Math Problem Statement

give just the answers

Question 1

By [𝑛][n]we denote the set {1,…,𝑛}{1,…,n}. A function 𝑓:[π‘š]β†’[𝑛]f:[m]β†’[n] is called monotone if 𝑓(𝑖)≀𝑓(𝑗)f(i)≀f(j)whenever 𝑖<𝑗i<j. Let 𝑆(π‘š,𝑛)S(m,n) be the number of monotone injective functions from [π‘š][m] to [𝑛][n]. What is the number of monotone injective functions from [100][100] to [80][80]?

1 point

Question 2

Let 𝑇(π‘š,𝑛)T(m,n) be the number of monotone functions 𝑓f from [π‘š][m] to [𝑛][n]. That is, functions 𝑓:[π‘š]β†’[𝑛]f:[m]β†’[n] such that 1β‰€π‘–β‰€π‘—β‰€π‘š1≀i≀j≀m implies 𝑓(𝑖)≀𝑓(𝑗)f(i)≀f(j). What is 𝑇(10,3)T(10,3)?

1 point

Question 3

Let 𝜏(π‘š,𝑛)=[𝑇(π‘š,𝑛)𝑇(𝑛,π‘š)]Ο„(m,n)=[T(n,m)T(m,n)​], where [π‘₯][x] means π‘₯x rounded down. What is 𝜏(10β‹…109,5β‹…109)Ο„(10β‹…109,5β‹…109)?

1 point

Question 4

We call a function 𝑓:𝐴→𝐡f:Aβ†’B almost injective if for every π‘βˆˆπ΅b∈B the number of elements π‘Ža with 𝑓(π‘Ž)=𝑏f(a)=b is at most two. Let 𝑀(𝑛)M(n) be the number of almost injective monotone functions from [𝑛][n] to [𝑛][n]. What is 𝑀(5)M(5)?

1 point

Question 5

A function {0,…,𝑛}→𝑍{0,…,n}β†’Z is a walk function if consecutive values differ by exactly 11, i.e., βˆ£π‘“(𝑖)βˆ’π‘“(π‘–βˆ’1)∣=1∣f(i)βˆ’f(iβˆ’1)∣=1 for all 1=1,…,𝑛1=1,…,n. How many walk functions are there from {0,…,6}{0,…,6} to 𝑍Z with 𝑓(0)=𝑓(6)=0f(0)=f(6)=0?

1 point

Question 6

A function {0,…,𝑛}→𝑍{0,…,n}β†’Z is a Lipschitz function if consecutive values differ by at most 11, i.e., βˆ£π‘“(𝑖)βˆ’π‘“(π‘–βˆ’1)βˆ£β‰€1∣f(i)βˆ’f(iβˆ’1)βˆ£β‰€1 for all 𝑖=1,…,𝑛i=1,…,n. Let 𝐿(𝑛)L(n) be the number Lipschitz functions 𝑓:{0,…,𝑛}→𝑍f:{0,…,n}β†’Z with 𝑓(0)=𝑓(𝑛)=0f(0)=f(n)=0. What is 𝐿(5)L(5)?

1 point

Solution

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Monotone functions
Injective functions
Walk functions
Lipschitz functions

Formulas

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Theorems

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Suitable Grade Level

Advanced Mathematics