- FindStat

Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000016
(load all 10 compositions to match this statistic)

Mapping

St000016: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => 0
[[1,2]]
 => 1
[[1],[2]]
 => 0
[[1,2,3]]
 => 3
[[1,3],[2]]
 => 1
[[1,2],[3]]
 => 1
[[1],[2],[3]]
 => 0
[[1,2,3,4]]
 => 6
[[1,3,4],[2]]
 => 3
[[1,2,4],[3]]
 => 3
[[1,2,3],[4]]
 => 3
[[1,3],[2,4]]
 => 3
[[1,2],[3,4]]
 => 3
[[1,4],[2],[3]]
 => 1
[[1,3],[2],[4]]
 => 1
[[1,2],[3],[4]]
 => 1
[[1],[2],[3],[4]]
 => 0
[[1,2,3,4,5]]
 => 10
[[1,3,4,5],[2]]
 => 6
[[1,2,4,5],[3]]
 => 6
[[1,2,3,5],[4]]
 => 6
[[1,2,3,4],[5]]
 => 6
[[1,3,5],[2,4]]
 => 5
[[1,2,5],[3,4]]
 => 5
[[1,3,4],[2,5]]
 => 5
[[1,2,4],[3,5]]
 => 5
[[1,2,3],[4,5]]
 => 5
[[1,4,5],[2],[3]]
 => 3
[[1,3,5],[2],[4]]
 => 3
[[1,2,5],[3],[4]]
 => 3
[[1,3,4],[2],[5]]
 => 3
[[1,2,4],[3],[5]]
 => 3
[[1,2,3],[4],[5]]
 => 3
[[1,4],[2,5],[3]]
 => 3
[[1,3],[2,5],[4]]
 => 3
[[1,2],[3,5],[4]]
 => 3
[[1,3],[2,4],[5]]
 => 3
[[1,2],[3,4],[5]]
 => 3
[[1,5],[2],[3],[4]]
 => 1
[[1,4],[2],[3],[5]]
 => 1
[[1,3],[2],[4],[5]]
 => 1
[[1,2],[3],[4],[5]]
 => 1
[[1],[2],[3],[4],[5]]
 => 0
[[1,2,3,4,5,6]]
 => 15
[[1,3,4,5,6],[2]]
 => 10
[[1,2,4,5,6],[3]]
 => 10
[[1,2,3,5,6],[4]]
 => 10
[[1,2,3,4,6],[5]]
 => 10
[[1,2,3,4,5],[6]]
 => 10
[[1,3,5,6],[2,4]]
 => 8

Description

The number of attacking pairs of a standard tableau. Note that this is actually a statistic on the underlying partition. A pair of cells

$(c, d)$ of a Young diagram (in English notation) is said to be attacking if one of the following conditions holds: 1.

$c$ and

$d$ lie in the same row with

$c$ strictly to the west of

$d$ . 2.

$c$ is in the row immediately to the south of

$d$ , and

$c$ lies strictly east of

$d$ .

Matching statistic: St000006

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000006: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [.,.]
 => [1,0]
 => 0
[[1,2]]
 => [1,2] => [.,[.,.]]
 => [1,1,0,0]
 => 0
[[1],[2]]
 => [2,1] => [[.,.],.]
 => [1,0,1,0]
 => 1
[[1,2,3]]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 0
[[1,3],[2]]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1
[[1,2],[3]]
 => [3,1,2] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1
[[1],[2],[3]]
 => [3,2,1] => [[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 3
[[1,2,3,4]]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,2,4],[3]]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,2,3],[4]]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,3],[2,4]]
 => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 3
[[1,2],[3,4]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 6
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 5
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => 10
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3

Description

The dinv of a Dyck path. Let

$a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path

$D$ (see [[St000012]]). The dinv statistic of

$D$ is

$\operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$ Equivalently,

$\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above

$D$ whose ''arm length'' is one larger or equal to its ''leg length''. There is a recursive definition of the

