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Your data matches 32 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [.,.]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> [1,0,1,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 8
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 10
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 6
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 8
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> 6
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> 4
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 6
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> 8
[5,5,4]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> 7
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001727
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 60%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 60%
Values
[1]
=> [1,0]
=> [1,0]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 4
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => ? = 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ? = 8
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 4
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => ? = 10
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ? = 6
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 2
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,6,4,5,1,2,8,3] => ? = 8
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 4
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 4
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [6,5,4,1,2,7,8,3] => ? = 6
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? = 2
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,4,1,2,6,7,8,3] => ? = 4
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 6
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [4,1,2,5,6,7,8,3] => ? = 2
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 5
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [8,1,2,7,6,3,4,5] => ? = 8
[5,5,4]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [7,1,2,6,3,4,8,5] => ? = 7
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => ? = 6
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Matching statistic: St001278
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 60%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001278: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 60%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 7 = 6 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 8 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 10 + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 6 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 8 + 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 4 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 6 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5 + 1
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 8 + 1
[5,5,4]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 7 + 1
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6 + 1
Description
The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra.
See theorem 5.8. in the reference for a motivation.
Matching statistic: St001769
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 40%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 40%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 2
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => 3
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 4
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 4
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 5
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 6
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => ? = 8
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => ? = 4
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => ? = 10
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [6,7,8,1,2,3,4,5] => ? = 6
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => ? = 2
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [7,8,9,1,2,3,4,5,6] => ? = 8
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => ? = 4
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [7,8,9,4,5,6,1,2,3] => ? = 4
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => [7,8,9,10,1,2,3,4,5,6] => ? = 6
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,10,1,2,3,4,5] => ? = 2
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? => ? = 4
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => [9,10,11,5,6,7,8,1,2,3,4] => ? = 6
[6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [7,8,9,10,11,12,1,2,3,4,5,6] => [7,8,9,10,11,12,1,2,3,4,5,6] => ? = 2
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? => ? = 5
[5,5,3]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13]]
=> ? => ? => ? = 8
[5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? => ? => ? = 7
[5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? => ? => ? = 6
Description
The reflection length of a signed permutation.
This is the minimal numbers of reflections needed to express a signed permutation.
Matching statistic: St001905
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 40%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001905: Parking functions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 40%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 2
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => 3
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 4
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 4
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 5
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 6
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => ? = 8
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => ? = 4
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => ? = 10
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [6,7,8,1,2,3,4,5] => ? = 6
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => ? = 2
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [7,8,9,1,2,3,4,5,6] => ? = 8
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => ? = 4
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [7,8,9,4,5,6,1,2,3] => ? = 4
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => [7,8,9,10,1,2,3,4,5,6] => ? = 6
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,10,1,2,3,4,5] => ? = 2
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? => ? = 4
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? => ? = 6
[6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [7,8,9,10,11,12,1,2,3,4,5,6] => [7,8,9,10,11,12,1,2,3,4,5,6] => ? = 2
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ? => ? = 5
[5,5,3]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13]]
=> ? => ? => ? = 8
[5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? => ? => ? = 7
[5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? => ? => ? = 6
Description
The number of preferred parking spots in a parking function less than the index of the car.
Let $(a_1,\dots,a_n)$ be a parking function. Then this statistic returns the number of indices $1\leq i\leq n$ such that $a_i < i$.
Matching statistic: St001207
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[1]
=> [[1]]
=> [[1]]
=> [1] => ? = 0
[2]
=> [[1,2]]
=> [[1,2]]
=> [1,2] => 0
[1,1]
=> [[1],[2]]
=> [[1],[2]]
=> [2,1] => 1
[3]
=> [[1,2,3]]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => 2
[4]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,1]
=> [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 3
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 2
[5]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ? = 0
[4,1]
=> [[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ? = 4
[3,2]
=> [[1,2,5],[3,4]]
=> [[1,3,4],[2,5]]
=> [2,5,1,3,4] => ? = 4
[6]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ? = 5
[4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => ? = 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => ? = 2
[7]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ? = 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? = 6
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => ? = 8
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => ? = 4
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3,4,5,6,7],[2,8]]
=> [2,8,1,3,4,5,6,7] => ? = 10
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [[1,2,5,6,7],[3,4,8]]
=> [3,4,8,1,2,5,6,7] => ? = 6
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,2,3,7],[4,5,6,8]]
=> [4,5,6,8,1,2,3,7] => ? = 2
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,2,5,6,7,8],[3,4,9]]
=> [3,4,9,1,2,5,6,7,8] => ? = 8
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,2,3,7,8],[4,5,6,9]]
=> [4,5,6,9,1,2,3,7,8] => ? = 4
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,2,5],[3,4,8],[6,7,9]]
=> [6,7,9,3,4,8,1,2,5] => ? = 4
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [[1,2,3,7,8,9],[4,5,6,10]]
=> [4,5,6,10,1,2,3,7,8,9] => ? = 6
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> [5,6,7,8,10,1,2,3,4,9] => ? = 2
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ?
=> ? => ? = 4
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,2,5,6],[3,4,9,10],[7,8,11]]
=> ? => ? = 6
[6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> [6,7,8,9,10,12,1,2,3,4,5,11] => ? = 2
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ?
=> ? => ? = 5
[5,5,3]
=> [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
=> ?
=> ? => ? = 8
[5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ?
=> ? => ? = 7
[5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ?
