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Your data matches 44 different statistics following compositions of up to 3 maps.
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Matching statistic: St000971
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => {{1}}
=> 1
[1,2] => [2,1] => [2,1] => {{1,2}}
=> 2
[2,1] => [1,2] => [1,2] => {{1},{2}}
=> 1
[1,2,3] => [3,2,1] => [2,3,1] => {{1,2,3}}
=> 3
[1,3,2] => [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[2,1,3] => [3,1,2] => [3,1,2] => {{1,2,3}}
=> 3
[2,3,1] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[3,1,2] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[3,2,1] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 1
[1,2,3,4] => [4,3,2,1] => [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,2,4,3] => [3,4,2,1] => [2,4,3,1] => {{1,2,4},{3}}
=> 3
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => {{1,2,3,4}}
=> 4
[1,3,4,2] => [2,4,3,1] => [3,2,4,1] => {{1,3,4},{2}}
=> 2
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[2,1,3,4] => [4,3,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 4
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => {{1,2,4},{3}}
=> 3
[2,3,1,4] => [4,1,3,2] => [4,3,1,2] => {{1,2,3,4}}
=> 4
[2,3,4,1] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 3
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[3,1,2,4] => [4,2,1,3] => [2,4,1,3] => {{1,2,3,4}}
=> 4
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 2
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => {{1,2,3,4}}
=> 4
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[4,1,2,3] => [3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 3
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 3
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 5
[1,2,3,5,4] => [4,5,3,2,1] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 4
[1,2,4,3,5] => [5,3,4,2,1] => [2,4,5,3,1] => {{1,2,3,4,5}}
=> 5
[1,2,4,5,3] => [3,5,4,2,1] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 3
[1,2,5,3,4] => [4,3,5,2,1] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 4
[1,2,5,4,3] => [3,4,5,2,1] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 3
[1,3,2,4,5] => [5,4,2,3,1] => [3,4,2,5,1] => {{1,2,3,4,5}}
=> 5
[1,3,2,5,4] => [4,5,2,3,1] => [3,5,2,4,1] => {{1,2,3,5},{4}}
=> 4
[1,3,4,2,5] => [5,2,4,3,1] => [3,4,5,1,2] => {{1,2,3,4,5}}
=> 5
[1,3,4,5,2] => [2,5,4,3,1] => [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 2
[1,3,5,2,4] => [4,2,5,3,1] => [3,5,4,1,2] => {{1,3,4},{2,5}}
=> 4
[1,3,5,4,2] => [2,4,5,3,1] => [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 2
[1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => {{1,2,3,4,5}}
=> 5
[1,4,2,5,3] => [3,5,2,4,1] => [4,5,3,2,1] => {{1,2,4,5},{3}}
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => [4,5,2,3,1] => {{1,2,3,4,5}}
=> 5
[1,4,3,5,2] => [2,5,3,4,1] => [4,2,5,3,1] => {{1,3,4,5},{2}}
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 1
[1,2] => [2,1] => [1,1,0,0]
=> [2] => 2
[2,1] => [1,2] => [1,0,1,0]
=> [1,1] => 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1] => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 3
[6,5,7,4,3,8,2,1] => [1,2,8,3,4,7,5,6] => ?
=> ? => ? = 1
[6,7,4,3,5,8,2,1] => [1,2,8,5,3,4,7,6] => ?
=> ? => ? = 1
[6,5,3,4,7,8,2,1] => [1,2,8,7,4,3,5,6] => ?
=> ? => ? = 1
[6,4,3,5,7,8,2,1] => [1,2,8,7,5,3,4,6] => ?
=> ? => ? = 1
[8,5,4,6,2,3,7,1] => [1,7,3,2,6,4,5,8] => ?
=> ? => ? = 1
[8,6,3,4,2,5,7,1] => [1,7,5,2,4,3,6,8] => ?
=> ? => ? = 1
[8,4,5,3,2,6,7,1] => [1,7,6,2,3,5,4,8] => ?
=> ? => ? = 1
[8,5,2,3,4,6,7,1] => [1,7,6,4,3,2,5,8] => ?
=> ? => ? = 1
[8,4,3,2,5,6,7,1] => [1,7,6,5,2,3,4,8] => ?
=> ? => ? = 1
[6,7,4,5,3,2,8,1] => [1,8,2,3,5,4,7,6] => ?
=> ? => ? = 1
[6,5,4,7,3,2,8,1] => [1,8,2,3,7,4,5,6] => ?
