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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001492
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(load all 5 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St000105
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => {{1}}
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000925
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 3
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [2,4,1,3,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [3,2,4,1,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,1,4] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [4,2,3,5,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [2,4,5,1,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,3,1,6,5,7] => [3,4,1,2,6,5,7] => ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,5,3,7] => [2,1,5,6,3,4,7] => ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,6,4,3,5,1,7] => [4,3,5,6,1,2,7] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,5,6] => ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,3,1,6,7,5] => [3,4,1,2,7,5,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,5,7,3] => [2,1,5,7,3,4,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => [3,5,7,1,2,4,6] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,6,4,3,5,7,1] => [4,3,5,7,1,2,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,6,5,3] => [2,1,6,5,7,3,4] => ? = 4 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => [3,6,5,7,1,2,4] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,7,4,3,6,5,1] => [4,3,6,5,7,1,2] => ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => [5,6,3,4,7,1,2] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => [2,1,4,3,6,7,5] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,3,1,7,6,5] => [3,4,1,2,6,7,5] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,7,5,6,3] => [2,1,5,6,7,3,4] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,7,5,6,1] => [3,5,6,7,1,2,4] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,4,3,5,6,1] => [4,3,5,6,7,1,2] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,3,1,5,7,6] => [3,4,1,2,5,7,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [2,1,5,3,4,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,3,5,1,7,6] => [3,5,1,2,4,7,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,4,3,1,7,6] => [4,3,5,1,2,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,6,3] => [2,1,6,7,3,4,5] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => [3,6,7,1,2,4,5] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,5,4,3,7,6,1] => [4,3,6,7,1,2,5] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,7,4,5,3,6,1] => [5,3,4,6,7,1,2] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,3,4,6,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,3,5,1,6,7] => [3,5,1,2,4,6,7] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,4,3,1,6,7] => [4,3,5,1,2,6,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,5,4,3,6,1,7] => [4,3,6,1,2,5,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,4,5,3,1,7] => [5,3,4,6,1,2,7] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => [3,7,1,2,4,5,6] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => [4,3,7,1,2,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,6,4,5,3,7,1] => [5,3,4,7,1,2,6] => ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => [6,3,4,5,7,1,2] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,4,6,3,7] => [2,1,4,6,3,5,7] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,3,4,6,1,7] => [3,4,6,1,2,5,7] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => [2,1,5,4,6,3,7] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,6,3,5,4,1,7] => [3,5,4,6,1,2,7] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [4,5,3,6,1,2,7] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,4,6,7,3] => [2,1,4,7,3,5,6] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,3,4,6,7,1] => [3,4,7,1,2,5,6] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,5,4,7,3] => [2,1,5,4,7,3,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,6,3,5,4,7,1] => [3,5,4,7,1,2,6] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => [4,5,3,7,1,2,6] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => [2,1,6,4,5,7,3] => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,7,3,5,6,4,1] => [3,6,4,5,7,1,2] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => [4,6,3,5,7,1,2] => ? = 5 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000702
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 3
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 3
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => 3
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 3
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 3
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 4
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,1,4,3,2] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 4
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [4,2,1,3,6,5,7] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => [6,2,1,4,3,5,7] => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => [6,2,4,3,1,5,7] => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => [4,2,1,3,7,6,5] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [2,1,7,4,3,6,5] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [7,2,1,4,3,6,5] => ? = 6
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [7,2,4,3,1,6,5] => ? = 5
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => [2,1,7,4,6,5,3] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [7,2,1,4,6,5,3] => ? = 5
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [7,2,4,3,6,5,1] => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [7,2,6,4,3,5,1] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => ? = 4
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => [4,2,1,3,7,5,6] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [2,1,7,4,3,5,6] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [7,2,1,4,3,5,6] => ? = 6
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [7,2,4,3,1,5,6] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [4,2,1,3,5,7,6] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [5,2,1,4,3,7,6] => ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => [5,2,4,3,1,7,6] => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [2,1,7,4,5,3,6] => ? = 5
