Identifier
Values
[1] => ([],1) => ([],1) => 1
[1,2] => ([],2) => ([],1) => 1
[2,1] => ([(0,1)],2) => ([(0,1)],2) => 1
[1,2,3] => ([],3) => ([],1) => 1
[1,3,2] => ([(1,2)],3) => ([(0,1)],2) => 1
[2,1,3] => ([(1,2)],3) => ([(0,1)],2) => 1
[2,3,1] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,2] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,2,3,4] => ([],4) => ([],1) => 1
[1,2,4,3] => ([(2,3)],4) => ([(0,1)],2) => 1
[1,3,2,4] => ([(2,3)],4) => ([(0,1)],2) => 1
[1,3,4,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,4,2,3] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[2,1,3,4] => ([(2,3)],4) => ([(0,1)],2) => 1
[2,1,4,3] => ([(0,3),(1,2)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1,4] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[3,1,2,4] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,2,3,4,5] => ([],5) => ([],1) => 1
[1,2,3,5,4] => ([(3,4)],5) => ([(0,1)],2) => 1
[1,2,4,3,5] => ([(3,4)],5) => ([(0,1)],2) => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,3,2,4,5] => ([(3,4)],5) => ([(0,1)],2) => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,4,2,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,4,2,3,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,1,3,4,5] => ([(3,4)],5) => ([(0,1)],2) => 1
[2,1,3,5,4] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,4,3,5] => ([(1,4),(2,3)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,3,1,4,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[3,1,2,4,5] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,3,4,6,5] => ([(4,5)],6) => ([(0,1)],2) => 1
[1,2,3,5,4,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,2,4,3,5,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,4,5,3,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,3,2,4,5,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,2,5,4,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,3,4,2,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,4,2,3,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
[1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,1,3,4,5,6] => ([(4,5)],6) => ([(0,1)],2) => 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,3,5,4,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,1,4,3,5,6] => ([(2,5),(3,4)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,3,1,4,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[3,1,2,4,5,6] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[3,1,2,5,4,6] => ([(1,2),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 1
[3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1
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Description
The size of the image of the pop stack sorting operator.
The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
Map
connected vertex partitions
Description
Sends a graph to the lattice of its connected vertex partitions.
A connected vertex partition of a graph $G = (V,E)$ is a set partition of $V$ such that each part induced a connected subgraph of $G$. The connected vertex partitions of $G$ form a lattice under refinement. If $G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on $n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.
A connected vertex partition of a graph $G = (V,E)$ is a set partition of $V$ such that each part induced a connected subgraph of $G$. The connected vertex partitions of $G$ form a lattice under refinement. If $G = K_n$ is a complete graph, the resulting lattice is the lattice of set partitions on $n$ elements.
In the language of matroid theory, this map sends a graph to the lattice of flats of its graphic matroid. The resulting lattice is a geometric lattice, i.e. it is atomistic and semimodular.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
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