searching the database
Your data matches 154 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000993
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000245
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3 = 2 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3 = 2 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3 = 2 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 4 = 3 + 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => 5 = 4 + 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,5],[3,7],[4],[6]]
=> [6,4,3,7,1,2,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,6],[3,4],[5],[7]]
=> [7,5,3,4,1,2,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,5],[2,6],[3],[7]]
=> [7,3,2,6,1,4,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,5],[3,6],[4],[7]]
=> [7,4,3,6,1,2,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,4],[2,6],[5],[7]]
=> [7,5,2,6,1,3,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,4],[3,6],[5],[7]]
=> [7,5,3,6,1,2,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3 = 2 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3 = 2 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3 = 2 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 4 = 3 + 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [[[[.,.],.],.],[.,[.,.]]]
=> [6,5,1,2,3,4] => 3 = 2 + 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [6,4,2,1,3,5] => 2 = 1 + 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [6,4,1,2,3,5] => 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => 4 = 3 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => 5 = 4 + 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [7,6,4,2,1,3,5] => ? = 1 + 1
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,5],[3,7],[4],[6]]
=> [6,4,3,7,1,2,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,6],[3,4],[5],[7]]
=> [7,5,3,4,1,2,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,4,5],[2,6],[3],[7]]
=> [7,3,2,6,1,4,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,5],[3,6],[4],[7]]
=> [7,4,3,6,1,2,5] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,3,4],[2,6],[5],[7]]
=> [7,5,2,6,1,3,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
[[1,2,4],[3,6],[5],[7]]
=> [7,5,3,6,1,2,4] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => ? = 2 + 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St001491
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Values
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 3
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,4],[2,5],[3,6]]
=> [2,2,2]
=> [2,2]
=> 1100 => 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> 1100 => 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> 1100 => 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> 1100 => 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> 1100 => 1
[[1,5],[2,6],[3],[4]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4],[2,6],[3],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3],[2,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2],[3,6],[4],[5]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4],[2,5],[3],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3],[2,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2],[3,5],[4],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3],[2,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 3
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 3
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 3
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 3
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => ? = 3
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => ? = 4
[[1,5,6,7],[2],[3],[4]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,4,6,7],[2],[3],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3,6,7],[2],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,6,7],[3],[4],[5]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,4,5,7],[2],[3],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3,5,7],[2],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,5,7],[3],[4],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3,4,7],[2],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,4,7],[3],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,3,7],[4],[5],[6]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,4,5,6],[2],[3],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3,5,6],[2],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,5,6],[3],[4],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,4,6],[3],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,3,6],[4],[5],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,3,4,5],[2],[6],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,4,5],[3],[6],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,3,5],[4],[6],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1]
=> [1,1,1]
=> 1110 => 2
[[1,4,7],[2,5],[3,6]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,3,7],[2,5],[4,6]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,2,7],[3,5],[4,6]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,3,7],[2,4],[5,6]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,2,7],[3,4],[5,6]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,4,6],[2,5],[3,7]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,3,6],[2,5],[4,7]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,2,6],[3,5],[4,7]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,3,6],[2,4],[5,7]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,2,6],[3,4],[5,7]]
=> [3,2,2]
=> [2,2]
=> 1100 => 1
[[1,5,7],[2,6],[3],[4]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4,7],[2,6],[3],[5]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,7],[2,6],[4],[5]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,7],[3,6],[4],[5]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4,7],[2,5],[3],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,7],[2,5],[4],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,7],[3,5],[4],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,7],[2,4],[5],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,7],[3,4],[5],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,5,6],[2,7],[3],[4]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4,6],[2,7],[3],[5]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,6],[2,7],[4],[5]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,6],[3,7],[4],[5]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4,5],[2,7],[3],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,5],[2,7],[4],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,5],[3,7],[4],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,4],[2,7],[5],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,4],[3,7],[5],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,3],[4,7],[5],[6]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4,6],[2,5],[3],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,6],[2,5],[4],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,6],[3,5],[4],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,6],[2,4],[5],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,6],[3,4],[5],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4,5],[2,6],[3],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,5],[2,6],[4],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,5],[3,6],[4],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,4],[2,6],[5],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,4],[3,6],[5],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,3],[4,6],[5],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,5],[2,4],[6],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,5],[3,4],[6],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3,4],[2,5],[6],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2,4],[3,5],[6],[7]]
=> [3,2,1,1]
=> [2,1,1]
=> 10110 => ? = 2
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001644
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [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)
=> 4 = 3 + 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [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)
=> 5 = 4 + 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2],[3,5],[4,7],[6]]
=> [6,4,7,3,5,1,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3],[2,4],[5,7],[6]]
=> [6,5,7,2,4,1,3] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2],[3,4],[5,7],[6]]
=> [6,5,7,3,4,1,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,4],[2,5],[3,6],[7]]
=> [7,3,6,2,5,1,4] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3],[2,5],[4,6],[7]]
=> [7,4,6,2,5,1,3] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2],[3,5],[4,6],[7]]
=> [7,4,6,3,5,1,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,3],[2,4],[5,6],[7]]
=> [7,5,6,2,4,1,3] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,4,6,7,8],[2],[3],[5]]
=> [5,3,2,1,4,6,7,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,6,7,8],[2],[4],[5]]
=> [5,4,2,1,3,6,7,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,6,7,8],[3],[4],[5]]
=> [5,4,3,1,2,6,7,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,4,5,7,8],[2],[3],[6]]
=> [6,3,2,1,4,5,7,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,5,7,8],[2],[4],[6]]
=> [6,4,2,1,3,5,7,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,5,7,8],[3],[4],[6]]
=> [6,4,3,1,2,5,7,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,4,7,8],[2],[5],[6]]
=> [6,5,2,1,3,4,7,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,4,7,8],[3],[5],[6]]
=> [6,5,3,1,2,4,7,8] => ? => ?