=> ? => ? = 1
[5,6,4,7,3,2,8,1] => [1,8,2,3,7,4,6,5] => ?
=> ? => ? = 1
[6,7,5,3,4,2,8,1] => [1,8,2,4,3,5,7,6] => ?
=> ? => ? = 1
[7,6,3,4,5,2,8,1] => [1,8,2,5,4,3,6,7] => ?
=> ? => ? = 1
[7,3,4,5,6,2,8,1] => [1,8,2,6,5,4,3,7] => ?
=> ? => ? = 1
[7,5,6,4,2,3,8,1] => [1,8,3,2,4,6,5,7] => ?
=> ? => ? = 1
[5,6,4,7,2,3,8,1] => [1,8,3,2,7,4,6,5] => ?
=> ? => ? = 1
[7,6,4,3,2,5,8,1] => [1,8,5,2,3,4,6,7] => ?
=> ? => ? = 1
[6,7,3,4,2,5,8,1] => [1,8,5,2,4,3,7,6] => ?
=> ? => ? = 1
[7,4,5,3,2,6,8,1] => [1,8,6,2,3,5,4,7] => ?
=> ? => ? = 1
[7,4,3,2,5,6,8,1] => [1,8,6,5,2,3,4,7] => ?
=> ? => ? = 1
[7,3,4,2,5,6,8,1] => [1,8,6,5,2,4,3,7] => ?
=> ? => ? = 1
[6,5,3,4,2,7,8,1] => [1,8,7,2,4,3,5,6] => ?
=> ? => ? = 1
[5,6,3,4,2,7,8,1] => [1,8,7,2,4,3,6,5] => ?
=> ? => ? = 1
[5,3,4,6,2,7,8,1] => [1,8,7,2,6,4,3,5] => ?
=> ? => ? = 1
[6,3,2,4,5,7,8,1] => [1,8,7,5,4,2,3,6] => ?
=> ? => ? = 1
[6,5,7,4,3,8,1,2] => [2,1,8,3,4,7,5,6] => ?
=> ? => ? = 2
[7,6,5,3,4,8,1,2] => [2,1,8,4,3,5,6,7] => ?
=> ? => ? = 2
[7,5,4,3,6,8,1,2] => [2,1,8,6,3,4,5,7] => ?
=> ? => ? = 2
[7,4,3,5,6,8,1,2] => [2,1,8,6,5,3,4,7] => ?
=> ? => ? = 2
[7,8,5,2,3,4,1,6] => [6,1,4,3,2,5,8,7] => ?
=> ? => ? = 6
[7,8,5,3,2,1,4,6] => [6,4,1,2,3,5,8,7] => ?
=> ? => ? = 6
[8,5,6,3,4,2,1,7] => [7,1,2,4,3,6,5,8] => ?
=> ? => ? = 7
[8,5,6,4,2,3,1,7] => [7,1,3,2,4,6,5,8] => ?
=> ? => ? = 7
[8,4,5,6,2,3,1,7] => [7,1,3,2,6,5,4,8] => ?
=> ? => ? = 7
[8,6,4,3,2,5,1,7] => [7,1,5,2,3,4,6,8] => ?
=> ? => ? = 7
[8,4,5,3,2,6,1,7] => [7,1,6,2,3,5,4,8] => ?
=> ? => ? = 7
[8,5,2,3,4,6,1,7] => [7,1,6,4,3,2,5,8] => ?
=> ? => ? = 7
[8,6,4,3,1,2,5,7] => [7,5,2,1,3,4,6,8] => ?
=> ? => ? = 7
[8,6,3,2,1,4,5,7] => [7,5,4,1,2,3,6,8] => ?
=> ? => ? = 7
[8,5,4,2,1,3,6,7] => [7,6,3,1,2,4,5,8] => ?
=> ? => ? = 7
[8,5,3,2,1,4,6,7] => [7,6,4,1,2,3,5,8] => ?
=> ? => ? = 7
[8,5,2,3,1,4,6,7] => [7,6,4,1,3,2,5,8] => ?
=> ? => ? = 7
[6,5,7,3,4,2,1,8] => [8,1,2,4,3,7,5,6] => ?
=> ? => ? = 8
[6,7,3,4,5,2,1,8] => [8,1,2,5,4,3,7,6] => ?
=> ? => ? = 8
[6,5,3,4,7,2,1,8] => [8,1,2,7,4,3,5,6] => ?
=> ? => ? = 8
[5,3,4,6,7,2,1,8] => [8,1,2,7,6,4,3,5] => ?
