- FindStat

Identifier

Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001617: Lattices ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying
$$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient Mp00196The modular quotient of a lattice..
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.

Map

skew partition

Description

The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.

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.