=> ? = 2 + 1
[[1,2,3,7,8],[4],[5],[6]]
=> [6,5,4,1,2,3,7,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,4,5,6,8],[2],[3],[7]]
=> [7,3,2,1,4,5,6,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,5,6,8],[2],[4],[7]]
=> [7,4,2,1,3,5,6,8] => ? => ?
=> ? = 2 + 1
[[1,2,5,6,8],[3],[4],[7]]
=> [7,4,3,1,2,5,6,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,4,6,8],[2],[5],[7]]
=> [7,5,2,1,3,4,6,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,4,6,8],[3],[5],[7]]
=> [7,5,3,1,2,4,6,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,3,6,8],[4],[5],[7]]
=> [7,5,4,1,2,3,6,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,4,5,8],[2],[6],[7]]
=> [7,6,2,1,3,4,5,8] => ? => ?
=> ? = 2 + 1
[[1,2,4,5,8],[3],[6],[7]]
=> [7,6,3,1,2,4,5,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,3,5,8],[4],[6],[7]]
=> [7,6,4,1,2,3,5,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,3,4,8],[5],[6],[7]]
=> [7,6,5,1,2,3,4,8] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,4,5,6,7],[2],[3],[8]]
=> [8,3,2,1,4,5,6,7] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,5,6,7],[2],[4],[8]]
=> [8,4,2,1,3,5,6,7] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,5,6,7],[3],[4],[8]]
=> [8,4,3,1,2,5,6,7] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,4,6,7],[2],[5],[8]]
=> [8,5,2,1,3,4,6,7] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,4,6,7],[3],[5],[8]]
=> [8,5,3,1,2,4,6,7] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,3,6,7],[4],[5],[8]]
=> [8,5,4,1,2,3,6,7] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,4,5,7],[2],[6],[8]]
=> [8,6,2,1,3,4,5,7] => ? => ?
=> ? = 2 + 1
[[1,2,4,5,7],[3],[6],[8]]
=> [8,6,3,1,2,4,5,7] => ? => ?
=> ? = 2 + 1
[[1,2,3,5,7],[4],[6],[8]]
=> [8,6,4,1,2,3,5,7] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,3,4,7],[5],[6],[8]]
=> [8,6,5,1,2,3,4,7] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,3,4,5,6],[2],[7],[8]]
=> [8,7,2,1,3,4,5,6] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,4,5,6],[3],[7],[8]]
=> [8,7,3,1,2,4,5,6] => ? => ?
=> ? = 2 + 1
[[1,2,3,5,6],[4],[7],[8]]
=> [8,7,4,1,2,3,5,6] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,3,4,6],[5],[7],[8]]
=> [8,7,5,1,2,3,4,6] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 1
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St000306
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 88%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 88%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[[1,6],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,4],[2,7],[3],[5],[6]]
=> [6,5,3,2,7,1,4] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3],[2,7],[4],[5],[6]]
=> [6,5,4,2,7,1,3] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,5],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,4],[2,6],[3],[5],[7]]
=> [7,5,3,2,6,1,4] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3],[2,6],[4],[5],[7]]
=> [7,5,4,2,6,1,3] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2],[3,6],[4],[5],[7]]
=> [7,5,4,3,6,1,2] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,4],[2,5],[3],[6],[7]]
=> [7,6,3,2,5,1,4] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3],[2,5],[4],[6],[7]]
=> [7,6,4,2,5,1,3] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2],[3,5],[4],[6],[7]]
=> [7,6,4,3,5,1,2] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,3],[2,4],[5],[6],[7]]
=> [7,6,5,2,4,1,3] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000632
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 75%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 75%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> ([],4)
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 3 = 2 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4)],5)
=> 3 = 2 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ([(2,3),(2,4)],5)
=> 3 = 2 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([],5)
=> ([],5)
=> 4 = 3 + 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> 3 = 2 + 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 3 = 2 + 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 3 = 2 + 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6)
=> ([(1,5),(5,2),(5,3),(5,4)],6)
=> 3 = 2 + 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(5,2),(5,3)],6)
=> 3 = 2 + 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(5,2)],6)
=> 3 = 2 + 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => ([(2,5),(3,5),(5,4)],6)
=> ([(2,3),(3,4),(3,5)],6)
=> 3 = 2 + 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(2,3),(2,4),(4,5)],6)
=> 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> ([(3,4),(4,5)],6)
=> 3 = 2 + 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> 2 = 1 + 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5)],6)
=> 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,5),(1,3),(2,4),(2,5)],6)
=> 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2 = 1 + 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> 3 = 2 + 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6)
=> 3 = 2 + 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5)],6)
=> 3 = 2 + 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => ([(0,5),(1,5),(2,5),(3,4)],6)
=> ([(0,5),(1,2),(1,3),(1,4)],6)
=> 3 = 2 + 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 3 = 2 + 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,3),(2,5)],6)
=> 3 = 2 + 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => ([(1,5),(2,5),(3,4)],6)
=> ([(1,5),(2,3),(2,4)],6)
=> 3 = 2 + 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => ([(2,5),(3,4),(3,5)],6)
=> ([(2,5),(3,4),(3,5)],6)
=> 3 = 2 + 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 4 = 3 + 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5)],6)