=> ? => ? = 6
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001583
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[1]
=> [1]
=> [[1]]
=> [1] => ? = 0
[2]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[3]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? = 0
[4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ? = 4
[3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ? = 4
[6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? = 0
[5,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ? = 5
[4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? = 6
[3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? = 2
[7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 0
[6,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 6
[5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 8
[4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? = 4
[6,2]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => ? = 10
[5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => ? = 6
[4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2
[6,3]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => ? = 8
[5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => ? = 4
[3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => ? = 4
[6,4]
=> [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> [9,7,5,10,3,8,2,6,1,4] => ? = 6
[5,5]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [9,10,7,8,5,6,3,4,1,2] => ? = 2
[6,5]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 4
[4,4,3]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 6
[6,6]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 2
[4,4,4]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 5
[5,5,3]
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? => ? = 8
[5,5,4]
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? => ? = 7
[5,5,5]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? => ? = 6
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St000091
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 30%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 30%
Values
[1]
=> [1,0]
=> 10 => [1,2] => 1 = 0 + 1
[2]
=> [1,0,1,0]
=> 1010 => [1,2,2] => 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> 1100 => [1,1,3] => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> 101010 => [1,2,2,2] => 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => [1,2,1,3] => 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => [1,2,2,2,2] => ? = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => [1,2,2,1,3] => ? = 3 + 1
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => [1,1,1,4] => 3 = 2 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => [1,2,2,2,2,2] => ? = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => [1,2,2,2,1,3] => ? = 4 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => [1,2,1,1,4] => ? = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => [1,2,2,2,2,2,2] => ? = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => [1,2,2,2,2,1,3] => ? = 5 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => [1,2,2,1,1,4] => ? = 6 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => [1,1,1,2,4] => ? = 2 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => [1,2,2,2,2,2,2,2] => ? = 0 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => [1,2,2,2,2,2,1,3] => ? = 6 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => [1,2,2,2,1,1,4] => ? = 8 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => [1,2,1,1,2,4] => ? = 4 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => [1,2,2,2,2,1,1,4] => ? = 10 + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => [1,2,2,1,1,2,4] => ? = 6 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => [1,1,1,2,2,4] => ? = 2 + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => [1,2,2,2,1,1,2,4] => ? = 8 + 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 101110101000 => [1,2,1,1,2,2,4] => ? = 4 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => [1,1,1,1,1,6] => ? = 4 + 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 10101110101000 => [1,2,2,1,1,2,2,4] => ? = 6 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 111010101000 => [1,1,1,2,2,2,4] => ? = 2 + 1
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> 10111010101000 => [1,2,1,1,2,2,2,4] => ? = 4 + 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 111011100000 => [1,1,1,2,1,1,6] => ? = 6 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 11101010101000 => [1,1,1,2,2,2,2,4] => ? = 2 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 111110100000 => [1,1,1,1,1,2,6] => ? = 5 + 1
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 11101011100000 => [1,1,1,2,2,1,1,6] => ? = 8 + 1
[5,5,4]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> 11101110100000 => [1,1,1,2,1,1,2,6] => ? = 7 + 1
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 11111010100000 => [1,1,1,1,1,2,2,6] => ? = 6 + 1
Description
The descent variation of a composition.
Defined in [1].
Matching statistic: St001632
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 18%●distinct values known / distinct values provided: 10%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 18%●distinct values known / distinct values provided: 10%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> ? = 0 + 1
[2]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1]
=> [[1],[2]]
=> [2,1] => ([],2)
=> ? = 1 + 1
[3]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 + 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 + 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2 + 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? = 4 + 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? = 4 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? = 5 + 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 6 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ? = 2 + 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 6 + 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 8 + 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 4 + 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? = 10 + 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
=> ? = 6 + 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
=> ? = 2 + 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
=> ? = 8 + 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
=> ? = 4 + 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => ([(0,8),(1,7),(2,6),(6,3),(7,4),(8,5)],9)
=> ? = 4 + 1
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => ([(0,8),(1,9),(4,6),(5,4),(6,3),(7,2),(8,7),(9,5)],10)
=> ? = 6 + 1
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => ([(0,9),(1,8),(4,6),(5,7),(6,2),(7,3),(8,4),(9,5)],10)
=> ? = 2 + 1
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ?
=> ? = 4 + 1
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ?
=> ? = 6 + 1
[6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [7,8,9,10,11,12,1,2,3,4,5,6] => ([(0,11),(1,10),(4,8),(5,9),(6,4),(7,5),(8,2),(9,3),(10,6),(11,7)],12)
=> ? = 2 + 1
[4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? => ?
=> ? = 5 + 1
[5,5,3]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13]]
=> ? => ?
=> ? = 8 + 1
[5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? => ?
=> ? = 7 + 1
[5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? => ?
=> ? = 6 + 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001645
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 60%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 60%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> ? = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 3 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ? = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? = 4 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? = 0 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? = 5 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 6 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? = 0 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ? = 6 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 8 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ? = 10 + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8 + 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,7,3,4,5,6,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(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)
=> ? = 6 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => ([(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)
=> 6 = 5 + 1
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8 + 1
[5,5,4]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7 + 1
[5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
Description
The pebbling number of a connected graph.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001778The largest greatest common divisor of an element and its image in a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001375The pancake length of a permutation. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001095The number of non-isomorphic posets with precisely one further covering relation. St001520The number of strict 3-descents. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001948The number of augmented double ascents of a permutation. St000736The last entry in the first row of a semistandard tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000075The orbit size of a standard tableau under promotion. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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