Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000772
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00295: Standard tableaux valley compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000772: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => ([],1)
=> 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,3,2,4] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,5,4,2,3] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,5,3,1,4] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,5,4,1,3] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,1,4,2,5] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,5,2,4] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,4,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,5,1,4] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,4,2,1,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,4,5,1,2] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,5,2,1,4] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[3,5,4,1,2] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,1,3,2,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,1,5,2,3] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,2,3,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,2,5,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,3,5,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,5,2,1,3] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[4,5,3,1,2] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,1,3,2,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,1,4,2,3] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,2,3,1,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,2,4,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[5,3,4,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Mp00061: Permutations to increasing treeBinary trees
Mp00009: Binary trees left rotateBinary trees
Mp00013: Binary trees to posetPosets
St001879: Posets ⟶ ℤResult quality: 8% values known / values provided: 8%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [.,.]
=> ([],1)
=> ? = 1 + 1
[2,1,3] => [[.,.],[.,.]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 1
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 1
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 1
[4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 1
[4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 1
[4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 1
[5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 1
[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 1
[5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 1
[5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 4 + 1
[1,2,3,6,4,5] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 4 + 1
[1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 1
[1,2,4,6,3,5] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 1
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 1
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 1
[1,2,6,4,3,5] => [.,[.,[[[.,.],.],[.,.]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 1
[1,2,6,5,3,4] => [.,[.,[[[.,.],.],[.,.]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 1
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 1
[1,3,4,6,2,5] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 1
[1,3,5,4,2,6] => [.,[[.,[[.,.],.]],[.,.]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 1
[1,3,5,6,2,4] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 1
[1,3,6,4,2,5] => [.,[[.,[[.,.],.]],[.,.]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 1
[1,3,6,5,2,4] => [.,[[.,[[.,.],.]],[.,.]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 1
[1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 1 + 1
[1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 1 + 1
[1,4,5,3,2,6] => [.,[[[.,[.,.]],.],[.,.]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 1
[1,4,5,6,2,3] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 1
[1,4,6,3,2,5] => [.,[[[.,[.,.]],.],[.,.]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 1
[2,3,4,5,1,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,3,4,6,1,5] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,3,5,4,1,6] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,3,5,6,1,4] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,3,6,4,1,5] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,3,6,5,1,4] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,4,5,3,1,6] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,4,5,6,1,3] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,4,6,3,1,5] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,4,6,5,1,3] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,5,4,3,1,6] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,5,6,3,1,4] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,5,6,4,1,3] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,6,4,3,1,5] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,6,5,3,1,4] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[2,6,5,4,1,3] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,4,5,2,1,6] => [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,4,5,6,1,2] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,4,6,2,1,5] => [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,4,6,5,1,2] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,5,4,2,1,6] => [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,5,6,2,1,4] => [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,5,6,4,1,2] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,6,4,2,1,5] => [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,6,5,2,1,4] => [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[3,6,5,4,1,2] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
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.
Mp00061: Permutations to increasing treeBinary trees
Mp00009: Binary trees left rotateBinary trees
Mp00013: Binary trees to posetPosets
St001880: Posets ⟶ ℤResult quality: 8% values known / values provided: 8%distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [.,.]
=> ([],1)
=> ? = 1 + 2
[2,1,3] => [[.,.],[.,.]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,1,2] => [[.,.],[.,.]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 + 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 3 + 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 2
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 2
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 2
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 2
[4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 2
[4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 2
[4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 + 2
[5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 2
[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 2
[5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 1 + 2
[5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 4 + 2
[1,2,3,6,4,5] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 4 + 2
[1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 2
[1,2,4,6,3,5] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 2
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 2
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 2
[1,2,6,4,3,5] => [.,[.,[[[.,.],.],[.,.]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 2
[1,2,6,5,3,4] => [.,[.,[[[.,.],.],[.,.]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? = 4 + 2
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 2
[1,3,4,6,2,5] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 2
[1,3,5,4,2,6] => [.,[[.,[[.,.],.]],[.,.]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 2
[1,3,5,6,2,4] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 2
[1,3,6,4,2,5] => [.,[[.,[[.,.],.]],[.,.]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 2
[1,3,6,5,2,4] => [.,[[.,[[.,.],.]],[.,.]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 2
[1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 1 + 2
[1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 1 + 2
[1,4,5,3,2,6] => [.,[[[.,[.,.]],.],[.,.]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 2
[1,4,5,6,2,3] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 2
[1,4,6,3,2,5] => [.,[[[.,[.,.]],.],[.,.]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 4 + 2
[2,3,4,5,1,6] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,3,4,6,1,5] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,3,5,4,1,6] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,3,5,6,1,4] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,3,6,4,1,5] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,3,6,5,1,4] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,4,5,3,1,6] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,4,5,6,1,3] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,4,6,3,1,5] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,4,6,5,1,3] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,5,4,3,1,6] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,5,6,3,1,4] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,5,6,4,1,3] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,6,4,3,1,5] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,6,5,3,1,4] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[2,6,5,4,1,3] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,4,5,2,1,6] => [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,4,5,6,1,2] => [[.,[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,[.,[.,.]]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,4,6,2,1,5] => [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,4,6,5,1,2] => [[.,[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,5,4,2,1,6] => [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,5,6,2,1,4] => [[[.,[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,5,6,4,1,2] => [[.,[[.,[.,.]],.]],[.,.]]