=> 4 = 3 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5)],6)
=> 4 = 3 + 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => ([(3,5),(4,5)],6)
=> ([(3,4),(3,5)],6)
=> 4 = 3 + 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(4,5)],6)
=> ([(4,5)],6)
=> 4 = 3 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([],6)
=> ([],6)
=> 5 = 4 + 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,4)],7)
=> ([(0,5),(5,6),(6,1),(6,2),(6,3),(6,4)],7)
=> 3 = 2 + 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => ([(0,6),(1,6),(2,6),(3,5),(5,4),(6,5)],7)
=> ([(0,5),(5,4),(5,6),(6,1),(6,2),(6,3)],7)
=> 3 = 2 + 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => ([(0,6),(1,6),(2,5),(3,5),(5,6),(6,4)],7)
=> ([(0,6),(5,3),(5,4),(6,1),(6,2),(6,5)],7)
=> 3 = 2 + 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7)
=> ([(0,6),(5,4),(6,1),(6,2),(6,3),(6,5)],7)
=> 3 = 2 + 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(6,4)],7)
=> ([(0,4),(0,5),(5,6),(6,1),(6,2),(6,3)],7)
=> 3 = 2 + 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> ([(0,4),(0,6),(5,2),(5,3),(6,1),(6,5)],7)
=> 3 = 2 + 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ([(0,4),(0,6),(5,3),(6,1),(6,2),(6,5)],7)
=> 3 = 2 + 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7)
=> ([(0,3),(0,4),(0,5),(5,6),(6,1),(6,2)],7)
=> 3 = 2 + 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(5,2),(6,1),(6,5)],7)
=> 3 = 2 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,6),(5,1),(6,5)],7)
=> 3 = 2 + 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => ([(1,6),(2,6),(3,6),(4,5),(6,4)],7)
=> ([(1,5),(5,6),(6,2),(6,3),(6,4)],7)
=> 3 = 2 + 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => ([(1,6),(2,5),(3,5),(5,6),(6,4)],7)
=> ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
=> ? = 2 + 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => ([(1,6),(2,6),(3,4),(4,6),(6,5)],7)
=> ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? = 2 + 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ([(1,6),(2,5),(3,5),(4,6),(5,4)],7)
=> ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? = 2 + 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ([(1,4),(1,6),(5,3),(6,2),(6,5)],7)
=> ? = 2 + 1
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(1,3),(1,4),(1,6),(5,2),(6,5)],7)
=> 3 = 2 + 1
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => ([(2,6),(3,6),(4,5),(6,4)],7)
=> ([(2,5),(5,6),(6,3),(6,4)],7)
=> 3 = 2 + 1
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ([(2,6),(5,4),(6,3),(6,5)],7)
=> ? = 2 + 1
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ([(2,4),(2,6),(5,3),(6,5)],7)
=> ? = 2 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
=> ([(3,4),(4,6),(6,5)],7)
=> 3 = 2 + 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => ([(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(4,5),(6,5)],7)
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5),(4,6)],7)
=> ? = 1 + 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ? = 1 + 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => ([(0,5),(1,4),(2,3),(2,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(4,3),(5,2),(5,6)],7)
=> ? = 1 + 1
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => ([(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(6,4)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 1 + 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => ([(0,4),(1,4),(1,5),(2,3),(2,5),(2,6),(4,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,5)],7)
=> ? = 1 + 1
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(2,6),(4,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
=> ? = 1 + 1
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(6,3)],7)
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(3,6)],7)
=> ? = 1 + 1
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 1 + 1
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => ([(0,6),(1,4),(2,3),(2,6),(4,5)],7)
=> ([(0,6),(1,4),(2,3),(2,6),(4,5)],7)
=> ? = 1 + 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(1,5),(3,6),(5,2),(5,6)],7)
=> ? = 1 + 1
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ([(0,4),(1,5),(1,6),(5,3),(6,2)],7)
=> ? = 1 + 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => ([(0,6),(1,3),(2,4),(2,6),(6,5)],7)
=> ([(0,6),(1,3),(2,4),(4,5),(4,6)],7)
=> ? = 1 + 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => ([(0,3),(1,5),(2,4),(2,6),(5,6)],7)
=> ([(0,6),(1,3),(2,4),(2,6),(4,5)],7)
=> ? = 1 + 1
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 1 + 1
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(6,4)],7)
=> ([(0,3),(1,4),(1,5),(1,6),(3,2),(3,4),(3,5),(3,6)],7)
=> ? = 2 + 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(6,4)],7)
=> ([(0,3),(0,6),(1,4),(1,5),(1,6),(3,2),(3,4),(3,5)],7)
=> ? = 2 + 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(3,4),(3,5),(4,6)],7)
=> ([(0,3),(0,5),(0,6),(1,4),(1,5),(1,6),(3,2),(3,4)],7)
=> ? = 2 + 1
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(3,2)],7)
=> ? = 2 + 1
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(6,4)],7)
=> ([(0,4),(1,3),(1,5),(1,6),(4,2),(4,5),(4,6)],7)
=> ? = 2 + 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => ([(0,6),(1,5),(1,6),(2,4),(3,4),(3,6),(4,5)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(1,6),(4,2),(4,5)],7)
=> ? = 2 + 1
[[1,2,5],[3,7],[4],[6]]
=> [6,4,3,7,1,2,5] => ([(0,5),(1,5),(1,6),(2,5),(2,6),(3,4),(4,6)],7)
=> ([(0,3),(0,5),(0,6),(1,4),(1,5),(1,6),(4,2)],7)
=> ? = 2 + 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => ([(0,6),(1,6),(2,5),(3,5),(3,6),(5,4)],7)
=> ([(0,5),(1,3),(1,4),(1,6),(5,2),(5,6)],7)
