# 2020-15 The number of cycles of fixed lengths in random permutations

Let $$m_0=n$$. For each $$i\geq 0$$, choose a number $$x$$ in $$\{1,\dots, m_i\}$$ uniformly at random and let $$m_{i+1}= m_i – x$$. This gives a random vector $$\mathbf{m}=(m_1,m_2, \dots)$$. For each $$1\leq k\leq n$$, let $$X_k$$ be the number of occurrences of $$k$$ in the vector $$\mathbf{m}$$.

For each $$1\leq k\leq n$$, let $$Y_k$$ be the number of cycles of length $$k$$ in a permutation of $$\{1,\dots, n\}$$ chosen uniformly at random. Prove that $$X_k$$ and $$Y_k$$ have the same distribution.

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# Solution: 2020-14 Connecting dots probabilistically

Say there are n points. For each pair of points, we add an edge with probability 1/3. Let $$P_n$$ be the probability of the resulting graph to be connected (meaning any two vertices can be joined by an edge path). What can you say about the limit of $$P_n$$ as n tends to infinity?

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-14.

Other solutions were submitted by 강한필 (전산학부 2016학번, +3), 김건우 (수리과학과 2017학번, +3), 이준호 (수리과학과 2016학번, +3), 김유일 (2020학번, +3).

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# 2020-14 Connecting dots probabilistically

Say there are n points. For each pair of points, we add an edge with probability 1/3. Let $$P_n$$ be the probability of the resulting graph to be connected (meaning any two vertices can be joined by an edge path). What can you say about the limit of $$P_n$$ as n tends to infinity?

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# Notice on POW 2020-13

POW 2020-13 is still open and anyone who first submits a correct solution will get the full credit.

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# 2020-13 An integral sequence

Let $$a_n$$ be a sequence defined recursively by $$a_0 = a_1 = \dots = a_5 = 1$$ and
$a_n = \frac{a_{n-1} a_{n-5} + a_{n-2} a_{n-4} + a_{n-3}^2}{a_{n-6}}$
for $$n \geq 6$$. Prove or disprove that every $$a_n$$ is an integer.

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# Solution: 2020-12 Draws on a chess tournament

There are $$n$$ people participating to a chess tournament and every two players play exactly one game against each other. The winner receives $$1$$ point and the loser gets $$0$$ point and if the game is a draw, each player receives $$0.5$$ points. Prove that if at least $$3/4$$ of the games are draws, then there are two players with the same total scores.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-12.

Another solution was submitted by 고성훈 (수리과학과 2018학번, +3).

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# Solution: 2020-11 Free group of rank 2

Show that there is a subgroup of a free group of ran 2 that is not finitely generated.

The best solution was submitted by 채지석 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-11.

Other solutions were submitted by 조한슬 (수리과학과 2017학번, +3), 최백규(생명과학과 대학원, +2).

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# 2020-12 Draws on a chess tournament

There are $$n$$ people participating to a chess tournament and every two players play exactly one game against each other. The winner receives $$1$$ point and the loser gets $$0$$ point and if the game is a draw, each player receives $$0.5$$ points. Prove that if at least $$3/4$$ of the games are draws, then there are two players with the same total scores.

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# Notice on POW 2020-11

POW 2020-11 is still open and anyone who first submits a correct solution will get the full credit.

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