=> ? => ? = 8
[7,5,4,6,2,3,1,8] => [8,1,3,2,6,4,5,7] => ?
=> ? => ? = 8
[6,5,4,7,2,3,1,8] => [8,1,3,2,7,4,5,6] => ?
=> ? => ? = 8
[7,6,3,4,2,5,1,8] => [8,1,5,2,4,3,6,7] => ?
=> ? => ? = 8
Description
The first part of an integer composition.
Matching statistic: St000439
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2 = 1 + 1
[1,2] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[2,1] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[6,5,7,4,3,8,2,1] => [1,2,8,3,4,7,5,6] => ?
=> ? = 1 + 1
[6,7,4,3,5,8,2,1] => [1,2,8,5,3,4,7,6] => ?
=> ? = 1 + 1
[6,5,3,4,7,8,2,1] => [1,2,8,7,4,3,5,6] => ?
=> ? = 1 + 1
[6,4,3,5,7,8,2,1] => [1,2,8,7,5,3,4,6] => ?
=> ? = 1 + 1
[8,5,4,6,2,3,7,1] => [1,7,3,2,6,4,5,8] => ?
=> ? = 1 + 1
[8,6,3,4,2,5,7,1] => [1,7,5,2,4,3,6,8] => ?
=> ? = 1 + 1
[8,4,5,3,2,6,7,1] => [1,7,6,2,3,5,4,8] => ?
=> ? = 1 + 1
[8,5,2,3,4,6,7,1] => [1,7,6,4,3,2,5,8] => ?
=> ? = 1 + 1
[8,4,3,2,5,6,7,1] => [1,7,6,5,2,3,4,8] => ?
=> ? = 1 + 1
[6,7,4,5,3,2,8,1] => [1,8,2,3,5,4,7,6] => ?
=> ? = 1 + 1
[6,5,4,7,3,2,8,1] => [1,8,2,3,7,4,5,6] => ?
=> ? = 1 + 1
[5,6,4,7,3,2,8,1] => [1,8,2,3,7,4,6,5] => ?
=> ? = 1 + 1
[6,7,5,3,4,2,8,1] => [1,8,2,4,3,5,7,6] => ?
=> ? = 1 + 1
[7,6,3,4,5,2,8,1] => [1,8,2,5,4,3,6,7] => ?
=> ? = 1 + 1
[7,3,4,5,6,2,8,1] => [1,8,2,6,5,4,3,7] => ?
=> ? = 1 + 1
[7,5,6,4,2,3,8,1] => [1,8,3,2,4,6,5,7] => ?
=> ? = 1 + 1
[5,6,4,7,2,3,8,1] => [1,8,3,2,7,4,6,5] => ?
=> ? = 1 + 1
[7,6,4,3,2,5,8,1] => [1,8,5,2,3,4,6,7] => ?
=> ? = 1 + 1
[6,7,3,4,2,5,8,1] => [1,8,5,2,4,3,7,6] => ?
=> ? = 1 + 1
[7,4,5,3,2,6,8,1] => [1,8,6,2,3,5,4,7] => ?
=> ? = 1 + 1
[7,4,3,2,5,6,8,1] => [1,8,6,5,2,3,4,7] => ?
=> ? = 1 + 1
[7,3,4,2,5,6,8,1] => [1,8,6,5,2,4,3,7] => ?
=> ? = 1 + 1
[6,5,3,4,2,7,8,1] => [1,8,7,2,4,3,5,6] => ?
=> ? = 1 + 1
[5,6,3,4,2,7,8,1] => [1,8,7,2,4,3,6,5] => ?
=> ? = 1 + 1
[5,3,4,6,2,7,8,1] => [1,8,7,2,6,4,3,5] => ?
=> ? = 1 + 1
[6,3,2,4,5,7,8,1] => [1,8,7,5,4,2,3,6] => ?
=> ? = 1 + 1
[6,5,7,4,3,8,1,2] => [2,1,8,3,4,7,5,6] => ?
=> ? = 2 + 1
[7,6,5,3,4,8,1,2] => [2,1,8,4,3,5,6,7] => ?
=> ? = 2 + 1
[7,5,4,3,6,8,1,2] => [2,1,8,6,3,4,5,7] => ?
=> ? = 2 + 1
[7,4,3,5,6,8,1,2] => [2,1,8,6,5,3,4,7] => ?
=> ? = 2 + 1
[7,8,5,2,3,4,1,6] => [6,1,4,3,2,5,8,7] => ?
