Identifier
Values
[1,3,2] => [2,1] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[1,3,2,4,5,6] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,2,3,5,6] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,5,2,3,4,6] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,6,2,3,4,5] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5,1,3,4,6] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,6,1,3,4,5] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,1,4,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[3,4,1,2,5,6] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,5,1,2,4,6] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,6,1,2,4,5] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,2,1,3,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[4,3,1,2,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[4,5,1,2,3,6] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,6,1,2,3,5] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[5,2,1,3,4,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[5,3,1,2,4,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[5,4,1,2,3,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[5,6,1,2,3,4] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[6,2,1,3,4,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[6,3,1,2,4,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[6,4,1,2,3,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[6,5,1,2,3,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[1,3,2,4,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,4,2,3,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,5,2,3,4,6,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,6,2,3,4,5,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,7,2,3,4,5,6] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,3,1,4,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,4,1,3,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,5,1,3,4,6,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,6,1,3,4,5,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[2,7,1,3,4,5,6] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,1,4,5,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[3,4,1,2,5,6,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,5,1,2,4,6,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,6,1,2,4,5,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,7,1,2,4,5,6] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,2,1,3,5,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[4,3,1,2,5,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[4,5,1,2,3,6,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,6,1,2,3,5,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[4,7,1,2,3,5,6] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[5,2,1,3,4,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[5,3,1,2,4,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[5,4,1,2,3,6,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[5,6,1,2,3,4,7] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[5,7,1,2,3,4,6] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[6,2,1,3,4,5,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[6,3,1,2,4,5,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[6,4,1,2,3,5,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[6,5,1,2,3,4,7] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[6,7,1,2,3,4,5] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[7,2,1,3,4,5,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[7,3,1,2,4,5,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[7,4,1,2,3,5,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[7,5,1,2,3,4,6] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
[7,6,1,2,3,4,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 2
search for individual values
searching the database for the individual values of this statistic
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
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.
Map
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.