Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤ
Values
[1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
[3] => [[3],[]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
[1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,1] => [[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4) => ([(0,2),(2,1)],3) => 1
[2,2] => [[3,2],[1]] => ([(0,3),(1,2),(1,3)],4) => ([(0,2),(2,1)],3) => 1
[4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,2] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => ([(0,2),(2,1)],3) => 1
[1,1,2,1] => [[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,1,3] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[1,4] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => ([(0,2),(2,1)],3) => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => 1
[2,3] => [[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,2),(2,1)],3) => 1
[5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,1,1,2] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
[1,1,1,3] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,4] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => ([(0,2),(2,1)],3) => 1
[1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => ([(0,3),(2,1),(3,2)],4) => 1
[1,3,2] => [[4,3,1],[2]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => ([(0,2),(2,1)],3) => 1
[1,5] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => ([(0,3),(2,1),(3,2)],4) => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => ([(0,3),(2,1),(3,2)],4) => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,2),(2,1)],3) => 1
[2,2,2] => [[4,3,2],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => ([(0,3),(2,1),(3,2)],4) => 1
[2,3,1] => [[4,4,2],[3,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,2),(2,1)],3) => 1
[2,4] => [[5,2],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[4,1,1] => [[4,4,4],[3,3]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[4,2] => [[5,4],[3]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[5,1] => [[5,5],[4]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => ([(0,3),(2,1),(3,2)],4) => 1
[6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[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) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) => ([(0,2),(2,1)],3) => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7) => ([(0,2),(2,1)],3) => 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
[1,1,2,3] => [[4,2,1,1],[1]] => ([(0,5),(0,6),(1,4),(1,6),(4,2),(5,3)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,3,2] => [[4,3,1,1],[2]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,4,1] => [[4,4,1,1],[3]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[1,2,4] => [[5,2,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) => ([(0,2),(2,1)],3) => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7) => ([(0,2),(2,1)],3) => 1
[1,3,3] => [[5,3,1],[2]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,2] => [[5,4,1],[3]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[1,5,1] => [[5,5,1],[4]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7) => ([(0,2),(2,1)],3) => 1
[1,6] => [[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7) => ([(0,2),(2,1)],3) => 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[2,1,4] => [[5,2,2],[1,1]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7) => ([(0,2),(2,1)],3) => 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7) => ([(0,2),(2,1)],3) => 1
[2,2,3] => [[5,3,2],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[2,4,1] => [[5,5,2],[4,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[3,1,3] => [[5,3,3],[2,2]] => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,2] => [[5,4,3],[3,2]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2
[3,3,1] => [[5,5,3],[4,2]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[4,1,2] => [[5,4,4],[3,3]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7) => ([(0,2),(2,1)],3) => 1
[4,2,1] => [[5,5,4],[4,3]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7) => ([(0,2),(2,1)],3) => 1
[6,1] => [[6,6],[5]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]] => ([(0,6),(0,7),(1,3),(1,7),(4,5),(5,2),(6,4)],8) => ([(0,3),(2,1),(3,2)],4) => 1
[1,7] => [[7,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => ([(0,3),(1,6),(2,6),(2,7),(3,5),(4,7),(5,4)],8) => ([(0,3),(2,1),(3,2)],4) => 1
[2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]] => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8) => ([(0,2),(2,1)],3) => 1
[2,3,2,1] => [[5,5,4,2],[4,3,1]] => ([(0,5),(1,6),(2,5),(2,7),(3,4),(3,6),(4,7)],8) => ([(0,2),(2,1)],3) => 1
[7,1] => [[7,7],[6]] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[1,8] => [[8,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[8,1] => [[8,8],[7]] => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
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Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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.
Map
antichains of maximal size
Description
The lattice of antichains of maximal size in a poset.
The set of antichains of maximal size can be ordered by setting $A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B$, where $\mathop{\downarrow} A$ is the order ideal generated by $A$.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
The set of antichains of maximal size can be ordered by setting $A \leq B \leftrightarrow \mathop{\downarrow} A \subseteq \mathop{\downarrow} B$, where $\mathop{\downarrow} A$ is the order ideal generated by $A$.
This is a sublattice of the lattice of all antichains with respect to the same order relation. In particular, it is distributive.
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.
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