Identifier
Values
['A',1] => ([],1) => [2] => 0
['A',2] => ([(0,2),(1,2)],3) => [3,2] => 2
['B',2] => ([(0,3),(1,3),(3,2)],4) => [4,2] => 3
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => 5
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Description
The number of standard desarrangement tableaux of shape equal to the given partition.
A standard desarrangement tableau is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation).
This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also:
  • St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition.: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition
  • St000500Eigenvalues of the random-to-random operator acting on the regular representation.: Eigenvalues of the random-to-random operator acting on the regular representation.
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
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.