Identifier
Values
[[2,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[2],[4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[3],[4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2],[4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,3],[4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,4],[3]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2,3],[2]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,3,3],[2]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2],[3,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[2,2],[3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1],[3],[5]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1],[4],[5]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2],[4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,3],[4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,4],[3]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1],[3,4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1],[4,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2],[2,4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,3],[2,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2],[2],[4]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => ([(0,1)],2) => 1
[[1,3],[2],[4]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,3],[3],[4]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,2,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1,3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2,3],[2]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,3,3],[2]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2],[3,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,2,2],[3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2,3],[2],[3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,3,3],[2],[3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1],[3],[5]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1],[4],[5]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2],[2],[5]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,2],[4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1,3],[4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,4],[3]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1],[3,4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1],[4,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2],[2,4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,3],[2,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,4],[2,3]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2],[2],[4]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => ([(0,1)],2) => 1
[[1,1,3],[2],[4]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,3],[3],[4]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2,2],[2],[4]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
[[1,2],[2,4],[3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,2],[3,3],[4]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,3],[2,4],[3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,1,2,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1,1,3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,2,3],[2]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1,3,3],[2]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,2],[3,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,2,3],[2,2]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,2,2],[3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,3,3],[2,2]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2,3],[2],[3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,3,3],[2],[3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2],[2,3,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,2,2],[2,3,3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1],[2],[4],[6]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1],[2],[5],[6]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1],[3],[5]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1],[4],[5]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2],[2],[5]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
[[1,1],[2,3],[5]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1],[2,4],[5]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1],[2,5],[4]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2],[2],[3],[5]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => ([(0,1)],2) => 1
[[1,2],[2],[4],[5]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
[[1,3],[2],[3],[5]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,1,2],[4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1,1,3],[4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,1,4],[3]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,1],[3,4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1,1],[4,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,2],[2,4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1,3],[2,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,4],[2,3]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1,2],[2],[4]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => ([(0,1)],2) => 1
[[1,1,1,3],[2],[4]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1,3],[3],[4]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2,2],[2],[4]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => 1
[[1,1,1],[2,3,4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,1],[2,4,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2],[2,2,4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,3],[2,2,4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,2],[2,2],[4]] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(2,3)],4) => ([(0,1)],2) => 1
[[1,1,2],[2,4],[3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,3],[2,2],[4]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,1,2],[3,3],[4]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,3],[2,3],[4]] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,3],[2,4],[3]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,1,4],[2,3],[3]] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
[[1,2,3],[2],[3],[4]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(3,4)],5) => ([(0,1)],2) => 1
[[1,3,3],[2],[3],[4]] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(4,5)],6) => ([(0,1)],2) => 1
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Description
The normalized isoperimetric number of a graph.
The isoperimetric number, or Cheeger constant, of a graph $G$ is
$$ i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\}, $$
where
$$ \partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}. $$
This statistic is $i(G)\cdot\lfloor n/2\rfloor$.
The isoperimetric number, or Cheeger constant, of a graph $G$ is
$$ i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\}, $$
where
$$ \partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}. $$
This statistic is $i(G)\cdot\lfloor n/2\rfloor$.
Map
subcrystal
Description
The underlying poset of the subcrystal obtained by applying the raising operators to a semistandard tableau.
Map
incomparability graph
Description
The incomparability graph of a poset.
Map
core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
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