = -b \hat{i} + a\hat{j}$$ How can I count how many lattice ...
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### Lattice paths and Catalan Numbers

Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner. How many such routes are there through a ...
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### Guaranteeing an integer lattice point centroid

My question is this: Writing $n(4)$ to be the minimum number of integer lattice points in the plane so that some four of them must determine an integer lattice point centroid, show that $n(4)=13$. I ...
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### A problem involving the lattice grid.

Suppose that $22$ points are arbitrarily chosen from a $7\times 7$ lattice grid. We are to prove that there exists at least one rectangle in any $4$ points chosen from the above $22$. A general ...
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### A certain labeling of lattice points on the plane [duplicate]

Possible Duplicate: A stronger version of discrete “Liouville’s theorem” Let each lattice point of the plane be labeled by a positive real number . Each of these numbers is the arithmetic ...
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### Proving Binomial Identities Using Bijections To Lattice Paths

How can I derive a bijection to show that the following equality holds? $2\displaystyle\sum\limits_{j=0}^{n-1} \binom{n-1+j}{j} = \binom{2n}{n}$ In class, we've been deriving bijections using ...
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### Number of matrices with weakly increasing rows and columns

I'm curious as to how many matrices there are of size $m \times n$ with elements of the set $\{1, \ldots , k\}$ such that each row and column is weakly increasing? The answer should be expressable as ...
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