Counting inversions of random elements in coxeter groups The 2019 Stack Overflow Developer Survey Results Are InRecurrence representation(s): $a(n+1)=a(n)(n-1/2)+o(1/n)$ and $a(n+1)=a(n)(n-1/2+o(1/n))$Finite/Infinite Coxeter GroupsA question on Coxeter groupsProof about coxeter groupsGroups - InversionsA question about Coxeter groups.Number of conjugacy classes or bounds for the index of stabilizers on finite Coxeter groups.Reference request: Inversions and $sum_win Wq^ell(w)$ for arbitrary Coxeter groupsThe ratio of Raw Maxima of Mahonian numbers in terms of groupsCoxeter length in the symmetric group equals number of inversions
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Counting inversions of random elements in coxeter groups
The 2019 Stack Overflow Developer Survey Results Are InRecurrence representation(s): $a(n+1)=a(n)(n-1/2)+o(1/n)$ and $a(n+1)=a(n)(n-1/2+o(1/n))$Finite/Infinite Coxeter GroupsA question on Coxeter groupsProof about coxeter groupsGroups - InversionsA question about Coxeter groups.Number of conjugacy classes or bounds for the index of stabilizers on finite Coxeter groups.Reference request: Inversions and $sum_win Wq^ell(w)$ for arbitrary Coxeter groupsThe ratio of Raw Maxima of Mahonian numbers in terms of groupsCoxeter length in the symmetric group equals number of inversions
$begingroup$
I am trying to find a general interperetation to the following facts (pls be patient to read it).
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian numbers $T(n,k)$ (the number of permutations of 1..n with k inversions).
According to Richard Stanley $$ left| Pleft( fracmathrminv(pi)-frac 12nchoose 2sqrtn(n-1)(2n+5)/72leq xright)-Phi(x)right| leq fracCsqrtn, $$
where $Phi(x)$ denotes the standard normal distribution. From this it is immediate that $M(n+1)/M(n)=n-frac 12+o(1)$Looking at combinatorial proof for the property of Kendall-Mann numbers numbers at MO
$M(n) approx c n!/n^3/2$ and
$$fracM(n+1)M(n) approx frac(n+1)(n+1)^-3/2n^-3/2 = n (1+1/n)^-1/2 approx n-1/2.$$
This is pretty the same result as #1.Reading through Counting inversions and descents of random elements in finite Coxeter groups I noticed Corollary 3.2 (page 6 of the article) that mean and variance of W-Mahonian distribution depend on the types of groups, i.e. $A_n$, $B_n$, $D_n$. By and large it's about $n^3/2$ like for #1 and #2.
This results in the similar 'structure': $approx n-1/2$ .
So I wonder why? I am looking for a general explanation to the facts.
I guess this is due to Cayley theorem but I need a more detailed explanation.
combinatorics group-theory finite-groups coxeter-groups permutation-cycles
asked Apr 5 at 20:20
Mikhail GaichenkovMikhail Gaichenkov
6610
$endgroup$
add a comment |
$begingroup$
I am trying to find a general interperetation to the following facts (pls be patient to read it).
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian numbers $T(n,k)$ (the number of permutations of 1..n with k inversions).
According to Richard Stanley $$ left| Pleft( fracmathrminv(pi)-frac 12nchoose 2sqrtn(n-1)(2n+5)/72leq xright)-Phi(x)right| leq fracCsqrtn, $$
where $Phi(x)$ denotes the standard normal distribution. From this it is immediate that $M(n+1)/M(n)=n-frac 12+o(1)$Looking at combinatorial proof for the property of Kendall-Mann numbers numbers at MO
$M(n) approx c n!/n^3/2$ and
$$fracM(n+1)M(n) approx frac(n+1)(n+1)^-3/2n^-3/2 = n (1+1/n)^-1/2 approx n-1/2.$$
This is pretty the same result as #1.Reading through Counting inversions and descents of random elements in finite Coxeter groups I noticed Corollary 3.2 (page 6 of the article) that mean and variance of W-Mahonian distribution depend on the types of groups, i.e. $A_n$, $B_n$, $D_n$. By and large it's about $n^3/2$ like for #1 and #2.