=> ? = 2 + 1
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(4,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,6),(5,2)],7)
=> ? = 2 + 1
[[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(6,2)],7)
=> ? = 2 + 1
[[1,4,5],[2,6],[3],[7]]
=> [7,3,2,6,1,4,5] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(6,4)],7)
=> ([(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 2 + 1
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => ([(1,5),(2,4),(2,6),(3,4),(3,5),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,5),(3,4),(3,6)],7)
=> ? = 2 + 1
[[1,2,5],[3,6],[4],[7]]
=> [7,4,3,6,1,2,5] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(4,6)],7)
=> ([(1,5),(1,6),(2,3),(2,5),(2,6),(3,4)],7)
=> ? = 2 + 1
[[1,3,4],[2,6],[5],[7]]
=> [7,5,2,6,1,3,4] => ([(1,6),(2,5),(3,5),(3,6),(6,4)],7)
=> ([(1,4),(2,3),(2,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[[1,2,4],[3,6],[5],[7]]
=> [7,5,3,6,1,2,4] => ([(1,5),(2,5),(2,6),(3,4),(4,6)],7)
=> ([(1,3),(1,6),(2,4),(2,6),(4,5)],7)
=> ? = 2 + 1
[[1,2,3],[4,6],[5],[7]]
=> [7,5,4,6,1,2,3] => ([(1,6),(2,6),(3,4),(4,5)],7)
=> ([(1,6),(2,4),(2,5),(6,3)],7)
=> ? = 2 + 1
[[1,3,4],[2,5],[6],[7]]
=> [7,6,2,5,1,3,4] => ([(2,6),(3,4),(3,6),(6,5)],7)
=> ([(2,6),(3,4),(4,5),(4,6)],7)
=> ? = 2 + 1
[[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ([(2,4),(3,5),(5,6)],7)
=> ? = 2 + 1
[[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 1 + 1
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => ([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6)],7)
=> ? = 1 + 1
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => ([(0,5),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6)],7)
=> ([(0,4),(0,5),(1,4),(1,5),(1,6),(2,3),(2,6)],7)
=> ? = 1 + 1
[[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4)],7)
=> ([(0,5),(0,6),(1,3),(2,4),(2,5),(2,6)],7)
=> ? = 1 + 1
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,3),(1,6),(2,4),(2,5),(2,6)],7)
=> ? = 1 + 1
[[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5)],7)
=> ? = 1 + 1
[[1,2],[3,5],[4,7],[6]]
=> [6,4,7,3,5,1,2] => ([(0,6),(1,5),(2,5),(2,6),(3,4)],7)
=> ([(0,5),(1,4),(1,6),(2,3),(2,6)],7)
=> ? = 1 + 1
[[1,3],[2,4],[5,7],[6]]
=> [6,5,7,2,4,1,3] => ([(0,5),(1,5),(2,6),(3,4),(3,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4)],7)
=> ? = 1 + 1
[[1,2],[3,4],[5,7],[6]]
=> [6,5,7,3,4,1,2] => ([(0,6),(1,6),(2,5),(3,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4)],7)
=> ? = 1 + 1
[[1,4],[2,5],[3,6],[7]]
=> [7,3,6,2,5,1,4] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 1 + 1
[[1,3],[2,5],[4,6],[7]]
=> [7,4,6,2,5,1,3] => ([(1,5),(2,5),(2,6),(3,4),(3,6)],7)
=> ([(1,5),(2,5),(2,6),(3,4),(3,6)],7)
=> ? = 1 + 1
[[1,2],[3,5],[4,6],[7]]
=> [7,4,6,3,5,1,2] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ([(1,6),(2,5),(3,4),(3,6)],7)
=> ? = 1 + 1
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000153
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 4 = 2 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 4 = 2 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 4 = 2 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 4 = 2 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 4 = 2 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 5 = 3 + 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [[[[.,.],.],.],[.,[.,.]]]
=> [1,2,3,6,5,4] => 4 = 2 + 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5] => 3 = 1 + 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5] => 3 = 1 + 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5] => 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [[[.,[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5] => 3 = 1 + 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5] => 3 = 1 + 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => 4 = 2 + 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => 4 = 2 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => 4 = 2 + 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => 4 = 2 + 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => 4 = 2 + 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => 4 = 2 + 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => 4 = 2 + 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => 4 = 2 + 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [[[[.,.],.],[.,.]],[.,.]]
=> [1,2,4,3,6,5] => 4 = 2 + 2
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
=> [1,2,3,4,6,5] => 5 = 3 + 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]]
=> [1,2,3,4,6,5] => 5 = 3 + 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]]
=> [1,2,3,4,6,5] => 5 = 3 + 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]]
=> [1,2,3,4,6,5] => 5 = 3 + 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]]
=> [1,2,3,4,6,5] => 5 = 3 + 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => 6 = 4 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,7,6,5,4] => ? = 2 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 1 + 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => ? = 2 + 2
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => ? = 2 + 2
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => ? = 2 + 2
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => ? = 2 + 2
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => ? = 2 + 2
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => ? = 2 + 2
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => ? = 2 + 2
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => ? = 2 + 2
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [1,2,4,3,7,6,5] => ? = 2 + 2
[[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,7,6] => 6 = 4 + 2