$(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2]. Let

$a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path

$D$ with

$a_i > 0$ for

$2\leq i\leq r$ (so that the path touches the diagonal for the first time after

$r$ steps). Assume that

$D$ has

$v$ entries where

$a_i=0$ . Let

$D'$ be the path with the area sequence

$(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$ , then the statistics are related by

$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$

Matching statistic: St001171

Mapping

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001171: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 67%

Values

[[1]]
 => [1]
 => [1,0]
 => [1] => 0
[[1,2]]
 => [2]
 => [1,0,1,0]
 => [1,2] => 0
[[1],[2]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 1
[[1,2,3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[[1,3],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[1,2],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[[1],[2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3
[[1,2,3,4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[[1,2,4],[3]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[[1,2,3],[4]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[[1,3],[2,4]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3
[[1,2],[3,4]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 6
[[1,2,3,4,5]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,3,4,5],[2]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,2,4,5],[3]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,2,3,5],[4]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,2,3,4],[5]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,3,5],[2,4]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 3
[[1,2,5],[3,4]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 3
[[1,3,4],[2,5]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 3
[[1,2,4],[3,5]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 3
[[1,2,3],[4,5]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 3
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 5
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 5
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 5
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 5
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 5
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,6,6,6,6,10}
[[1,2,3,4,5,6]]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5,6],[2,4]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,5,6],[3,4]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,6],[2,5]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,6],[3,5]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,6],[4,5]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5],[2,6]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5],[3,6]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5],[4,6]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4],[5,6]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,4,5,6],[2],[3]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5,6],[2],[4]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,5,6],[3],[4]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,6],[2],[5]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,6],[3],[5]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,6],[4],[5]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5],[2],[6]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5],[3],[6]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5],[4],[6]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4],[5],[6]]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5],[2,4,6]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 8
[[1,2,5],[3,4,6]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 8
[[1,3,4],[2,5,6]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 8
[[1,2,4],[3,5,6]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 8
[[1,2,3],[4,5,6]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 8
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,4],[2,5],[3,6]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 6
[[1,3],[2,5],[4,6]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 6
[[1,2],[3,5],[4,6]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 6
[[1,3],[2,4],[5,6]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 6
[[1,2],[3,4],[5,6]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 6

Description

The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ .

Matching statistic: St001232
(load all 2 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 22%

Values

[[1]]
 => [1] => [.,.]
 => [1,0]
 => 0
[[1,2]]
 => [1,2] => [.,[.,.]]
 => [1,0,1,0]
 => 1
[[1],[2]]
 => [2,1] => [[.,.],.]
 => [1,1,0,0]
 => 0
[[1,2,3]]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => ? = 3
[[1,3],[2]]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[[1,2],[3]]
 => [3,1,2] => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => 1
[[1],[2],[3]]
 => [3,2,1] => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => 0
[[1,2,3,4]]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,6}
[[1,3,4],[2]]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,6}
[[1,2,4],[3]]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,6}
[[1,2,3],[4]]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,6}
[[1,3],[2,4]]
 => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {3,3,3,3,3,6}
[[1,2],[3,4]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {3,3,3,3,3,6}
[[1,4],[2],[3]]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5,6],[2],[4]]
 => [4,2,1,3,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,6],[2],[5]]
 => [5,2,1,3,4,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,6],[3],[5]]
 => [5,3,1,2,4,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => [[[.,.],.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St000455

