Identifier
Values
[1] => [1] => [1,0] => [1,0] => 0
[1,1] => [2] => [1,1,0,0] => [1,1,0,0] => 0
[2] => [1,1] => [1,0,1,0] => [1,0,1,0] => 0
[1,1,1] => [3] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => 0
[1,2] => [1,2] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 0
[2,1] => [2,1] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 0
[3] => [1,1,1] => [1,0,1,0,1,0] => [1,0,1,0,1,0] => 0
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 0
[1,1,2] => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 0
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 1
[1,3] => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => 0
[2,1,1] => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 0
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 0
[3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 0
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 0
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 0
[1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 1
[1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 0
[1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 0
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 1
[1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => 0
[2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 0
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 0
[2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 0
[2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => 0
[3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 0
[3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0] => 0
[4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 0
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,1,2] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 0
[1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1
[1,1,1,3] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 0
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 2
[1,1,2,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 0
[1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 1
[1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 0
[1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 1
[1,2,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 1
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 1
[1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => 0
[1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,1,0,0] => 0
[1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => 0
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => 1
[1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0
[2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 0
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 0
[2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 0
[2,1,3] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 0
[2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 1
[2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 0
[2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => 2
[2,4] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 0
[3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 0
[3,1,2] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 0
[3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 0
[3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 0
[4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 0
[4,2] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 0
[5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
[6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
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Description
The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
conjugate
Description
The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.
Map
Cori-Le Borgne involution
Description
The Cori-Le Borgne involution on Dyck paths.
Append an additional down step to the Dyck path and consider its (literal) reversal. The image of the involution is then the unique rotation of this word which is a Dyck word followed by an additional down step. Alternatively, it is the composite $\zeta\circ\mathrm{rev}\circ\zeta^{(-1)}$, where $\zeta$ is Mp00030zeta map.