This results in the similar 'structure': $approx n-1/2$ .
So I wonder why? I am looking for a general explanation to the facts.
I guess this is due to Cayley theorem but I need a more detailed explanation.
combinatorics group-theory finite-groups coxeter-groups permutation-cycles
asked Apr 5 at 20:20
Mikhail GaichenkovMikhail Gaichenkov
6610
$endgroup$
add a comment |
$begingroup$
I am trying to find a general interperetation to the following facts (pls be patient to read it).
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian numbers $T(n,k)$ (the number of permutations of 1..n with k inversions).
According to Richard Stanley $$ left| Pleft( fracmathrminv(pi)-frac 12nchoose 2sqrtn(n-1)(2n+5)/72leq xright)-Phi(x)right| leq fracCsqrtn, $$
where $Phi(x)$ denotes the standard normal distribution. From this it is immediate that $M(n+1)/M(n)=n-frac 12+o(1)$Looking at combinatorial proof for the property of Kendall-Mann numbers numbers at MO
$M(n) approx c n!/n^3/2$ and
$$fracM(n+1)M(n) approx frac(n+1)(n+1)^-3/2n^-3/2 = n (1+1/n)^-1/2 approx n-1/2.$$
This is pretty the same result as #1.Reading through Counting inversions and descents of random elements in finite Coxeter groups I noticed Corollary 3.2 (page 6 of the article) that mean and variance of W-Mahonian distribution depend on the types of groups, i.e. $A_n$, $B_n$, $D_n$. By and large it's about $n^3/2$ like for #1 and #2.
This results in the similar 'structure': $approx n-1/2$ .
So I wonder why? I am looking for a general explanation to the facts.
I guess this is due to Cayley theorem but I need a more detailed explanation.
combinatorics group-theory finite-groups coxeter-groups permutation-cycles
asked Apr 5 at 20:20
Mikhail GaichenkovMikhail Gaichenkov
6610
$endgroup$
I am trying to find a general interperetation to the following facts (pls be patient to read it).
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian numbers $T(n,k)$ (the number of permutations of 1..n with k inversions).
According to Richard Stanley $$ left| Pleft( fracmathrminv(pi)-frac 12nchoose 2sqrtn(n-1)(2n+5)/72leq xright)-Phi(x)right| leq fracCsqrtn, $$
where $Phi(x)$ denotes the standard normal distribution. From this it is immediate that $M(n+1)/M(n)=n-frac 12+o(1)$Looking at combinatorial proof for the property of Kendall-Mann numbers numbers at MO
$M(n) approx c n!/n^3/2$ and
$$fracM(n+1)M(n) approx frac(n+1)(n+1)^-3/2n^-3/2 = n (1+1/n)^-1/2 approx n-1/2.$$
This is pretty the same result as #1.Reading through Counting inversions and descents of random elements in finite Coxeter groups I noticed Corollary 3.2 (page 6 of the article) that mean and variance of W-Mahonian distribution depend on the types of groups, i.e. $A_n$, $B_n$, $D_n$. By and large it's about $n^3/2$ like for #1 and #2.
This results in the similar 'structure': $approx n-1/2$ .
So I wonder why? I am looking for a general explanation to the facts.
I guess this is due to Cayley theorem but I need a more detailed explanation.
combinatorics group-theory finite-groups coxeter-groups permutation-cycles
combinatorics group-theory finite-groups coxeter-groups permutation-cycles
asked Apr 5 at 20:20
Mikhail GaichenkovMikhail Gaichenkov
6610
asked Apr 5 at 20:20
Mikhail GaichenkovMikhail Gaichenkov
6610
edited Apr 6 at 19:27
Mikhail Gaichenkov
asked Apr 5 at 20:20
Mikhail GaichenkovMikhail Gaichenkov
6610
asked Apr 5 at 20:20
Mikhail GaichenkovMikhail Gaichenkov
6610
asked Apr 5 at 20:20
Mikhail GaichenkovMikhail Gaichenkov
6610
6610
add a comment |
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