Mapping

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 22%

Values

[[1]]
 => [1] => ([],1)
 => ([(0,1)],2)
 => -1 = 0 - 1
[[1,2]]
 => [2] => ([],2)
 => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[[1],[2]]
 => [1,1] => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => -1 = 0 - 1
[[1,2,3]]
 => [3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
[[1,2],[3]]
 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[1],[2],[3]]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 0 - 1
[[1,2,3,4]]
 => [4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,6} - 1
[[1,2,4],[3]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,6} - 1
[[1,2,3],[4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,6} - 1
[[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,6} - 1
[[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,6} - 1
[[1,3],[2],[4]]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,6} - 1
[[1,2],[3],[4]]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => -1 = 0 - 1
[[1,2,3,4,5]]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2,4,5],[3]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2,3,5],[4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2,3,4],[5]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[1,3,5],[2,4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,3,4],[2,5]]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2,4],[3,5]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2,3],[4,5]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,3,5],[2],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2,5],[3],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,3,4],[2],[5]]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2,4],[3],[5]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2,3],[4],[5]]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[1,4],[2,5],[3]]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2],[3,5],[4]]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,3],[2,4],[5]]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2],[3,4],[5]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,5],[2],[3],[4]]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,4],[2],[3],[5]]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,3],[2],[4],[5]]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} - 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => -1 = 0 - 1
[[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,4,5,6],[3]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,3,5,6],[4]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,3,4,6],[5]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,3,4,5],[6]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[[1,3,5,6],[2,4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,3,4,6],[2,5]]
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,4,6],[3,5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,3,6],[4,5]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,3,4,5],[2,6]]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,4,5],[3,6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,3,5],[4,6]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,3,4],[5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,3,5,6],[2],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,5,6],[3],[4]]
 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,3,4,6],[2],[5]]
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,4,6],[3],[5]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,3,6],[4],[5]]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,3,4,5],[2],[6]]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,4,5],[3],[6]]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,3,5],[4],[6]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} - 1
[[1,2,3,4],[5],[6]]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0 = 1 - 1
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => -1 = 0 - 1

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St000264

Mapping

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 14%●distinct values known / distinct values provided: 11%

Values

[[1]]
 => [1] => [1] => ([],1)
 => ? = 0
[[1,2]]
 => [2] => [1] => ([],1)
 => ? ∊ {0,1}
[[1],[2]]
 => [2] => [1] => ([],1)
 => ? ∊ {0,1}
[[1,2,3]]
 => [3] => [1] => ([],1)
 => ? ∊ {0,1,1,3}
[[1,3],[2]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,3}
[[1,2],[3]]
 => [3] => [1] => ([],1)
 => ? ∊ {0,1,1,3}
[[1],[2],[3]]
 => [3] => [1] => ([],1)
 => ? ∊ {0,1,1,3}
[[1,2,3,4]]
 => [4] => [1] => ([],1)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1,3,4],[2]]
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1,2,4],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1,2,3],[4]]
 => [4] => [1] => ([],1)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1,3],[2,4]]
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1,2],[3,4]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1,4],[2],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1,3],[2],[4]]
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1,2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1],[2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? ∊ {0,1,1,1,3,3,3,3,3,6}
[[1,2,3,4,5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4,5],[2]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4,5],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,4],[5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,5],[2,4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,5],[3,4]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4],[2,5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4],[3,5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3],[4,5]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4,5],[2],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,5],[2],[4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,5],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4],[2],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4],[3],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4],[2,5],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2,5],[4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2,4],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3,4],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2],[4],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1],[2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,4,5,6]]
 => [6] => [1] => ([],1)
 => ? ∊ {0,1,1,1,1,1,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5,6],[3]]
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,1,1,1,1,1,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,1,1,1,1,1,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,5],[6]]
 => [6] => [1] => ([],1)
 => ? ∊ {0,1,1,1,1,1,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5,6],[2,4]]
 => [2,2,2] => [3] => ([],3)
 => ? ∊ {0,1,1,1,1,1,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,4,6],[3,5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,4,6],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,4],[3,5,6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4,6],[2,5],[3]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,6],[3,4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,4],[3,6],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4,6],[2],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2],[3,4],[5,6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,4],[2,6],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