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [7,2,1,4,5,3,6] => ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [7,2,4,3,5,1,6] => ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [7,2,5,4,3,1,6] => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => ? = 5
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [5,2,1,4,3,6,7] => ? = 6
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2,4,3,1,6,7] => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => [6,2,1,4,5,3,7] => ? = 6
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => [6,2,4,3,5,1,7] => ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => [6,2,5,4,3,1,7] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => [2,1,7,4,5,6,3] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [7,2,1,4,5,6,3] => ? = 6
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [7,2,4,3,5,6,1] => ? = 5
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => [7,2,5,4,3,6,1] => ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => [7,2,6,4,5,3,1] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => [2,1,6,3,5,4,7] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [6,2,1,3,5,4,7] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [2,1,6,5,4,3,7] => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => [6,2,1,5,4,3,7] => ? = 5
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => [6,2,5,3,4,1,7] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [2,1,7,3,5,6,4] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [7,2,1,3,5,6,4] => ? = 6
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000672
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,1,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,1,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,1,2,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,1,2,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,1,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,1,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,1,2,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [2,4,1,3,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [3,2,4,1,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [2,5,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [3,2,5,1,4] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [4,2,3,5,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [2,4,5,1,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [3,2,4,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,3,1,4,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,3,1,6,5,7] => [3,4,1,2,6,5,7] => ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,5,3,7] => [2,1,5,6,3,4,7] => ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,6,4,3,5,1,7] => [4,3,5,6,1,2,7] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,5,6] => ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,3,1,6,7,5] => [3,4,1,2,7,5,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,5,7,3] => [2,1,5,7,3,4,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => [3,5,7,1,2,4,6] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,6,4,3,5,7,1] => [4,3,5,7,1,2,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,6,5,3] => [2,1,6,5,7,3,4] => ? = 4 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => [3,6,5,7,1,2,4] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,7,4,3,6,5,1] => [4,3,6,5,7,1,2] => ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => [5,6,3,4,7,1,2] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => [2,1,4,3,6,7,5] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,3,1,7,6,5] => [3,4,1,2,6,7,5] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,7,5,6,3] => [2,1,5,6,7,3,4] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,7,5,6,1] => [3,5,6,7,1,2,4] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,4,3,5,6,1] => [4,3,5,6,7,1,2] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,3,1,5,7,6] => [3,4,1,2,5,7,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => [2,1,5,3,4,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,3,5,1,7,6] => [3,5,1,2,4,7,6] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,4,3,1,7,6] => [4,3,5,1,2,7,6] => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,6,3] => [2,1,6,7,3,4,5] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => [3,6,7,1,2,4,5] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,5,4,3,7,6,1] => [4,3,6,7,1,2,5] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,7,4,5,3,6,1] => [5,3,4,6,7,1,2] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,3,4,6,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,3,5,1,6,7] => [3,5,1,2,4,6,7] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,4,3,1,6,7] => [4,3,5,1,2,6,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,5,4,3,6,1,7] => [4,3,6,1,2,5,7] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,4,5,3,1,7] => [5,3,4,6,1,2,7] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => [3,7,1,2,4,5,6] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => [4,3,7,1,2,5,6] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,6,4,5,3,7,1] => [5,3,4,7,1,2,6] => ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => [6,3,4,5,7,1,2] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,4,6,3,7] => [2,1,4,6,3,5,7] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,3,4,6,1,7] => [3,4,6,1,2,5,7] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => [2,1,5,4,6,3,7] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,6,3,5,4,1,7] => [3,5,4,6,1,2,7] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [4,5,3,6,1,2,7] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,4,6,7,3] => [2,1,4,7,3,5,6] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,3,4,6,7,1] => [3,4,7,1,2,5,6] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,5,4,7,3] => [2,1,5,4,7,3,6] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,6,3,5,4,7,1] => [3,5,4,7,1,2,6] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => [4,5,3,7,1,2,6] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,7,5,6,4,3] => [2,1,6,4,5,7,3] => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,7,3,5,6,4,1] => [3,6,4,5,7,1,2] => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => [4,6,3,5,7,1,2] => ? = 5 - 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: St000213
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 4