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,7,6] => 6 = 4 + 2
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,7,6] => 6 = 4 + 2
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,7,6] => 6 = 4 + 2
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,7,6] => 6 = 4 + 2
[[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,7,6] => 6 = 4 + 2
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => 7 = 5 + 2
[[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 3 = 1 + 2
[[1,3,7,8],[2,5],[4,6]]
=> [4,6,2,5,1,3,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 3 = 1 + 2
[[1,2,7,8],[3,5],[4,6]]
=> [4,6,3,5,1,2,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 3 = 1 + 2
[[1,3,7,8],[2,4],[5,6]]
=> [5,6,2,4,1,3,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 3 = 1 + 2
[[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 3 = 1 + 2
[[1,4,6,8],[2,5],[3,7]]
=> [3,7,2,5,1,4,6,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 3 = 1 + 2
[[1,2,6,8],[3,5],[4,7]]
=> [4,7,3,5,1,2,6,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [2,1,4,3,8,7,6,5] => 3 = 1 + 2
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000308
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 62%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 62%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 2 + 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 4 = 2 + 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [4,1,2,3,5] => 4 = 2 + 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [3,1,2,4,5] => 4 = 2 + 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [2,1,3,4,5] => 4 = 2 + 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 5 = 3 + 2
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => [6,5,1,2,3,4] => 4 = 2 + 2
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [5,3,2,1,4,6] => [6,4,1,2,3,5] => 4 = 2 + 2
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [5,4,2,1,3,6] => [6,3,1,2,4,5] => 4 = 2 + 2
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [5,4,3,1,2,6] => [6,2,1,3,4,5] => 4 = 2 + 2
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [6,3,2,1,4,5] => [5,4,1,2,3,6] => 4 = 2 + 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6,4,2,1,3,5] => [5,3,1,2,4,6] => 4 = 2 + 2
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [6,4,3,1,2,5] => [5,2,1,3,4,6] => 4 = 2 + 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [6,5,2,1,3,4] => [4,3,1,2,5,6] => 4 = 2 + 2
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [6,5,3,1,2,4] => [4,2,1,3,5,6] => 4 = 2 + 2
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [6,5,4,1,2,3] => [3,2,1,4,5,6] => 4 = 2 + 2
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 3 = 1 + 2
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 3 = 1 + 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 3 = 1 + 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 3 = 1 + 2
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => 3 = 1 + 2
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,2,6,1,5] => [5,1,6,2,3,4] => 4 = 2 + 2
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [4,3,2,6,1,5] => [5,1,6,2,3,4] => 4 = 2 + 2
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [4,3,2,6,1,5] => [5,1,6,2,3,4] => 4 = 2 + 2
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [4,3,2,6,1,5] => [5,1,6,2,3,4] => 4 = 2 + 2
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [6,3,2,5,1,4] => [4,1,5,2,3,6] => 4 = 2 + 2
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [6,3,2,5,1,4] => [4,1,5,2,3,6] => 4 = 2 + 2
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [6,3,2,5,1,4] => [4,1,5,2,3,6] => 4 = 2 + 2
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [6,5,2,4,1,3] => [3,1,4,2,5,6] => 4 = 2 + 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [6,5,2,4,1,3] => [3,1,4,2,5,6] => 4 = 2 + 2
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => [6,1,2,3,4,5] => 5 = 3 + 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [6,4,3,2,1,5] => [5,1,2,3,4,6] => 5 = 3 + 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [6,5,3,2,1,4] => [4,1,2,3,5,6] => 5 = 3 + 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [6,5,4,2,1,3] => [3,1,2,4,5,6] => 5 = 3 + 2
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [6,5,4,3,1,2] => [2,1,3,4,5,6] => 5 = 3 + 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6 = 4 + 2
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [7,6,5,1,2,3,4] => 4 = 2 + 2
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [5,3,2,1,4,6,7] => [7,6,4,1,2,3,5] => 4 = 2 + 2
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [5,4,2,1,3,6,7] => [7,6,3,1,2,4,5] => 4 = 2 + 2
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [5,4,3,1,2,6,7] => [7,6,2,1,3,4,5] => 4 = 2 + 2
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [6,3,2,1,4,5,7] => [7,5,4,1,2,3,6] => 4 = 2 + 2
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [6,4,2,1,3,5,7] => [7,5,3,1,2,4,6] => 4 = 2 + 2
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [6,4,3,1,2,5,7] => [7,5,2,1,3,4,6] => 4 = 2 + 2
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [6,5,2,1,3,4,7] => [7,4,3,1,2,5,6] => 4 = 2 + 2
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [6,5,3,1,2,4,7] => [7,4,2,1,3,5,6] => 4 = 2 + 2
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [6,5,4,1,2,3,7] => [7,3,2,1,4,5,6] => 4 = 2 + 2
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [6,5,4,1,2,3,7] => 4 = 2 + 2
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [7,4,2,1,3,5,6] => [6,5,3,1,2,4,7] => 4 = 2 + 2
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [7,4,3,1,2,5,6] => [6,5,2,1,3,4,7] => 4 = 2 + 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [7,5,2,1,3,4,6] => [6,4,3,1,2,5,7] => 4 = 2 + 2
