Identifier
Values
['A',1] => ([],1) => [1] => 10 => 1
['A',2] => ([(0,2),(1,2)],3) => [2,1] => 1010 => 2
['B',2] => ([(0,3),(1,3),(3,2)],4) => [3,1] => 10010 => 1
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [5,1] => 1000010 => 1
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => [3,2,1] => 101010 => 3
['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] => 10010010 => 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] => 10010010 => 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] => 10101010 => 4
['B',4] => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16) => [7,5,3,1] => 10010010010 => 1
['C',4] => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16) => [7,5,3,1] => 10010010010 => 1
['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] => 100110010 => 1
['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] => 1010101010 => 5
['A',6] => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21) => [6,5,4,3,2,1] => 101010101010 => 6
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Description
The exponent of a binary word.
This is the largest number $e$ such that $w$ is the concatenation of $e$ identical factors. This statistic is also called frequency.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
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.