- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001964: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,0,1,0,1,0,1,0,1,0] => [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

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

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

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

[1,1,0,1,0,0,1,0,1,0,1,0] => [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

[1,1,0,1,0,1,0,0,1,0,1,0] => [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

[1,1,0,1,0,1,0,1,0,0,1,0] => [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

[1,1,0,1,0,1,0,1,0,1,0,0] => [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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5$

$F_{4} = 14$

$F_{5} = 42$

Description

The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

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

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.

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.