- FindStat

Identifier

Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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search for individual values

searching the database for the individual values of this statistic

Description

The number of simple modules with projective dimension at most 1.

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

to touch composition

Description

Sends a parking function to the composition of its labelled Dyck path.