Identifier
Values
=>
Cc0002;cc-rep
[2]=>1
[1,1]=>1
[3]=>1
[2,1]=>2
[1,1,1]=>1
[4]=>1
[3,1]=>3
[2,2]=>1
[2,1,1]=>3
[1,1,1,1]=>1
[5]=>1
[4,1]=>4
[3,2]=>2
[3,1,1]=>6
[2,2,1]=>3
[2,1,1,1]=>4
[1,1,1,1,1]=>1
[6]=>1
[5,1]=>5
[4,2]=>3
[4,1,1]=>10
[3,3]=>1
[3,2,1]=>8
[3,1,1,1]=>10
[2,2,2]=>1
[2,2,1,1]=>6
[2,1,1,1,1]=>5
[1,1,1,1,1,1]=>1
[7]=>1
[6,1]=>6
[5,2]=>4
[5,1,1]=>15
[4,3]=>2
[4,2,1]=>15
[4,1,1,1]=>20
[3,3,1]=>6
[3,2,2]=>3
[3,2,1,1]=>20
[3,1,1,1,1]=>15
[2,2,2,1]=>4
[2,2,1,1,1]=>10
[2,1,1,1,1,1]=>6
[1,1,1,1,1,1,1]=>1
[8]=>1
[7,1]=>7
[6,2]=>5
[6,1,1]=>21
[5,3]=>3
[5,2,1]=>24
[5,1,1,1]=>35
[4,4]=>1
[4,3,1]=>15
[4,2,2]=>6
[4,2,1,1]=>45
[4,1,1,1,1]=>35
[3,3,2]=>3
[3,3,1,1]=>20
[3,2,2,1]=>15
[3,2,1,1,1]=>40
[3,1,1,1,1,1]=>21
[2,2,2,2]=>1
[2,2,2,1,1]=>10
[2,2,1,1,1,1]=>15
[2,1,1,1,1,1,1]=>7
[1,1,1,1,1,1,1,1]=>1
[9]=>1
[8,1]=>8
[7,2]=>6
[7,1,1]=>28
[6,3]=>4
[6,2,1]=>35
[6,1,1,1]=>56
[5,4]=>2
[5,3,1]=>27
[5,2,2]=>10
[5,2,1,1]=>84
[5,1,1,1,1]=>70
[4,4,1]=>10
[4,3,2]=>8
[4,3,1,1]=>60
[4,2,2,1]=>36
[4,2,1,1,1]=>105
[4,1,1,1,1,1]=>56
[3,3,3]=>1
[3,3,2,1]=>20
[3,3,1,1,1]=>50
[3,2,2,2]=>4
[3,2,2,1,1]=>45
[3,2,1,1,1,1]=>70
[3,1,1,1,1,1,1]=>28
[2,2,2,2,1]=>5
[2,2,2,1,1,1]=>20
[2,2,1,1,1,1,1]=>21
[2,1,1,1,1,1,1,1]=>8
[1,1,1,1,1,1,1,1,1]=>1
[10]=>1
[9,1]=>9
[8,2]=>7
[8,1,1]=>36
[7,3]=>5
[7,2,1]=>48
[7,1,1,1]=>84
[6,4]=>3
[6,3,1]=>42
[6,2,2]=>15
[6,2,1,1]=>140
[6,1,1,1,1]=>126
[5,5]=>1
[5,4,1]=>24
[5,3,2]=>15
[5,3,1,1]=>126
[5,2,2,1]=>70
[5,2,1,1,1]=>224
[5,1,1,1,1,1]=>126
[4,4,2]=>6
[4,4,1,1]=>50
[4,3,3]=>3
[4,3,2,1]=>64
[4,3,1,1,1]=>175
[4,2,2,2]=>10
[4,2,2,1,1]=>126
[4,2,1,1,1,1]=>210
[4,1,1,1,1,1,1]=>84
[3,3,3,1]=>10
[3,3,2,2]=>6
[3,3,2,1,1]=>75
[3,3,1,1,1,1]=>105
[3,2,2,2,1]=>24
[3,2,2,1,1,1]=>105
[3,2,1,1,1,1,1]=>112
[3,1,1,1,1,1,1,1]=>36
[2,2,2,2,2]=>1
[2,2,2,2,1,1]=>15
[2,2,2,1,1,1,1]=>35
[2,2,1,1,1,1,1,1]=>28
[2,1,1,1,1,1,1,1,1]=>9
[1,1,1,1,1,1,1,1,1,1]=>1
[11]=>1
[10,1]=>10
[9,2]=>8
[9,1,1]=>45
[8,3]=>6
[8,2,1]=>63
[8,1,1,1]=>120
[7,4]=>4
[7,3,1]=>60
[7,2,2]=>21
[7,2,1,1]=>216
[7,1,1,1,1]=>210
[6,5]=>2
[6,4,1]=>42
[6,3,2]=>24
[6,3,1,1]=>224
[6,2,2,1]=>120
[6,2,1,1,1]=>420
[6,1,1,1,1,1]=>252
[5,5,1]=>15
[5,4,2]=>15
[5,4,1,1]=>140
[5,3,3]=>6
[5,3,2,1]=>140
[5,3,1,1,1]=>420
[5,2,2,2]=>20
[5,2,2,1,1]=>280
[5,2,1,1,1,1]=>504
[5,1,1,1,1,1,1]=>210
[4,4,3]=>3
[4,4,2,1]=>60
