Identifier
-
Mp00100:
Dyck paths
—touch composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[1,0,1,0,1,0] => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 3
[1,1,0,1,0,0] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 3
[1,1,1,0,0,0] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 3
[1,0,1,0,1,0,1,0] => [1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,0,1,0,1,0,0] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,0,1,1,0,0,0] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,0,0,1,0,0] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,0,1,0,0,0] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,1,1,1,0,0,0,0] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,0,1,0,1,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,0,1,1,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,1,0,0,1,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,1,0,1,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,0,1,1,1,0,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,0,1,0,1,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,0,1,1,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,1,0,0,1,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,1,0,1,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,0,1,1,0,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,0,0,0,1,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,0,0,1,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,0,1,0,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,1,1,1,1,0,0,0,0,0] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,0,1,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,0,1,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,1,0,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,1,0,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,0,1,1,1,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,0,0,1,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,0,0,1,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,0,1,0,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,0,1,0,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,0,1,1,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,1,0,0,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,1,0,0,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,1,0,1,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,0,1,1,1,1,0,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,1,0,1,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,1,0,1,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,1,1,0,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,1,1,0,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,0,1,1,1,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,0,1,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,0,1,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,1,0,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,1,0,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,0,1,1,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,1,0,0,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,1,0,0,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,1,0,1,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,0,1,1,1,0,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,0,1,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,0,1,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,1,0,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,1,0,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,0,1,1,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,1,0,0,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,1,0,0,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,1,0,1,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,0,1,1,0,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,0,0,0,0,1,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,0,0,0,1,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,0,0,1,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,0,1,0,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,1,1,1,1,1,0,0,0,0,0,0] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,1,0,1,1,0,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,1,0,0,1,0,1,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
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Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Map
touch composition
Description
Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.
Map
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
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