Permutation with local maxima
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I'm trying solve this problem:
Permutation of numbers $1$ to $n$ has a local maximum on $j^{th}$ position if number on $j^{th}$ position is bigger than both its neighbours. For first and last number of a permutation, local maximum exists if that number is bigger than its only neighbour. Knowing that every of $n!$ permutations is equally expected, calculate expected number of local maxima.
I tried defining $X$ (variable) with $X_i$ equal to number of local maxima on $i^{th}$ position, but couldn't continue past that.
probability probability-distributions permutations expected-value
asked Dec 28 '16 at 21:48
CrabzmaticCrabzmatic
302214
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add a comment |
$begingroup$
I'm trying solve this problem:
Permutation of numbers $1$ to $n$ has a local maximum on $j^{th}$ position if number on $j^{th}$ position is bigger than both its neighbours. For first and last number of a permutation, local maximum exists if that number is bigger than its only neighbour. Knowing that every of $n!$ permutations is equally expected, calculate expected number of local maxima.
I tried defining $X$ (variable) with $X_i$ equal to number of local maxima on $i^{th}$ position, but couldn't continue past that.
probability probability-distributions permutations expected-value
asked Dec 28 '16 at 21:48
CrabzmaticCrabzmatic
302214
$endgroup$
$begingroup$
Putnam 2006-A4 See kskedlaya.org/putnam-archive/2006s.pdf
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– Catalin Zara
Dec 28 '16 at 22:03
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Hint: instead of counting by permutations (add the number of local maxima for each permutations), count by positions (add the number of permutations that have a local maximum at a specific position).
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– Catalin Zara
Dec 28 '16 at 22:13
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@CatalinZara The link that you provided answered my question and I'm very thankful for that. If you want you can post it as an answer so I can accept it.
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– Crabzmatic
Dec 28 '16 at 22:27
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I'm happy that you found an answer to your question, but I don't think providing a link really qualifies as an answer.
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– Catalin Zara
Dec 28 '16 at 22:30
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@CatalinZara You can always rewrite the answer from the link.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:42
add a comment |
$begingroup$
I'm trying solve this problem:
Permutation of numbers $1$ to $n$ has a local maximum on $j^{th}$ position if number on $j^{th}$ position is bigger than both its neighbours. For first and last number of a permutation, local maximum exists if that number is bigger than its only neighbour. Knowing that every of $n!$ permutations is equally expected, calculate expected number of local maxima.
I tried defining $X$ (variable) with $X_i$ equal to number of local maxima on $i^{th}$ position, but couldn't continue past that.
probability probability-distributions permutations expected-value
asked Dec 28 '16 at 21:48
CrabzmaticCrabzmatic
302214
$endgroup$
I'm trying solve this problem:
Permutation of numbers $1$ to $n$ has a local maximum on $j^{th}$ position if number on $j^{th}$ position is bigger than both its neighbours. For first and last number of a permutation, local maximum exists if that number is bigger than its only neighbour. Knowing that every of $n!$ permutations is equally expected, calculate expected number of local maxima.
I tried defining $X$ (variable) with $X_i$ equal to number of local maxima on $i^{th}$ position, but couldn't continue past that.
probability probability-distributions permutations expected-value
probability probability-distributions permutations expected-value
asked Dec 28 '16 at 21:48
CrabzmaticCrabzmatic
302214
asked Dec 28 '16 at 21:48
CrabzmaticCrabzmatic
302214
edited Dec 4 '18 at 22:06
Crabzmatic
asked Dec 28 '16 at 21:48
CrabzmaticCrabzmatic
302214
asked Dec 28 '16 at 21:48
CrabzmaticCrabzmatic
302214
asked Dec 28 '16 at 21:48
CrabzmaticCrabzmatic
302214
302214
$begingroup$
Putnam 2006-A4 See kskedlaya.org/putnam-archive/2006s.pdf
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:03
$begingroup$
Hint: instead of counting by permutations (add the number of local maxima for each permutations), count by positions (add the number of permutations that have a local maximum at a specific position).
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:13
$begingroup$
@CatalinZara The link that you provided answered my question and I'm very thankful for that. If you want you can post it as an answer so I can accept it.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:27
$begingroup$
I'm happy that you found an answer to your question, but I don't think providing a link really qualifies as an answer.
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:30
$begingroup$
@CatalinZara You can always rewrite the answer from the link.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:42
add a comment |
$begingroup$
Putnam 2006-A4 See kskedlaya.org/putnam-archive/2006s.pdf
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:03
$begingroup$
Hint: instead of counting by permutations (add the number of local maxima for each permutations), count by positions (add the number of permutations that have a local maximum at a specific position).
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:13
$begingroup$
@CatalinZara The link that you provided answered my question and I'm very thankful for that. If you want you can post it as an answer so I can accept it.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:27
$begingroup$
I'm happy that you found an answer to your question, but I don't think providing a link really qualifies as an answer.
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:30
$begingroup$
@CatalinZara You can always rewrite the answer from the link.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:42
$begingroup$
Putnam 2006-A4 See kskedlaya.org/putnam-archive/2006s.pdf
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:03
$begingroup$
Putnam 2006-A4 See kskedlaya.org/putnam-archive/2006s.pdf
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:03
$begingroup$
Hint: instead of counting by permutations (add the number of local maxima for each permutations), count by positions (add the number of permutations that have a local maximum at a specific position).
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:13
$begingroup$
Hint: instead of counting by permutations (add the number of local maxima for each permutations), count by positions (add the number of permutations that have a local maximum at a specific position).
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:13
$begingroup$
@CatalinZara The link that you provided answered my question and I'm very thankful for that. If you want you can post it as an answer so I can accept it.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:27
$begingroup$
@CatalinZara The link that you provided answered my question and I'm very thankful for that. If you want you can post it as an answer so I can accept it.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:27
$begingroup$
I'm happy that you found an answer to your question, but I don't think providing a link really qualifies as an answer.
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:30
$begingroup$
I'm happy that you found an answer to your question, but I don't think providing a link really qualifies as an answer.
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:30
$begingroup$
@CatalinZara You can always rewrite the answer from the link.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:42
$begingroup$
@CatalinZara You can always rewrite the answer from the link.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:42
add a comment |
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$begingroup$
Putnam 2006-A4 See kskedlaya.org/putnam-archive/2006s.pdf
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:03
$begingroup$
Hint: instead of counting by permutations (add the number of local maxima for each permutations), count by positions (add the number of permutations that have a local maximum at a specific position).
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:13
$begingroup$
@CatalinZara The link that you provided answered my question and I'm very thankful for that. If you want you can post it as an answer so I can accept it.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:27
$begingroup$
I'm happy that you found an answer to your question, but I don't think providing a link really qualifies as an answer.
$endgroup$
– Catalin Zara
Dec 28 '16 at 22:30
$begingroup$
@CatalinZara You can always rewrite the answer from the link.
$endgroup$
– Crabzmatic
Dec 28 '16 at 22:42