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,3,1,6,5,7] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,5,3,7] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,4,3,6,5,1,7] => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,6,4,3,5,1,7] => ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,3,1,6,7,5] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,5,7,3] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => ? = 6
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,6,4,3,5,7,1] => ? = 5
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,6,5,3] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => ? = 5
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,7,4,3,6,5,1] => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 4
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,4,3,1,7,6,5] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,7,5,6,3] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,7,5,6,1] => ? = 6
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,4,3,5,6,1] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,3,1,5,7,6] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [2,4,3,5,1,7,6] => ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,5,4,3,1,7,6] => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,6,3] => ? = 5
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => ? = 6
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,5,4,3,7,6,1] => ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,7,4,5,3,6,1] => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 5
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7] => ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => ? = 5
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,3,5,1,6,7] => ? = 6
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,4,3,1,6,7] => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [2,4,3,5,6,1,7] => ? = 6
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,5,4,3,6,1,7] => ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,4,5,3,1,7] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => ? = 5
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => ? = 6
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,5,4,3,6,7,1] => ? = 5
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,6,4,5,3,7,1] => ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,4,6,3,7] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,3,4,6,1,7] => ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,6,3,5,4,1,7] => ? = 5
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,4,6,7,3] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,3,4,6,7,1] => ? = 6
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,5,4,7,3] => ? = 4
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000155
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [3,2,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [4,3,1,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [4,3,2,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [3,4,2,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,4,3] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [2,4,1,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,3,5,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,2,1,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [4,3,1,5,2] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,3,5,4,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [3,2,5,1,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [4,3,5,1,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [5,3,4,1,2] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [3,5,4,2,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [5,4,3,2,1] => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [3,1,4,5,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,4,3,5,2] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,5,4,3,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [3,5,4,1,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,4,3,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,4,2,1,5] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [2,5,4,1,3] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,1] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [3,2,5,4,7,6,1] => ? = 4 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [4,2,1,3,6,5,7] => [4,3,5,2,7,6,1] => ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => [3,2,6,5,7,4,1] => ? = 5 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => [6,2,1,4,3,5,7] => [4,3,6,5,7,2,1] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => [6,2,4,3,1,5,7] => [6,3,5,4,7,2,1] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [3,2,5,4,1,7,6] => ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => [4,2,1,3,7,6,5] => [4,3,5,2,1,7,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [2,1,7,4,3,6,5] => [3,2,6,5,1,7,4] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [7,2,1,4,3,6,5] => [4,3,6,5,1,7,2] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [7,2,4,3,1,6,5] => [6,3,5,4,1,7,2] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => [2,1,7,4,6,5,3] => [3,2,1,5,7,6,4] => ? = 4 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [7,2,1,4,6,5,3] => [4,3,1,5,7,6,2] => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [7,2,4,3,6,5,1] => [1,3,5,4,7,6,2] => ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [7,2,6,4,3,5,1] => [1,3,6,5,7,4,2] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => [3,2,5,4,7,1,6] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => [4,2,1,3,7,5,6] => [4,3,5,2,7,1,6] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [2,1,7,4,3,5,6] => [3,2,6,5,7,1,4] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [7,2,1,4,3,5,6] => [4,3,6,5,7,1,2] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [7,2,4,3,1,5,6] => [6,3,5,4,7,1,2] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [3,2,5,4,6,1,7] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [4,2,1,3,5,7,6] => [4,3,5,2,6,1,7] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [3,2,6,5,4,1,7] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [5,2,1,4,3,7,6] => [4,3,6,5,2,1,7] => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => [5,2,4,3,1,7,6] => [6,3,5,4,2,1,7] => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [2,1,7,4,5,3,6] => [3,2,7,5,6,1,4] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [7,2,1,4,5,3,6] => [4,3,7,5,6,1,2] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [7,2,4,3,5,1,6] => [7,3,5,4,6,1,2] => ? = 5 