System of distinct representative professors and students problem
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So I was doing exercises from a Combinatorics textbook. I ran into this problem.
Suppose we have $n$ students, $s_1,s_2,dots,s_n,$ and $n$ professors,
$p_1,p_2,dots,p_n$. For each student $s_i$ we define a set of his/her favorite professors, $S_i$. Every student has at most $f$ favorite professors (where $fgeq 3$), while each professor $p_j$ appears in at least $f$ sets $S_{j_1},S_{j_2},dots,S_{j_f}$.
For a given value of $f$, let $pi_i (f)$ be the number of appearances of professor $p_i$ on the students' lists, and $sigma_j (f)= |S_j|$.
$1.$ What is $operatorname{max}{pi_i(f)| 1leq ileq n}$ and
$operatorname{min}{sigma_j(f)| 1leq jleq n}$ for a fixed value of $f$? Note that you are not looking the max or min over the whole range of values of $f$, but rather for any given value, and the range is all values of $i$. So, your answer is a function of $f$
$2.$ Every student needs three letters of recommendation. Can they select their favorite professors for the letters so that no professor writes more than three letters? Prove your claim or show a counter-example.
How would you solve this? I recognize that this is a system of distinct representatives kind of problem. But somehow I can't think of a way to find a solution. Please help!
combinatorics
asked Dec 3 '18 at 17:28
DummKorfDummKorf
335
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add a comment |
$begingroup$
So I was doing exercises from a Combinatorics textbook. I ran into this problem.
Suppose we have $n$ students, $s_1,s_2,dots,s_n,$ and $n$ professors,
$p_1,p_2,dots,p_n$. For each student $s_i$ we define a set of his/her favorite professors, $S_i$. Every student has at most $f$ favorite professors (where $fgeq 3$), while each professor $p_j$ appears in at least $f$ sets $S_{j_1},S_{j_2},dots,S_{j_f}$.
For a given value of $f$, let $pi_i (f)$ be the number of appearances of professor $p_i$ on the students' lists, and $sigma_j (f)= |S_j|$.
$1.$ What is $operatorname{max}{pi_i(f)| 1leq ileq n}$ and
$operatorname{min}{sigma_j(f)| 1leq jleq n}$ for a fixed value of $f$? Note that you are not looking the max or min over the whole range of values of $f$, but rather for any given value, and the range is all values of $i$. So, your answer is a function of $f$
$2.$ Every student needs three letters of recommendation. Can they select their favorite professors for the letters so that no professor writes more than three letters? Prove your claim or show a counter-example.
How would you solve this? I recognize that this is a system of distinct representatives kind of problem. But somehow I can't think of a way to find a solution. Please help!
combinatorics
asked Dec 3 '18 at 17:28
DummKorfDummKorf
335
$endgroup$
add a comment |
$begingroup$
So I was doing exercises from a Combinatorics textbook. I ran into this problem.
Suppose we have $n$ students, $s_1,s_2,dots,s_n,$ and $n$ professors,
$p_1,p_2,dots,p_n$. For each student $s_i$ we define a set of his/her favorite professors, $S_i$. Every student has at most $f$ favorite professors (where $fgeq 3$), while each professor $p_j$ appears in at least $f$ sets $S_{j_1},S_{j_2},dots,S_{j_f}$.
For a given value of $f$, let $pi_i (f)$ be the number of appearances of professor $p_i$ on the students' lists, and $sigma_j (f)= |S_j|$.
$1.$ What is $operatorname{max}{pi_i(f)| 1leq ileq n}$ and
$operatorname{min}{sigma_j(f)| 1leq jleq n}$ for a fixed value of $f$? Note that you are not looking the max or min over the whole range of values of $f$, but rather for any given value, and the range is all values of $i$. So, your answer is a function of $f$
$2.$ Every student needs three letters of recommendation. Can they select their favorite professors for the letters so that no professor writes more than three letters? Prove your claim or show a counter-example.
How would you solve this? I recognize that this is a system of distinct representatives kind of problem. But somehow I can't think of a way to find a solution. Please help!
combinatorics
asked Dec 3 '18 at 17:28
DummKorfDummKorf
335
$endgroup$
So I was doing exercises from a Combinatorics textbook. I ran into this problem.
Suppose we have $n$ students, $s_1,s_2,dots,s_n,$ and $n$ professors,
$p_1,p_2,dots,p_n$. For each student $s_i$ we define a set of his/her favorite professors, $S_i$. Every student has at most $f$ favorite professors (where $fgeq 3$), while each professor $p_j$ appears in at least $f$ sets $S_{j_1},S_{j_2},dots,S_{j_f}$.
For a given value of $f$, let $pi_i (f)$ be the number of appearances of professor $p_i$ on the students' lists, and $sigma_j (f)= |S_j|$.
