Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000914: Posets ⟶ ℤ
Values
[1,2,1] => [[2,2,1],[1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[2,2] => [[3,2],[1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,1,2,1] => [[2,2,1,1],[1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,2,2] => [[3,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
[1,3,1] => [[3,3,1],[2]] => ([(0,1)],2) => ([(0,1)],2) => 1
[2,1,2] => [[3,2,2],[1,1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[2,2,1] => [[3,3,2],[2,1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
[2,3] => [[4,2],[1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[3,2] => [[4,3],[2]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,1)],2) => ([(0,1)],2) => 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
[1,1,3,1] => [[3,3,1,1],[2]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,1)],2) => ([(0,1)],2) => 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) => 1
[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
[1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,1)],2) => ([(0,1)],2) => 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) => 1
[1,4,1] => [[4,4,1],[3]] => ([(0,1)],2) => ([(0,1)],2) => 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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) => 1
[2,1,3] => [[4,2,2],[1,1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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) => 1
[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) => 1
[2,4] => [[5,2],[1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[3,1,2] => [[4,3,3],[2,2]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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
[4,2] => [[5,4],[3]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => ([(0,1)],2) => ([(0,1)],2) => 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
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => ([(0,1)],2) => ([(0,1)],2) => 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) => 1
[1,1,4,1] => [[4,4,1,1],[3]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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) => 1
[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) => 1
[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
[1,2,3,1] => [[4,4,2,1],[3,1]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 1
[1,2,4] => [[5,2,1],[1]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 1
[1,3,2,1] => [[4,4,3,1],[3,2]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 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) => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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) => 1
[1,5,1] => [[5,5,1],[4]] => ([(0,1)],2) => ([(0,1)],2) => 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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) => 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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) => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 1
[2,1,4] => [[5,2,2],[1,1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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) => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],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) => 1
[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) => 1
[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) => 1
[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) => 1
[2,5] => [[6,2],[1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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) => 1
[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) => 1
[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) => 1
[3,4] => [[6,3],[2]] => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
[4,1,2] => [[5,4,4],[3,3]] => ([(0,1)],2) => ([(0,1)],2) => 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
[5,2] => [[6,5],[4]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => ([(0,1)],2) => ([(0,1)],2) => 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) => 1
[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) => 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) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],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) => 1
[1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]] => ([(0,1)],2) => ([(0,1)],2) => 1
[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) => 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
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 1
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,4,1,2] => [[5,4,4,1],[3,3]] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 1
[1,4,2,1] => [[5,5,4,1],[4,3]] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 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) => 1
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Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$ \sum_{x\leq y} \mu(x,y). $$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$ \sum_{x\leq y} \mu(x,y). $$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
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
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.
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 poset
Description
Return the poset corresponding to the lattice.
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