[4,4,1,1,1]=>175
[4,3,3,1]=>36
[4,3,2,2]=>20
[4,3,2,1,1]=>280
[4,3,1,1,1,1]=>420
[4,2,2,2,1]=>70
[4,2,2,1,1,1]=>336
[4,2,1,1,1,1,1]=>378
[4,1,1,1,1,1,1,1]=>120
[3,3,3,2]=>4
[3,3,3,1,1]=>50
[3,3,2,2,1]=>45
[3,3,2,1,1,1]=>210
[3,3,1,1,1,1,1]=>196
[3,2,2,2,2]=>5
[3,2,2,2,1,1]=>84
[3,2,2,1,1,1,1]=>210
[3,2,1,1,1,1,1,1]=>168
[3,1,1,1,1,1,1,1,1]=>45
[2,2,2,2,2,1]=>6
[2,2,2,2,1,1,1]=>35
[2,2,2,1,1,1,1,1]=>56
[2,2,1,1,1,1,1,1,1]=>36
[2,1,1,1,1,1,1,1,1,1]=>10
[1,1,1,1,1,1,1,1,1,1,1]=>1
[12]=>1
[11,1]=>11
[10,2]=>9
[10,1,1]=>55
[9,3]=>7
[9,2,1]=>80
[9,1,1,1]=>165
[8,4]=>5
[8,3,1]=>81
[8,2,2]=>28
[8,2,1,1]=>315
[8,1,1,1,1]=>330
[7,5]=>3
[7,4,1]=>64
[7,3,2]=>35
[7,3,1,1]=>360
[7,2,2,1]=>189
[7,2,1,1,1]=>720
[7,1,1,1,1,1]=>462
[6,6]=>1
[6,5,1]=>35
[6,4,2]=>27
[6,4,1,1]=>280
[6,3,3]=>10
[6,3,2,1]=>256
[6,3,1,1,1]=>840
[6,2,2,2]=>35
[6,2,2,1,1]=>540
[6,2,1,1,1,1]=>1050
[6,1,1,1,1,1,1]=>462
[5,5,2]=>10
[5,5,1,1]=>105
[5,4,3]=>8
[5,4,2,1]=>175
[5,4,1,1,1]=>560
[5,3,3,1]=>84
[5,3,2,2]=>45
[5,3,2,1,1]=>700
[5,3,1,1,1,1]=>1134
[5,2,2,2,1]=>160
[5,2,2,1,1,1]=>840
[5,2,1,1,1,1,1]=>1008
[5,1,1,1,1,1,1,1]=>330
[4,4,4]=>1
[4,4,3,1]=>45
[4,4,2,2]=>20
[4,4,2,1,1]=>315
[4,4,1,1,1,1]=>490
[4,3,3,2]=>15
[4,3,3,1,1]=>210
[4,3,2,2,1]=>175
[4,3,2,1,1,1]=>896
[4,3,1,1,1,1,1]=>882
[4,2,2,2,2]=>15
[4,2,2,2,1,1]=>280
[4,2,2,1,1,1,1]=>756
[4,2,1,1,1,1,1,1]=>630
[4,1,1,1,1,1,1,1,1]=>165
[3,3,3,3]=>1
[3,3,3,2,1]=>40
[3,3,3,1,1,1]=>175
[3,3,2,2,2]=>10
[3,3,2,2,1,1]=>189
[3,3,2,1,1,1,1]=>490
[3,3,1,1,1,1,1,1]=>336
[3,2,2,2,2,1]=>35
[3,2,2,2,1,1,1]=>224
[3,2,2,1,1,1,1,1]=>378
[3,2,1,1,1,1,1,1,1]=>240
[3,1,1,1,1,1,1,1,1,1]=>55
[2,2,2,2,2,2]=>1
[2,2,2,2,2,1,1]=>21
[2,2,2,2,1,1,1,1]=>70
[2,2,2,1,1,1,1,1,1]=>84
[2,2,1,1,1,1,1,1,1,1]=>45
[2,1,1,1,1,1,1,1,1,1,1]=>11
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
References
[1] Fulton, W., Harris, J. Representation theory MathSciNet:1153249
Code
def statistic(L):
return SemistandardTableaux(shape=L,max_entry=len(L)).cardinality()
def statistic_alt1(L):
return prod( QQ(len(L)+j-i)/L.hook_length(i,j) for i,j in L.cells() )
def statistic_alt2(L):
return SymmetricFunctions(QQ).schur()(L).expand(len(L))([1]*len(L))
Created
Mar 07, 2017 at 09:15 by Christian Stump
Updated
Mar 07, 2017 at 09:15 by Christian Stump
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!