Identifier
Values
['A',1] => ([],1) => [1] => 1
['A',2] => ([(0,2),(1,2)],3) => [2,1] => 0
['B',2] => ([(0,3),(1,3),(3,2)],4) => [3,1] => 1
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [5,1] => 1
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => [3,2,1] => 0
['B',3] => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9) => [5,3,1] => 1
['C',3] => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9) => [5,3,1] => 1
['A',4] => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10) => [4,3,2,1] => 0
['D',4] => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12) => [5,3,3,1] => 4
['A',5] => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15) => [5,4,3,2,1] => 0
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
click to show known generating functions       
Description
The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions.
For example, $p_{22} = s_{1111} - s_{211} + 2s_{22} - s_{31} + s_4$, so the statistic on the partition $22$ is 2.
This is also the sum of the character values at the given conjugacy class over all irreducible characters of the symmetric group. [2]
For a permutation $\pi$ of given cycle type, this is also the number of permutations whose square equals $\pi$. [2]
Map
to root poset
Description
The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where $\alpha \prec \beta$ if $\beta - \alpha$ is a simple root.
Map
Greene-Kleitman invariant
Description
The Greene-Kleitman invariant of a poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.