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [7,2,5,4,3,1,6] => [7,3,6,5,4,1,2] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [3,2,5,4,6,7,1] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => [4,2,1,3,5,6,7] => [4,3,5,2,6,7,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [3,2,6,5,4,7,1] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [5,2,1,4,3,6,7] => [4,3,6,5,2,7,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => [5,2,4,3,1,6,7] => [6,3,5,4,2,7,1] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [3,2,7,5,6,4,1] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => [6,2,1,4,5,3,7] => [4,3,7,5,6,2,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => [6,2,4,3,5,1,7] => [7,3,5,4,6,2,1] => ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => [6,2,5,4,3,1,7] => [7,3,6,5,4,2,1] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => [2,1,7,4,5,6,3] => [3,2,1,5,6,7,4] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [7,2,1,4,5,6,3] => [4,3,1,5,6,7,2] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [7,2,4,3,5,6,1] => [1,3,5,4,6,7,2] => ? = 5 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => [7,2,5,4,3,6,1] => [1,3,6,5,4,7,2] => ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => [7,2,6,4,5,3,1] => [1,3,7,5,6,4,2] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => [2,1,6,3,5,4,7] => [3,2,5,7,6,4,1] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [6,2,1,3,5,4,7] => [4,3,5,7,6,2,1] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [2,1,6,5,4,3,7] => [3,2,7,6,5,4,1] => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => [6,2,1,5,4,3,7] => [4,3,7,6,5,2,1] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => [6,2,5,3,4,1,7] => [7,3,5,6,4,2,1] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [2,1,7,3,5,6,4] => [3,2,5,1,6,7,4] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [7,2,1,3,5,6,4] => [4,3,5,1,6,7,2] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => [2,1,7,5,4,6,3] => [3,2,1,6,5,7,4] => ? = 4 - 1
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St001508
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001508: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001508: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 5
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 5
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 5
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 5
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 5
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 5
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 5
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 4
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 5
Description
The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary.
Given two lattice paths $U,L$ from $(0,0)$ to $(d,n-d)$, [1] describes a bijection between lattice paths weakly between $U$ and $L$ and subsets of $\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid $M[U,L]$.
This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly above the diagonal and relative to the diagonal boundary.
Matching statistic: St001863
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 71%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,2,3,1] => 3
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ? = 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1,5] => ? = 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,5,3] => ? = 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,4,3,5,1] => ? = 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,5,3,4,1] => ? = 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ? = 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,5,4] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,3,5,4,1] => ? = 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ? = 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ? = 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => ? = 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => ? = 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [5,3,4,2,1] => ? = 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => ? = 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,5,4,2] => 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 4
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [5,2,4,3,1] => ? = 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ? = 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,3,5,1] => ? = 4
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,5,2] => 4
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => ? = 4
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => 4
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 4
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => 4
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => ? = 4
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,5,3,4,2] => 4
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => 4
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => [2,4,3,1,6,5] => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => [2,1,4,6,5,3] => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => [2,4,3,6,5,1] => ? = 5
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => [2,6,4,3,5,1] => ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => [2,4,3,1,5,6] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,4,5,3,6] => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,5,1,6] => [2,4,3,5,1,6] => ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [2,5,4,3,1,6] => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => [2,1,4,5,6,3] => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => [2,4,3,5,6,1] => ? = 5
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => [2,5,4,3,6,1] => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => [2,6,4,5,3,1] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => [2,1,5,4,6,3] => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => [2,5,3,4,6,1] => ? = 5
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => [2,6,3,5,4,1] => ? = 4
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [2,6,5,4,3,1] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => [2,1,5,4,3,6] => ? = 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,5,3,4,1,6] => [2,5,3,4,1,6] => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => ? = 4
Description
The number of weak excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
The following 1 statistic also match your data. Click on any of them to see the details.
St001626The number of maximal proper sublattices of a lattice.
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