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [3,6,2,5,1,4,7] => [7,4,1,5,2,6,3] => ? = 1 + 2
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [3,6,2,5,1,4,7] => [7,4,1,5,2,6,3] => ? = 1 + 2
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [3,6,2,5,1,4,7] => [7,4,1,5,2,6,3] => ? = 1 + 2
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [3,6,2,5,1,4,7] => [7,4,1,5,2,6,3] => ? = 1 + 2
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [3,6,2,5,1,4,7] => [7,4,1,5,2,6,3] => ? = 1 + 2
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,7,2,5,1,4,6] => [6,4,1,5,2,7,3] => ? = 1 + 2
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [3,7,2,5,1,4,6] => [6,4,1,5,2,7,3] => ? = 1 + 2
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [3,7,2,5,1,4,6] => [6,4,1,5,2,7,3] => ? = 1 + 2
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [3,7,2,6,1,4,5] => [5,4,1,6,2,7,3] => ? = 1 + 2
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [4,7,2,6,1,3,5] => [5,3,1,6,2,7,4] => ? = 1 + 2
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [4,7,3,6,1,2,5] => [5,2,1,6,3,7,4] => ? = 1 + 2
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [4,7,2,6,1,3,5] => [5,3,1,6,2,7,4] => ? = 1 + 2
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [4,7,3,6,1,2,5] => [5,2,1,6,3,7,4] => ? = 1 + 2
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [4,7,3,6,1,2,5] => [5,2,1,6,3,7,4] => ? = 1 + 2
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [4,7,2,6,1,3,5] => [5,3,1,6,2,7,4] => ? = 1 + 2
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [4,7,3,6,1,2,5] => [5,2,1,6,3,7,4] => ? = 1 + 2
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [4,7,3,6,1,2,5] => [5,2,1,6,3,7,4] => ? = 1 + 2
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [4,3,2,6,1,5,7] => [7,5,1,6,2,3,4] => ? = 2 + 2
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [4,3,2,6,1,5,7] => [7,5,1,6,2,3,4] => ? = 2 + 2
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [4,3,2,6,1,5,7] => [7,5,1,6,2,3,4] => ? = 2 + 2
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [4,3,2,6,1,5,7] => [7,5,1,6,2,3,4] => ? = 2 + 2
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [6,3,2,5,1,4,7] => [7,4,1,5,2,3,6] => ? = 2 + 2
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [6,3,2,5,1,4,7] => [7,4,1,5,2,3,6] => ? = 2 + 2
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [6,3,2,5,1,4,7] => [7,4,1,5,2,3,6] => ? = 2 + 2
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [4,3,2,7,1,5,6] => [6,5,1,7,2,3,4] => ? = 2 + 2
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [5,3,2,7,1,4,6] => [6,4,1,7,2,3,5] => ? = 2 + 2
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [5,4,2,7,1,3,6] => [6,3,1,7,2,4,5] => ? = 2 + 2
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [5,4,3,7,1,2,6] => [6,2,1,7,3,4,5] => ? = 2 + 2
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [5,3,2,7,1,4,6] => [6,4,1,7,2,3,5] => ? = 2 + 2
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [5,4,2,7,1,3,6] => [6,3,1,7,2,4,5] => ? = 2 + 2
[[1,2,5],[3,7],[4],[6]]
=> [6,4,3,7,1,2,5] => [5,4,3,7,1,2,6] => [6,2,1,7,3,4,5] => ? = 2 + 2
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [5,4,2,7,1,3,6] => [6,3,1,7,2,4,5] => ? = 2 + 2
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [5,4,3,7,1,2,6] => [6,2,1,7,3,4,5] => ? = 2 + 2
[[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [5,4,3,7,1,2,6] => [6,2,1,7,3,4,5] => ? = 2 + 2
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [7,3,2,5,1,4,6] => [6,4,1,5,2,3,7] => ? = 2 + 2
[[1,3,6],[2,5],[4],[7]]
=> [7,4,2,5,1,3,6] => [7,3,2,5,1,4,6] => [6,4,1,5,2,3,7] => ? = 2 + 2
[[1,2,6],[3,5],[4],[7]]
=> [7,4,3,5,1,2,6] => [7,3,2,5,1,4,6] => [6,4,1,5,2,3,7] => ? = 2 + 2
[[1,4,5],[2,6],[3],[7]]
=> [7,3,2,6,1,4,5] => [7,3,2,6,1,4,5] => [5,4,1,6,2,3,7] => ? = 2 + 2
[[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => [7,4,2,6,1,3,5] => [5,3,1,6,2,4,7] => ? = 2 + 2
[[1,2,5],[3,6],[4],[7]]
=> [7,4,3,6,1,2,5] => [7,4,3,6,1,2,5] => [5,2,1,6,3,4,7] => ? = 2 + 2
[[1,3,4],[2,6],[5],[7]]
=> [7,5,2,6,1,3,4] => [7,4,2,6,1,3,5] => [5,3,1,6,2,4,7] => ? = 2 + 2
[[1,2,4],[3,6],[5],[7]]
=> [7,5,3,6,1,2,4] => [7,4,3,6,1,2,5] => [5,2,1,6,3,4,7] => ? = 2 + 2
[[1,2,3],[4,6],[5],[7]]
=> [7,5,4,6,1,2,3] => [7,4,3,6,1,2,5] => [5,2,1,6,3,4,7] => ? = 2 + 2
[[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => [4,3,7,2,6,1,5] => [5,1,6,2,7,3,4] => ? = 1 + 2
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [4,3,7,2,6,1,5] => [5,1,6,2,7,3,4] => ? = 1 + 2
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [4,3,7,2,6,1,5] => [5,1,6,2,7,3,4] => ? = 1 + 2
[[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => [4,3,7,2,6,1,5] => [5,1,6,2,7,3,4] => ? = 1 + 2
[[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => [4,3,7,2,6,1,5] => [5,1,6,2,7,3,4] => ? = 1 + 2
[[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [4,3,7,2,6,1,5] => [5,1,6,2,7,3,4] => ? = 1 + 2
[[1,2],[3,5],[4,7],[6]]
=> [6,4,7,3,5,1,2] => [4,3,7,2,6,1,5] => [5,1,6,2,7,3,4] => ? = 1 + 2
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000662
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,5,3,2,1] => [5,4,3,1,2] => 3 = 2 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => [5,4,2,3,1] => 3 = 2 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => [5,3,4,2,1] => 3 = 2 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [4,5,3,2,1] => 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 4 = 3 + 1
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 3 = 2 + 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [3,5,6,4,2,1] => [6,5,2,4,1,3] => 3 = 2 + 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [2,5,6,4,3,1] => [6,3,5,4,1,2] => 3 = 2 + 1
