- FindStat

Identifier

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001878: Lattices ⟶ ℤ

Values

[1,2,2] => [[3,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[2,2,1] => [[3,3,2],[2,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,2,2] => [[3,2,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1

[1,2,3] => [[4,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,3,2] => [[4,3,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[2,2,2] => [[4,3,2],[2,1]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

[2,3,1] => [[4,4,2],[3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[3,2,1] => [[4,4,3],[3,2]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[3,3] => [[5,3],[2]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1

[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1

[1,1,2,3] => [[4,2,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,3,2] => [[4,3,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1

[1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1

[1,2,2,2] => [[4,3,2,1],[2,1]] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1

[1,2,3,1] => [[4,4,2,1],[3,1]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1

[1,2,4] => [[5,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,3,2,1] => [[4,4,3,1],[3,2]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1

[1,3,3] => [[5,3,1],[2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

[1,4,2] => [[5,4,1],[3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1

[2,1,2,2] => [[4,3,2,2],[2,1,1]] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

[2,1,3,1] => [[4,4,2,2],[3,1,1]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[2,2,1,2] => [[4,3,3,2],[2,2,1]] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

[2,2,2,1] => [[4,4,3,2],[3,2,1]] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 1

[2,2,3] => [[5,3,2],[2,1]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2

[2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,3,2] => [[5,4,2],[3,1]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2

[2,4,1] => [[5,5,2],[4,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[3,1,3] => [[5,3,3],[2,2]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[3,2,2] => [[5,4,3],[3,2]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2

[3,3,1] => [[5,5,3],[4,2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

[3,4] => [[6,3],[2]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[4,2,1] => [[5,5,4],[4,3]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[4,3] => [[6,4],[3]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 1

[1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1

[1,1,1,2,2,1] => [[3,3,2,1,1,1],[2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1

[1,1,1,3,2] => [[4,3,1,1,1],[2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2

[1,1,2,3,1] => [[4,4,2,1,1],[3,1]] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1

[1,1,3,1,2] => [[4,3,3,1,1],[2,2]] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2

[1,1,3,2,1] => [[4,4,3,1,1],[3,2]] => ([(0,5),(1,7),(2,6),(3,6),(4,3),(4,7),(5,1),(5,4),(7,2)],8) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1

[1,1,3,3] => [[5,3,1,1],[2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1

[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]] => ([(0,5),(1,7),(2,6),(3,6),(4,3),(4,7),(5,1),(5,4),(7,2)],8) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1

[1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1

[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]] => ([(0,5),(1,7),(2,6),(3,6),(4,3),(4,7),(5,1),(5,4),(7,2)],8) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1

[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]] => ([(0,5),(1,7),(2,6),(3,6),(4,3),(4,7),(5,1),(5,4),(7,2)],8) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 1

[1,3,1,3] => [[5,3,3,1],[2,2]] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2

[1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 1

[1,4,1,2] => [[5,4,4,1],[3,3]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[1,4,2,1] => [[5,5,4,1],[4,3]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1

[1,4,3] => [[6,4,1],[3]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2

[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2

[2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1

[2,1,2,3] => [[5,3,2,2],[2,1,1]] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2

[2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2

[2,1,3,2] => [[5,4,2,2],[3,1,1]] => ([(0,3),(0,5),(2,7),(3,6),(4,2),(4,6),(5,4),(6,7),(7,1)],8) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2

[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

[2,2,1,3] => [[5,3,3,2],[2,2,1]] => ([(0,3),(0,5),(2,7),(3,6),(4,2),(4,6),(5,4),(6,7),(7,1)],8) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2

[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,3,1,2] => [[5,4,4,2],[3,3,1]] => ([(0,3),(0,5),(2,7),(3,6),(4,2),(4,6),(5,4),(6,7),(7,1)],8) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2

[2,4,1,1] => [[5,5,5,2],[4,4,1]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[2,4,2] => [[6,5,2],[4,1]] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2

[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

[3,1,1,3] => [[5,3,3,3],[2,2,2]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1

[3,1,2,2] => [[5,4,3,3],[3,2,2]] => ([(0,3),(0,5),(2,7),(3,6),(4,2),(4,6),(5,4),(6,7),(7,1)],8) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 2

[3,1,3,1] => [[5,5,3,3],[4,2,2]] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2

[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1

[3,2,1,2] => [[5,4,4,3],[3,3,2]] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2

[3,3,1,1] => [[5,5,5,3],[4,4,2]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

[3,4,1] => [[6,6,3],[5,2]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2

[4,1,2,1] => [[5,5,4,4],[4,3,3]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2

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searching the database for the individual values of this statistic

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.

Map

dominating sublattice

Description

Return the sublattice of the dominance order induced by the support of the expansion of the skew Schur function into Schur functions.
Consider the expansion of the skew Schur function

$s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the subposet of the dominance order whose elements are the partitions

$\nu$ with

$c^\lambda_{\mu, \nu} > 0$ form a lattice.
The example

$\lambda = (5^2,4^2,1)$ and

$\mu=(3,2)$ shows that this lattice is not a sublattice of the dominance order.

Map

The modular quotient of a lattice.

Description

The modular quotient of a lattice.
This is the largest quotient of a lattice which is modular.