Matching statistic: St001582

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 44%

Values

[[1]]
 => [1] => [1] => [1] => ? = 0
[[1,2]]
 => [1,2] => [1,2] => [1,2] => 1
[[1],[2]]
 => [2,1] => [2,1] => [2,1] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,2,3] => 3
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,2],[3]]
 => [3,1,2] => [2,3,1] => [3,1,2] => 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [3,2,1] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 6
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 3
[[1,2,4],[3]]
 => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 3
[[1,3],[2,4]]
 => [2,4,1,3] => [3,4,1,2] => [1,4,2,3] => 3
[[1,2],[3,4]]
 => [3,4,1,2] => [2,4,1,3] => [1,3,4,2] => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,4,2,1] => [4,3,1,2] => 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [2,4,3,1] => [4,2,3,1] => 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,3,4,1,5] => [4,1,2,3,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [2,3,4,5,1] => [5,1,2,3,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [3,4,1,2,5] => [1,4,2,3,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [2,4,1,3,5] => [1,3,4,2,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [3,4,5,1,2] => [1,5,2,3,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [2,4,5,1,3] => [1,3,5,2,4] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,3,5,1,4] => [1,3,4,5,2] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [3,4,2,1,5] => [4,3,1,2,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [2,4,3,1,5] => [4,2,3,1,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [3,4,5,2,1] => [5,4,1,2,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [2,4,5,3,1] => [5,2,4,1,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [2,3,5,4,1] => [5,2,3,4,1] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [4,5,1,3,2] => [2,5,4,1,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [3,5,1,4,2] => [2,5,3,4,1] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [2,5,1,4,3] => [2,4,5,3,1] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [3,4,1,5,2] => [4,5,1,2,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [2,4,1,5,3] => [4,2,5,1,3] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,5,3,2,1] => [5,4,3,1,2] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [3,5,4,2,1] => [5,4,2,3,1] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [2,5,4,3,1] => [5,3,4,2,1] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [2,3,4,5,1,6] => [5,1,2,3,4,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [3,4,1,2,5,6] => [1,4,2,3,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [2,4,1,3,5,6] => [1,3,4,2,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => [3,4,5,1,2,6] => [1,5,2,3,4,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => [2,4,5,1,3,6] => [1,3,5,2,4,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => [2,3,5,1,4,6] => [1,3,4,5,2,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [3,4,5,6,1,2] => [1,6,2,3,4,5] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => [2,4,5,6,1,3] => [1,3,6,2,4,5] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => [2,3,5,6,1,4] => [1,3,4,6,2,5] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [2,3,4,6,1,5] => [1,3,4,5,6,2] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5,6],[2],[4]]
 => [4,2,1,3,5,6] => [3,4,2,1,5,6] => [4,3,1,2,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [2,4,3,1,5,6] => [4,2,3,1,5,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,6],[2],[5]]
 => [5,2,1,3,4,6] => [3,4,5,2,1,6] => [5,4,1,2,3,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,6],[3],[5]]
 => [5,3,1,2,4,6] => [2,4,5,3,1,6] => [5,2,4,1,3,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [2,3,5,4,1,6] => [5,2,3,4,1,6] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => [3,4,5,6,2,1] => [6,5,1,2,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5],[3],[6]]
 => [6,3,1,2,4,5] => [2,4,5,6,3,1] => [6,2,5,1,3,4] => ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}

Description

The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.