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [1,5,6,4,3,2] => [4,6,5,3,1,2] => 3 = 2 + 1
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [3,4,6,5,2,1] => [6,5,2,3,4,1] => 3 = 2 + 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,5,3,1] => [6,3,5,2,4,1] => 3 = 2 + 1
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [1,4,6,5,3,2] => [4,6,5,2,3,1] => 3 = 2 + 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [2,3,6,5,4,1] => [6,3,4,5,2,1] => 3 = 2 + 1
[[1,2,4],[3],[5],[6]]
=> [6,5,3,1,2,4] => [1,3,6,5,4,2] => [4,6,3,5,2,1] => 3 = 2 + 1
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => [4,5,6,3,2,1] => 3 = 2 + 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,2,5,1,4] => [1,3,5,6,4,2] => 2 = 1 + 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,6,2,5,1,3] => [1,3,6,4,5,2] => 2 = 1 + 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [4,6,3,5,1,2] => [1,6,2,4,5,3] => 2 = 1 + 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [5,6,2,4,1,3] => [1,3,6,5,2,4] => 2 = 1 + 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 2 = 1 + 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,6,3,5,2,1] => [6,5,1,3,4,2] => 3 = 2 + 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [5,6,3,4,2,1] => [6,5,1,4,2,3] => 3 = 2 + 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [5,6,2,4,3,1] => [6,2,5,4,1,3] => 3 = 2 + 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [5,6,1,4,3,2] => [3,6,5,4,1,2] => 3 = 2 + 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,6,2,5,4,1] => [6,2,4,5,3,1] => 3 = 2 + 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [4,6,2,5,3,1] => [6,2,5,3,4,1] => 3 = 2 + 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [4,6,1,5,3,2] => [3,6,5,2,4,1] => 3 = 2 + 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [2,6,1,5,4,3] => [3,5,6,4,2,1] => 3 = 2 + 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,6,1,5,4,2] => [3,6,4,5,2,1] => 3 = 2 + 1
[[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,6,4,3,2,1] => [6,5,4,3,1,2] => 4 = 3 + 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,6,5,3,2,1] => [6,5,4,2,3,1] => 4 = 3 + 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [3,6,5,4,2,1] => [6,5,3,4,2,1] => 4 = 3 + 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [2,6,5,4,3,1] => [6,4,5,3,2,1] => 4 = 3 + 1
[[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => [5,6,4,3,2,1] => 4 = 3 + 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5 = 4 + 1
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => 3 = 2 + 1
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [3,5,6,7,4,2,1] => [7,6,2,5,1,3,4] => ? = 2 + 1
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [2,5,6,7,4,3,1] => [7,3,6,5,1,2,4] => ? = 2 + 1
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,5,6,7,4,3,2] => [4,7,6,5,1,2,3] => ? = 2 + 1
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [3,4,6,7,5,2,1] => [7,6,2,3,5,1,4] => ? = 2 + 1
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [2,4,6,7,5,3,1] => [7,3,6,2,5,1,4] => ? = 2 + 1
[[1,2,5,7],[3],[4],[6]]
=> [6,4,3,1,2,5,7] => [1,4,6,7,5,3,2] => [4,7,6,2,5,1,3] => ? = 2 + 1
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [2,3,6,7,5,4,1] => [7,3,4,6,5,1,2] => ? = 2 + 1
[[1,2,4,7],[3],[5],[6]]
=> [6,5,3,1,2,4,7] => [1,3,6,7,5,4,2] => [4,7,3,6,5,1,2] => ? = 2 + 1
[[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => [1,2,6,7,5,4,3] => [4,5,7,6,3,1,2] => ? = 2 + 1
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [3,4,5,7,6,2,1] => [7,6,2,3,4,5,1] => ? = 2 + 1
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [2,4,5,7,6,3,1] => [7,3,6,2,4,5,1] => ? = 2 + 1
[[1,2,5,6],[3],[4],[7]]
=> [7,4,3,1,2,5,6] => [1,4,5,7,6,3,2] => [4,7,6,2,3,5,1] => ? = 2 + 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,3,5,7,6,4,1] => [7,3,4,6,2,5,1] => ? = 2 + 1
[[1,2,4,6],[3],[5],[7]]
=> [7,5,3,1,2,4,6] => [1,3,5,7,6,4,2] => [4,7,3,6,2,5,1] => ? = 2 + 1
[[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => [4,5,7,6,2,3,1] => ? = 2 + 1
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [2,3,4,7,6,5,1] => [7,3,4,5,6,2,1] => ? = 2 + 1
[[1,2,4,5],[3],[6],[7]]
=> [7,6,3,1,2,4,5] => [1,3,4,7,6,5,2] => [4,7,3,5,6,2,1] => ? = 2 + 1
[[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => [1,2,4,7,6,5,3] => [4,5,7,3,6,2,1] => ? = 2 + 1
[[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => ? = 2 + 1
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [3,6,7,2,5,1,4] => [1,3,5,7,6,2,4] => 2 = 1 + 1
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [4,6,7,2,5,1,3] => [1,3,7,4,6,2,5] => 2 = 1 + 1
[[1,2,7],[3,5],[4,6]]
=> [4,6,3,5,1,2,7] => [4,6,7,3,5,1,2] => [1,7,2,4,6,3,5] => ? = 1 + 1
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [5,6,7,2,4,1,3] => [1,3,7,6,2,4,5] => 2 = 1 + 1
[[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [5,6,7,3,4,1,2] => [1,7,2,6,3,4,5] => ? = 1 + 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [3,5,7,2,6,1,4] => [1,3,5,7,4,6,2] => 2 = 1 + 1
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [4,5,7,2,6,1,3] => [1,3,7,4,5,6,2] => 2 = 1 + 1
[[1,2,6],[3,5],[4,7]]
=> [4,7,3,5,1,2,6] => [4,5,7,3,6,1,2] => [1,7,2,4,5,6,3] => ? = 1 + 1
[[1,3,6],[2,4],[5,7]]
=> [5,7,2,4,1,3,6] => [2,5,7,4,6,1,3] => [1,4,7,2,5,6,3] => ? = 1 + 1
[[1,2,6],[3,4],[5,7]]
=> [5,7,3,4,1,2,6] => [3,5,7,4,6,1,2] => [1,7,3,2,5,6,4] => ? = 1 + 1
[[1,4,5],[2,6],[3,7]]
=> [3,7,2,6,1,4,5] => [3,4,7,2,6,1,5] => [1,3,5,6,7,4,2] => 2 = 1 + 1
[[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [2,4,7,3,6,1,5] => [1,4,3,6,7,5,2] => 2 = 1 + 1
