- FindStat

Identifier

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000914: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Map

subcrystal

Description

The underlying poset of the subcrystal obtained by applying the raising operators to a semistandard tableau.