Identifier
Values
[1,0,1,0,1,0] => [1,2,3] => [.,[.,[.,.]]] => ([(0,2),(2,1)],3) => 2
[1,0,1,1,0,0] => [1,3,2] => [.,[[.,.],.]] => ([(0,2),(2,1)],3) => 2
[1,1,0,1,0,0] => [2,3,1] => [[.,[.,.]],.] => ([(0,2),(2,1)],3) => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => ([(0,3),(2,1),(3,2)],4) => 3
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [.,[.,[[.,.],.]]] => ([(0,3),(2,1),(3,2)],4) => 3
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [.,[[.,[.,.]],.]] => ([(0,3),(2,1),(3,2)],4) => 3
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [[.,[.,[.,.]]],.] => ([(0,3),(2,1),(3,2)],4) => 3
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => [.,[[.,[.,[.,[.,[.,.]]]]],.]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6
search for individual values
searching the database for the individual values of this statistic
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
to poset
Description
Return the poset obtained by interpreting the tree as a Hasse diagram.