[[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => [1,4,7,3,6,2,5] => [3,1,4,6,7,5,2] => ? = 1 + 1
[[1,3,4],[2,6],[5,7]]
=> [5,7,2,6,1,3,4] => [2,5,7,3,6,1,4] => [1,4,3,7,5,6,2] => 2 = 1 + 1
[[1,2,4],[3,6],[5,7]]
=> [5,7,3,6,1,2,4] => [1,5,7,3,6,2,4] => [3,1,4,7,5,6,2] => ? = 1 + 1
[[1,2,3],[4,6],[5,7]]
=> [5,7,4,6,1,2,3] => [1,5,7,4,6,2,3] => [3,1,7,2,5,6,4] => ? = 1 + 1
[[1,3,5],[2,4],[6,7]]
=> [6,7,2,4,1,3,5] => [2,6,7,4,5,1,3] => [1,4,7,2,6,3,5] => ? = 1 + 1
[[1,2,5],[3,4],[6,7]]
=> [6,7,3,4,1,2,5] => [3,6,7,4,5,1,2] => [1,7,3,2,6,4,5] => ? = 1 + 1
[[1,3,4],[2,5],[6,7]]
=> [6,7,2,5,1,3,4] => [2,6,7,3,5,1,4] => [1,4,3,7,6,2,5] => 2 = 1 + 1
[[1,2,4],[3,5],[6,7]]
=> [6,7,3,5,1,2,4] => [1,6,7,3,5,2,4] => [3,1,4,7,6,2,5] => ? = 1 + 1
[[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,6,7,4,5,2,3] => [3,1,7,2,6,4,5] => ? = 1 + 1
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [4,6,7,3,5,2,1] => [7,6,1,3,5,2,4] => ? = 2 + 1
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [5,6,7,3,4,2,1] => [7,6,1,5,2,3,4] => ? = 2 + 1
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [5,6,7,2,4,3,1] => [7,2,6,5,1,3,4] => ? = 2 + 1
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [5,6,7,1,4,3,2] => [3,7,6,5,1,2,4] => ? = 2 + 1
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [3,6,7,2,5,4,1] => [7,2,4,6,5,1,3] => ? = 2 + 1
[[1,3,7],[2,5],[4],[6]]
=> [6,4,2,5,1,3,7] => [4,6,7,2,5,3,1] => [7,2,6,3,5,1,4] => ? = 2 + 1
[[1,2,7],[3,5],[4],[6]]
=> [6,4,3,5,1,2,7] => [4,6,7,1,5,3,2] => [3,7,6,2,5,1,4] => ? = 2 + 1
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [2,6,7,1,5,4,3] => [3,5,7,6,4,1,2] => ? = 2 + 1
[[1,2,7],[3,4],[5],[6]]
=> [6,5,3,4,1,2,7] => [3,6,7,1,5,4,2] => [3,7,4,6,5,1,2] => ? = 2 + 1
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [4,5,7,3,6,2,1] => [7,6,1,3,4,5,2] => ? = 2 + 1
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [3,5,7,4,6,2,1] => [7,6,2,1,4,5,3] => ? = 2 + 1
[[1,3,6],[2,7],[4],[5]]
=> [5,4,2,7,1,3,6] => [2,5,7,4,6,3,1] => [7,3,6,1,4,5,2] => ? = 2 + 1
[[1,2,6],[3,7],[4],[5]]
=> [5,4,3,7,1,2,6] => [1,5,7,4,6,3,2] => [4,7,6,1,3,5,2] => ? = 2 + 1
[[1,4,5],[2,7],[3],[6]]
=> [6,3,2,7,1,4,5] => [3,6,7,4,5,2,1] => [7,6,2,1,5,3,4] => ? = 2 + 1
[[1,3,5],[2,7],[4],[6]]
=> [6,4,2,7,1,3,5] => [2,6,7,4,5,3,1] => [7,3,6,1,5,2,4] => ? = 2 + 1
[[1,2,5],[3,7],[4],[6]]
=> [6,4,3,7,1,2,5] => [1,6,7,4,5,3,2] => [4,7,6,1,5,2,3] => ? = 2 + 1
[[1,3,4],[2,7],[5],[6]]
=> [6,5,2,7,1,3,4] => [2,6,7,3,5,4,1] => [7,3,2,6,5,1,4] => ? = 2 + 1
[[1,2,4],[3,7],[5],[6]]
=> [6,5,3,7,1,2,4] => [1,6,7,3,5,4,2] => [4,7,2,6,5,1,3] => ? = 2 + 1
[[1,2,3],[4,7],[5],[6]]
=> [6,5,4,7,1,2,3] => [1,6,7,2,5,4,3] => [4,3,7,6,5,1,2] => ? = 2 + 1
[[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => 4 = 3 + 1
[[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => 5 = 4 + 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 6 = 5 + 1
[[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => 3 = 2 + 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
The following 144 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000007The number of saliances of the permutation. St000703The number of deficiencies of a permutation. St000366The number of double descents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000711The number of big exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St000702The number of weak deficiencies of a permutation. St000956The maximal displacement of a permutation. St000470The number of runs in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001427The number of descents of a signed permutation. St000354The number of recoils of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001489The maximum of the number of descents and the number of inverse descents. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000489The number of cycles of a permutation of length at most 3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001566The length of the longest arithmetic progression in a permutation. St000021The number of descents of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000325The width of the tree associated to a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St000317The cycle descent number of a permutation. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001812The biclique partition number of a graph. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth of a graph. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000482The (zero)-forcing number of a graph. St000536The pathwidth of a graph. St000619The number of cyclic descents of a permutation. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000822The Hadwiger number of the graph. St000831The number of indices that are either descents or recoils. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001061The number of indices that are both descents and recoils of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001391The disjunction number of a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000087The number of induced subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000166The depth minus 1 of an ordered tree. St000172The Grundy number of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000485The length of the longest cycle of a permutation. St000636The hull number of a graph. St000637The length of the longest cycle in a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St000990The first ascent of a permutation. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000094The depth of an ordered tree. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001706The number of closed sets in a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!