Matching statistic: St000259

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%

Values

[[1]]
 => [1] => [1] => ([],1)
 => 0
[[1,2]]
 => [1,2] => [1,2] => ([],2)
 => ? = 0
[[1],[2]]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[[1,2,3]]
 => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {0,1,3}
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,1,3}
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,1,3}
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,3,3,3,3,6}
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,3,3,3,3,6}
[[1,2,4],[3]]
 => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,1,1,3,3,3,3,6}
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,3,3,3,3,6}
[[1,3],[2,4]]
 => [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,3,3,3,3,6}
[[1,2],[3,4]]
 => [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,3,3,3,3,6}
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,1,3,3,3,3,6}
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,1,1,3,3,3,3,6}
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10}
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15}
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3
[[1,3,4],[2],[5],[6]]
 => [6,5,2,1,3,4] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 3
[[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [5,4,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St001621

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 22%

Values

[[1]]
 => [1] => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,2]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => ([],3)
 => ([(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)
 => ? = 3 + 1
[[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2 = 1 + 1
[[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,3],[2,4]]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2 = 1 + 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ?
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => ?
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,4),(0,6),(1,8),(3,7),(4,9),(5,2),(6,3),(6,9),(7,8),(8,5),(9,1),(9,7)],10)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
 => ([(0,4),(0,6),(2,10),(3,9),(4,7),(5,2),(5,8),(6,3),(6,7),(7,5),(7,9),(8,10),(9,8),(10,1)],11)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,9),(2,8),(3,7),(4,2),(4,11),(5,4),(5,10),(6,1),(6,7),(7,5),(7,9),(9,10),(10,11),(11,8)],12)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,10),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,8),(12,11),(12,13),(13,8),(13,10)],14)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,5,6],[2],[4]]
 => [4,2,1,3,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,4,6],[2],[5]]
 => [5,2,1,3,4,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1

Description

The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.

Matching statistic: St001624

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 22%

Values

[[1]]
 => [1] => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[[1,2]]
 => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[1],[2]]
 => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[1,3],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[[1],[2],[3]]
 => [3,2,1] => ([],3)
 => ([(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)
 => ? = 3 + 1
[[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2 = 1 + 1
[[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 1 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,3],[2,4]]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? ∊ {1,3,3,3,3,3,6} + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2 = 1 + 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ?
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => ?
 => ? ∊ {1,1,1,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,6,6,6,6,10} + 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,4),(0,6),(1,8),(3,7),(4,9),(5,2),(6,3),(6,9),(7,8),(8,5),(9,1),(9,7)],10)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
 => ([(0,4),(0,6),(2,10),(3,9),(4,7),(5,2),(5,8),(6,3),(6,7),(7,5),(7,9),(8,10),(9,8),(10,1)],11)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(2,11),(3,10),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(8,11),(9,8),(10,2),(10,8),(11,1)],12)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,3),(0,6),(1,9),(2,8),(3,7),(4,2),(4,11),(5,4),(5,10),(6,1),(6,7),(7,5),(7,9),(9,10),(10,11),(11,8)],12)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,5),(0,6),(1,8),(2,9),(3,10),(4,1),(4,11),(5,2),(5,7),(6,3),(6,7),(7,9),(7,10),(9,12),(10,4),(10,12),(11,8),(12,11)],13)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,5),(0,6),(1,11),(2,4),(2,13),(3,7),(4,10),(5,1),(5,12),(6,2),(6,12),(8,9),(9,7),(10,3),(10,9),(11,8),(12,11),(12,13),(13,8),(13,10)],14)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,5,6],[2],[4]]
 => [4,2,1,3,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(0,3),(0,4),(0,6),(1,10),(1,11),(3,8),(3,9),(4,7),(4,9),(5,2),(6,1),(6,7),(6,8),(7,10),(7,13),(8,11),(8,13),(9,13),(10,12),(11,12),(12,5),(13,12)],14)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1
[[1,3,4,6],[2],[5]]
 => [5,2,1,3,4,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
 => ? ∊ {1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,8,8,8,8,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,15} + 1

Description

The breadth of a lattice. The '''breadth''' of a lattice is the least integer

$b$ such that any join

$x_1\vee x_2\vee\cdots\vee x_n$ , with

$n > b$ , can be expressed as a join over a proper subset of

$\{x_1,x_2,\ldots,x_n\}$ .

The following 1 statistic also match your data. Click on any of them to see the details.

St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.