$1.$ What is $operatorname{max}{pi_i(f)| 1leq ileq n}$ and
$operatorname{min}{sigma_j(f)| 1leq jleq n}$ for a fixed value of $f$? Note that you are not looking the max or min over the whole range of values of $f$, but rather for any given value, and the range is all values of $i$. So, your answer is a function of $f$
$2.$ Every student needs three letters of recommendation. Can they select their favorite professors for the letters so that no professor writes more than three letters? Prove your claim or show a counter-example.
How would you solve this? I recognize that this is a system of distinct representatives kind of problem. But somehow I can't think of a way to find a solution. Please help!
combinatorics
combinatorics
asked Dec 3 '18 at 17:28
DummKorfDummKorf
335
asked Dec 3 '18 at 17:28
DummKorfDummKorf
335
asked Dec 3 '18 at 17:28
DummKorfDummKorf
335
asked Dec 3 '18 at 17:28
DummKorfDummKorf
335
asked Dec 3 '18 at 17:28
DummKorfDummKorf
335
335
add a comment |
add a comment |
1 Answer
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Let $f(S_{j},p_{i})=begin{cases} 1 & S_{j} text{ likes }p_{i}\ 0 & text{otherwise}end{cases}$.$sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})leq nf$ as every student has at most $f$ favorite professors and $sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})geq nf$ as each professor is the favourite of at least $f$ students. Therefore, each student has exactly $f$ favorite professors and each professor is the favourite of exactly $f$ students.
Hence, $pi_i (f)=sigma_j (f)= f$.
The answer to the second part is in the affirmative.
answered Dec 3 '18 at 18:14
Anubhab GhosalAnubhab Ghosal
81618
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$begingroup$
Let $f(S_{j},p_{i})=begin{cases} 1 & S_{j} text{ likes }p_{i}\ 0 & text{otherwise}end{cases}$.$sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})leq nf$ as every student has at most $f$ favorite professors and $sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})geq nf$ as each professor is the favourite of at least $f$ students. Therefore, each student has exactly $f$ favorite professors and each professor is the favourite of exactly $f$ students.
Hence, $pi_i (f)=sigma_j (f)= f$.
The answer to the second part is in the affirmative.
answered Dec 3 '18 at 18:14
Anubhab GhosalAnubhab Ghosal
81618
$endgroup$
add a comment |
$begingroup$
Let $f(S_{j},p_{i})=begin{cases} 1 & S_{j} text{ likes }p_{i}\ 0 & text{otherwise}end{cases}$.$sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})leq nf$ as every student has at most $f$ favorite professors and $sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})geq nf$ as each professor is the favourite of at least $f$ students. Therefore, each student has exactly $f$ favorite professors and each professor is the favourite of exactly $f$ students.
Hence, $pi_i (f)=sigma_j (f)= f$.
The answer to the second part is in the affirmative.
answered Dec 3 '18 at 18:14
Anubhab GhosalAnubhab Ghosal
81618
$endgroup$
add a comment |
$begingroup$
Let $f(S_{j},p_{i})=begin{cases} 1 & S_{j} text{ likes }p_{i}\ 0 & text{otherwise}end{cases}$.$sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})leq nf$ as every student has at most $f$ favorite professors and $sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})geq nf$ as each professor is the favourite of at least $f$ students. Therefore, each student has exactly $f$ favorite professors and each professor is the favourite of exactly $f$ students.
Hence, $pi_i (f)=sigma_j (f)= f$.
The answer to the second part is in the affirmative.
answered Dec 3 '18 at 18:14
Anubhab GhosalAnubhab Ghosal
81618
$endgroup$
Let $f(S_{j},p_{i})=begin{cases} 1 & S_{j} text{ likes }p_{i}\ 0 & text{otherwise}end{cases}$.$sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})leq nf$ as every student has at most $f$ favorite professors and $sumlimits_{j=1}^nsum limits_{i=1}^nf(S_{j},p_{i})geq nf$ as each professor is the favourite of at least $f$ students. Therefore, each student has exactly $f$ favorite professors and each professor is the favourite of exactly $f$ students.
Hence, $pi_i (f)=sigma_j (f)= f$.
The answer to the second part is in the affirmative.
answered Dec 3 '18 at 18:14
Anubhab GhosalAnubhab Ghosal
81618
answered Dec 3 '18 at 18:14
Anubhab GhosalAnubhab Ghosal
81618
answered Dec 3 '18 at 18:14
Anubhab GhosalAnubhab Ghosal
81618
answered Dec 3 '18 at 18:14
Anubhab GhosalAnubhab Ghosal
81618
81618
add a comment |
add a comment |
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