0
votes
0answers
12 views

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 ...
12
votes
2answers
327 views

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 $\binom{n}{k}$ ...
3
votes
1answer
84 views

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 ...
0
votes
1answer
21 views

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 $a$ ...
0
votes
2answers
338 views

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 ...
1
vote
2answers
406 views

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 ...
2
votes
0answers
37 views

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 ...
0
votes
2answers
123 views

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 ...
2
votes
2answers
2k views

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 ...
4
votes
2answers
190 views

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 ...
2
votes
2answers
176 views

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 ...
2
votes
0answers
49 views

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 ...
0
votes
1answer
208 views

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 ...
1
vote
1answer
141 views

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 ...
2
votes
1answer
125 views

Bounding the number of integer solutions of the following inequality

Let $r\geq 1$ be a real number, $-1\leq x\leq 1$ a real number and $y>2$ a real number. We consider this data to be fixed. How can I obtain an upper bound on the number of $(a,b,c,d)\in ...
0
votes
1answer
51 views

If E is a subset of a lattice closed under addition then is the intersection of E with the opposite of some translate finite?

this seems intuitive to me but I'm struggling to prove it (is it false?). Let $E$ be a subset of a lattice (free abelian group of finite rank) closed under addition, containing the origin and such ...
3
votes
1answer
2k views

How can I find the number of the shortest paths between two points on a 2D lattice grid?

How do you find the number of the shortest distances between two points on a grid where you can only move one unit up, down, left, or right? Is there a formula for this? Eg. The shortest path between ...
2
votes
1answer
247 views

Closest point to line in lattice

Given an $n$ x $n$ integer grid, I look at all possible lines through two grid points and I am interested in the minimum distance from any grid point (not on the line) to any line. I my guess is that ...
1
vote
1answer
660 views

Lattice Paths to $(2n, 2n)$ that do not go through $(n,n)$

I know that from the origin to $(x,y)$ there are $${x+y \choose x} = {x+y \choose y}$$ Lattice paths. The question is finding the number of paths from the origin to $(2n, 2n)$ not passing through ...
0
votes
1answer
356 views

Lattice Paths Question

You know, how we can have lattice paths, where we can move either one block north, or one block east, and we have the find all the possible ways of reaching the point (x.y) from (0,0). That is ...
4
votes
1answer
219 views

Proving this identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$ using lattice paths

How can I prove the identity $\sum_k\frac{1}{k}\binom{2k-2}{k-1}\binom{2n-2k+1}{n-k}=\binom{2n}{n-1}$? I have to prove it using lattice paths, it should be related to Catalan numbers The $n$th ...
8
votes
1answer
314 views

“Center-of-Mass” of lattice polygons (generalization of Pick's theorem)

Call a polygon with integer coordinates (in the Euclidean plane) a 'lattice polygon'. Pick's Theorem allows you to efficiently compute the number of lattice points inside this polygon given just its ...
2
votes
1answer
120 views

Mathematical Results from Counting Points in Lattices

I'm preparing a talk on lattice point enumeration in polytopes (Ehrhart-Macdonald Theory), and I'd like to have an introduction with a few motivational problems/results which arise from the ...