=> ? = 6 + 1
[7,8,5,3,2,1,4,6] => [6,4,1,2,3,5,8,7] => ?
=> ? = 6 + 1
[8,5,6,3,4,2,1,7] => [7,1,2,4,3,6,5,8] => ?
=> ? = 7 + 1
[8,5,6,4,2,3,1,7] => [7,1,3,2,4,6,5,8] => ?
=> ? = 7 + 1
[8,4,5,6,2,3,1,7] => [7,1,3,2,6,5,4,8] => ?
=> ? = 7 + 1
[8,6,4,3,2,5,1,7] => [7,1,5,2,3,4,6,8] => ?
=> ? = 7 + 1
[8,4,5,3,2,6,1,7] => [7,1,6,2,3,5,4,8] => ?
=> ? = 7 + 1
[8,5,2,3,4,6,1,7] => [7,1,6,4,3,2,5,8] => ?
=> ? = 7 + 1
[8,6,4,3,1,2,5,7] => [7,5,2,1,3,4,6,8] => ?
=> ? = 7 + 1
[8,6,3,2,1,4,5,7] => [7,5,4,1,2,3,6,8] => ?
=> ? = 7 + 1
[8,5,4,2,1,3,6,7] => [7,6,3,1,2,4,5,8] => ?
=> ? = 7 + 1
[8,5,3,2,1,4,6,7] => [7,6,4,1,2,3,5,8] => ?
=> ? = 7 + 1
[8,5,2,3,1,4,6,7] => [7,6,4,1,3,2,5,8] => ?
=> ? = 7 + 1
[6,5,7,3,4,2,1,8] => [8,1,2,4,3,7,5,6] => ?
=> ? = 8 + 1
[6,7,3,4,5,2,1,8] => [8,1,2,5,4,3,7,6] => ?
=> ? = 8 + 1
[6,5,3,4,7,2,1,8] => [8,1,2,7,4,3,5,6] => ?
=> ? = 8 + 1
[5,3,4,6,7,2,1,8] => [8,1,2,7,6,4,3,5] => ?
=> ? = 8 + 1
[7,5,4,6,2,3,1,8] => [8,1,3,2,6,4,5,7] => ?
=> ? = 8 + 1
[6,5,4,7,2,3,1,8] => [8,1,3,2,7,4,5,6] => ?
=> ? = 8 + 1
[7,6,3,4,2,5,1,8] => [8,1,5,2,4,3,6,7] => ?
=> ? = 8 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[6,5,7,4,3,8,2,1] => [1,2,8,3,4,7,5,6] => ?
=> ?
=> ? = 1
[6,7,4,3,5,8,2,1] => [1,2,8,5,3,4,7,6] => ?
=> ?
=> ? = 1
[6,5,3,4,7,8,2,1] => [1,2,8,7,4,3,5,6] => ?
=> ?
=> ? = 1
[6,4,3,5,7,8,2,1] => [1,2,8,7,5,3,4,6] => ?
=> ?
=> ? = 1
[8,5,4,6,2,3,7,1] => [1,7,3,2,6,4,5,8] => ?
=> ?
=> ? = 1
[8,6,3,4,2,5,7,1] => [1,7,5,2,4,3,6,8] => ?
=> ?
=> ? = 1
[8,4,5,3,2,6,7,1] => [1,7,6,2,3,5,4,8] => ?
=> ?
=> ? = 1
[8,5,2,3,4,6,7,1] => [1,7,6,4,3,2,5,8] => ?
=> ?
=> ? = 1
[8,4,3,2,5,6,7,1] => [1,7,6,5,2,3,4,8] => ?
=> ?
=> ? = 1
[6,7,4,5,3,2,8,1] => [1,8,2,3,5,4,7,6] => ?
=> ?
=> ? = 1
[6,5,4,7,3,2,8,1] => [1,8,2,3,7,4,5,6] => ?
=> ?
=> ? = 1
[5,6,4,7,3,2,8,1] => [1,8,2,3,7,4,6,5] => ?
=> ?
=> ? = 1
[6,7,5,3,4,2,8,1] => [1,8,2,4,3,5,7,6] => ?
=> ?
=> ? = 1
[7,6,3,4,5,2,8,1] => [1,8,2,5,4,3,6,7] => ?
=> ?
=> ? = 1
[7,3,4,5,6,2,8,1] => [1,8,2,6,5,4,3,7] => ?
=> ?
=> ? = 1
[7,5,6,4,2,3,8,1] => [1,8,3,2,4,6,5,7] => ?
