Identifier
Values
[1,0] => [1,0] => [1,1,0,0] => [1,1,0,0] => 2
[1,0,1,0] => [1,1,0,0] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => 3
[1,1,0,0] => [1,0,1,0] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 4
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 4
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => 5
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => 5
[1,1,0,1,0,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 5
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => 7
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 5
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 6
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 6
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 6
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 8
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 7
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 6
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 6
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 7
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 8
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => 8
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 8
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 11
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 7
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 7
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 7
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 9
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 7
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 8
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 7
[1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 7
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 9
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => 9
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 8
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 7
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 9
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 9
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 8
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 10
[1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 7
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 7
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 7
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => 8
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 9
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 8
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 8
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 9
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 10
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 9
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 9
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 10
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 9
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 9
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 9
[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 10
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 11
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => 12
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 12
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 12
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 16
[] => [] => [1,0] => [1,0] => 1
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Description
Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Map
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
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.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.