Expected number of parts of a uniformly selected partition of $n$
up vote
0
down vote
favorite
I have a very basic question on partition theory, which I feel should be very well known. Suppose that you fix a natural number $n$ and select a random partition $P$ of $n$ by choosing uniformly from the set of all partitions of $n$ (so that each partition has probability $frac{1}{p(n)}$ of being selected, where $p(n)$ stands for the number of partitions of $n$). My question is that, what is the best known estimate/approximation of the expected number of parts of $P$ and the length of the largest part of $P$?
Since the there is an inherent symmetry between the number of parts and the length of the largest part of a partition (which is more clear if one views the Young's diagram), and since the area of the Young's diagram is $n$, I suspect that the expectation is roughly of the order of $sqrt{n}$, although this is a heuristic guess. Can anyone give me the best known estimate?
combinatorics number-theory probability-theory integer-partitions
asked Nov 19 at 4:08
abcd
817
add a comment |
up vote
0
down vote
favorite
I have a very basic question on partition theory, which I feel should be very well known. Suppose that you fix a natural number $n$ and select a random partition $P$ of $n$ by choosing uniformly from the set of all partitions of $n$ (so that each partition has probability $frac{1}{p(n)}$ of being selected, where $p(n)$ stands for the number of partitions of $n$). My question is that, what is the best known estimate/approximation of the expected number of parts of $P$ and the length of the largest part of $P$?
Since the there is an inherent symmetry between the number of parts and the length of the largest part of a partition (which is more clear if one views the Young's diagram), and since the area of the Young's diagram is $n$, I suspect that the expectation is roughly of the order of $sqrt{n}$, although this is a heuristic guess. Can anyone give me the best known estimate?
combinatorics number-theory probability-theory integer-partitions
asked Nov 19 at 4:08
abcd
817
1
An asymptotic formula is known: $sqrt{frac{3n}{2pi}}(log n+2gamma-logtfrac{pi}6)+{rm{O}}(log^3 n)$. Source: resolver.sub.uni-goettingen.de/purl?PPN362162050_0081
– Andrew Woods
Nov 19 at 7:36
Thanks, Andrew!
– abcd
Nov 19 at 20:54
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a very basic question on partition theory, which I feel should be very well known. Suppose that you fix a natural number $n$ and select a random partition $P$ of $n$ by choosing uniformly from the set of all partitions of $n$ (so that each partition has probability $frac{1}{p(n)}$ of being selected, where $p(n)$ stands for the number of partitions of $n$). My question is that, what is the best known estimate/approximation of the expected number of parts of $P$ and the length of the largest part of $P$?
Since the there is an inherent symmetry between the number of parts and the length of the largest part of a partition (which is more clear if one views the Young's diagram), and since the area of the Young's diagram is $n$, I suspect that the expectation is roughly of the order of $sqrt{n}$, although this is a heuristic guess. Can anyone give me the best known estimate?
combinatorics number-theory probability-theory integer-partitions
asked Nov 19 at 4:08
abcd
817
I have a very basic question on partition theory, which I feel should be very well known. Suppose that you fix a natural number $n$ and select a random partition $P$ of $n$ by choosing uniformly from the set of all partitions of $n$ (so that each partition has probability $frac{1}{p(n)}$ of being selected, where $p(n)$ stands for the number of partitions of $n$). My question is that, what is the best known estimate/approximation of the expected number of parts of $P$ and the length of the largest part of $P$?
Since the there is an inherent symmetry between the number of parts and the length of the largest part of a partition (which is more clear if one views the Young's diagram), and since the area of the Young's diagram is $n$, I suspect that the expectation is roughly of the order of $sqrt{n}$, although this is a heuristic guess. Can anyone give me the best known estimate?
combinatorics number-theory probability-theory integer-partitions
combinatorics number-theory probability-theory integer-partitions
asked Nov 19 at 4:08
abcd
817
asked Nov 19 at 4:08
abcd
817
asked Nov 19 at 4:08
abcd
817
asked Nov 19 at 4:08
abcd
817
asked Nov 19 at 4:08
abcd
817
817
1
An asymptotic formula is known: $sqrt{frac{3n}{2pi}}(log n+2gamma-logtfrac{pi}6)+{rm{O}}(log^3 n)$. Source: resolver.sub.uni-goettingen.de/purl?PPN362162050_0081
– Andrew Woods
Nov 19 at 7:36
Thanks, Andrew!
– abcd
Nov 19 at 20:54
add a comment |
1
An asymptotic formula is known: $sqrt{frac{3n}{2pi}}(log n+2gamma-logtfrac{pi}6)+{rm{O}}(log^3 n)$. Source: resolver.sub.uni-goettingen.de/purl?PPN362162050_0081
– Andrew Woods
Nov 19 at 7:36
Thanks, Andrew!
– abcd
Nov 19 at 20:54
1
1
An asymptotic formula is known: $sqrt{frac{3n}{2pi}}(log n+2gamma-logtfrac{pi}6)+{rm{O}}(log^3 n)$. Source: resolver.sub.uni-goettingen.de/purl?PPN362162050_0081
– Andrew Woods
Nov 19 at 7:36
An asymptotic formula is known: $sqrt{frac{3n}{2pi}}(log n+2gamma-logtfrac{pi}6)+{rm{O}}(log^3 n)$. Source: resolver.sub.uni-goettingen.de/purl?PPN362162050_0081
– Andrew Woods
Nov 19 at 7:36
Thanks, Andrew!
– abcd
Nov 19 at 20:54
Thanks, Andrew!
– abcd
Nov 19 at 20:54
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004510%2fexpected-number-of-parts-of-a-uniformly-selected-partition-of-n%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
An asymptotic formula is known: $sqrt{frac{3n}{2pi}}(log n+2gamma-logtfrac{pi}6)+{rm{O}}(log^3 n)$. Source: resolver.sub.uni-goettingen.de/purl?PPN362162050_0081
– Andrew Woods
Nov 19 at 7:36
Thanks, Andrew!
– abcd
Nov 19 at 20:54