=> [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,6,4,2,1,5] => [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,6,5,2,1,4] => [[[.,[[.,.],.]],.],[.,.]]
=> [[[[.,[[.,.],.]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[3,6,5,4,1,2] => [[.,[[[.,.],.],.]],[.,.]]
=> [[[.,[[[.,.],.],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Mp00061: Permutations to increasing treeBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St001948: Permutations ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 60%
Values
[1] => [.,.]
=> [1,0]
=> [1] => ? = 1 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 2 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 2 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2 = 3 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2 = 3 - 1
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2 = 3 - 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 2 = 3 - 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 2 = 3 - 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 2 = 3 - 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 0 = 1 - 1
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 0 = 1 - 1
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 2 = 3 - 1
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2 = 3 - 1
[3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 2 = 3 - 1
[3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 2 = 3 - 1
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 0 = 1 - 1
[4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 0 = 1 - 1
[4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 0 = 1 - 1
[4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 2 = 3 - 1
[4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 2 = 3 - 1
[5,1,3,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 0 = 1 - 1
[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 0 = 1 - 1
[5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 0 = 1 - 1
[5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ? = 4 - 1
[1,2,3,6,4,5] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ? = 4 - 1
[1,2,4,5,3,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ? = 4 - 1
[1,2,4,6,3,5] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ? = 4 - 1
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ? = 4 - 1
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ? = 4 - 1
[1,2,6,4,3,5] => [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ? = 4 - 1
[1,2,6,5,3,4] => [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ? = 4 - 1
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 1 - 1
[1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 1 - 1
[1,3,4,5,2,6] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ? = 4 - 1
[1,3,4,6,2,5] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ? = 4 - 1
[1,3,5,4,2,6] => [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ? = 4 - 1
[1,3,5,6,2,4] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ? = 4 - 1
[1,3,6,4,2,5] => [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ? = 4 - 1
[1,3,6,5,2,4] => [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ? = 4 - 1
[1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 1 - 1
[1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 1 - 1
[1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ? = 1 - 1
[1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ? = 1 - 1
[1,4,5,3,2,6] => [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ? = 4 - 1
[1,4,5,6,2,3] => [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ? = 4 - 1
[1,4,6,3,2,5] => [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ? = 4 - 1
[1,4,6,5,2,3] => [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ? = 4 - 1
[1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 1 - 1
[1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 1 - 1
[1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ? = 1 - 1
[1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ? = 1 - 1
[1,5,4,3,2,6] => [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ? = 4 - 1
[1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ? = 1 - 1
[1,5,6,3,2,4] => [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ? = 4 - 1
[1,5,6,4,2,3] => [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ? = 4 - 1
[1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 1 - 1
[1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 1 - 1
[1,6,3,4,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ? = 1 - 1
[1,6,3,5,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ? = 1 - 1
[1,6,4,3,2,5] => [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ? = 4 - 1
[1,6,4,5,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ? = 1 - 1
[1,6,5,3,2,4] => [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ? = 4 - 1
[1,6,5,4,2,3] => [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ? = 4 - 1
[2,1,3,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => ? = 2 - 1
[2,1,3,6,4,5] => [[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => ? = 2 - 1
[2,1,4,5,3,6] => [[.,.],[[.,[.,.]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => ? = 2 - 1
[2,1,4,6,3,5] => [[.,.],[[.,[.,.]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => ? = 2 - 1
[2,1,5,4,3,6] => [[.,.],[[[.,.],.],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => ? = 2 - 1
[2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => ? = 2 - 1
[2,1,6,4,3,5] => [[.,.],[[[.,.],.],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => ? = 2 - 1
[2,1,6,5,3,4] => [[.,.],[[[.,.],.],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,3,4,6] => ? = 2 - 1
[2,3,1,5,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => ? = 1 - 1
Description
The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Matching statistic: St000454
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00295: Standard tableaux valley compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000454: Graphs ⟶ ℤResult quality: 1% values known / values provided: 1%distinct values known / distinct values provided: 40%
Values
[1] => [[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 - 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 - 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 - 1
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 - 1
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 - 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 - 1
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 - 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 - 1
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 - 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,5,3,2,4] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,5,4,2,3] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,5,3,1,4] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,5,4,1,3] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,4,2,5] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,1,5,2,4] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,2,4,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,2,5,1,4] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,4,2,1,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,4,5,1,2] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,5,2,1,4] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,5,4,1,2] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,3,2,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,1,5,2,3] => [[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,2,3,1,5] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,2,5,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,3,5,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[4,5,2,1,3] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,5,3,1,2] => [[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,1,3,2,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,1,4,2,3] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,2,3,1,4] => [[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,2,4,1,3] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,3,4,1,2] => [[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,5,4,6] => [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,3,6,4,5] => [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,4,5,3,6] => [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,4,6,3,5] => [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,5,4,3,6] => [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,5,6,3,4] => [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,6,4,3,5] => [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,2,6,5,3,4] => [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,2,5,4,6] => [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,3,2,6,4,5] => [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,3,4,5,2,6] => [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,4,6,2,5] => [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,5,4,2,6] => [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,5,6,2,4] => [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,6,4,2,5] => [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,3,6,5,2,4] => [[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,2,5,3,6] => [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,4,2,6,3,5] => [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,4,3,5,2,6] => [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,4,3,6,2,5] => [[1,2,4],[3,6],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,4,5,3,2,6] => [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,5,6,2,3] => [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,6,3,2,5] => [[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[1,4,6,5,2,3] => [[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.