=> ?
=> ? = 1
[5,6,4,7,2,3,8,1] => [1,8,3,2,7,4,6,5] => ?
=> ?
=> ? = 1
[7,6,4,3,2,5,8,1] => [1,8,5,2,3,4,6,7] => ?
=> ?
=> ? = 1
[6,7,3,4,2,5,8,1] => [1,8,5,2,4,3,7,6] => ?
=> ?
=> ? = 1
[7,4,5,3,2,6,8,1] => [1,8,6,2,3,5,4,7] => ?
=> ?
=> ? = 1
[7,4,3,2,5,6,8,1] => [1,8,6,5,2,3,4,7] => ?
=> ?
=> ? = 1
[7,3,4,2,5,6,8,1] => [1,8,6,5,2,4,3,7] => ?
=> ?
=> ? = 1
[6,5,3,4,2,7,8,1] => [1,8,7,2,4,3,5,6] => ?
=> ?
=> ? = 1
[5,6,3,4,2,7,8,1] => [1,8,7,2,4,3,6,5] => ?
=> ?
=> ? = 1
[5,3,4,6,2,7,8,1] => [1,8,7,2,6,4,3,5] => ?
=> ?
=> ? = 1
[6,3,2,4,5,7,8,1] => [1,8,7,5,4,2,3,6] => ?
=> ?
=> ? = 1
[6,5,7,4,3,8,1,2] => [2,1,8,3,4,7,5,6] => ?
=> ?
=> ? = 2
[7,6,5,3,4,8,1,2] => [2,1,8,4,3,5,6,7] => ?
=> ?
=> ? = 2
[7,5,4,3,6,8,1,2] => [2,1,8,6,3,4,5,7] => ?
=> ?
=> ? = 2
[7,4,3,5,6,8,1,2] => [2,1,8,6,5,3,4,7] => ?
=> ?
=> ? = 2
[7,8,5,2,3,4,1,6] => [6,1,4,3,2,5,8,7] => ?
=> ?
=> ? = 6
[7,8,5,3,2,1,4,6] => [6,4,1,2,3,5,8,7] => ?
=> ?
=> ? = 6
[8,5,6,3,4,2,1,7] => [7,1,2,4,3,6,5,8] => ?
=> ?
=> ? = 7
[8,5,6,4,2,3,1,7] => [7,1,3,2,4,6,5,8] => ?
=> ?
=> ? = 7
[8,4,5,6,2,3,1,7] => [7,1,3,2,6,5,4,8] => ?
=> ?
=> ? = 7
[8,6,4,3,2,5,1,7] => [7,1,5,2,3,4,6,8] => ?
=> ?
=> ? = 7
[8,4,5,3,2,6,1,7] => [7,1,6,2,3,5,4,8] => ?
=> ?
=> ? = 7
[8,5,2,3,4,6,1,7] => [7,1,6,4,3,2,5,8] => ?
=> ?
=> ? = 7
[8,6,4,3,1,2,5,7] => [7,5,2,1,3,4,6,8] => ?
=> ?
=> ? = 7
[8,6,3,2,1,4,5,7] => [7,5,4,1,2,3,6,8] => ?
=> ?
=> ? = 7
[8,5,4,2,1,3,6,7] => [7,6,3,1,2,4,5,8] => ?
=> ?
=> ? = 7
[8,5,3,2,1,4,6,7] => [7,6,4,1,2,3,5,8] => ?
=> ?
=> ? = 7
[8,5,2,3,1,4,6,7] => [7,6,4,1,3,2,5,8] => ?
=> ?
=> ? = 7
[6,5,7,3,4,2,1,8] => [8,1,2,4,3,7,5,6] => ?
=> ?
=> ? = 8
[6,7,3,4,5,2,1,8] => [8,1,2,5,4,3,7,6] => ?
=> ?
=> ? = 8
[6,5,3,4,7,2,1,8] => [8,1,2,7,4,3,5,6] => ?
=> ?
=> ? = 8
[5,3,4,6,7,2,1,8] => [8,1,2,7,6,4,3,5] => ?
=> ?
=> ? = 8
[7,5,4,6,2,3,1,8] => [8,1,3,2,6,4,5,7] => ?
=> ?
=> ? = 8
[6,5,4,7,2,3,1,8] => [8,1,3,2,7,4,5,6] => ?
=> ?
=> ? = 8
[7,6,3,4,2,5,1,8] => [8,1,5,2,4,3,6,7] => ?
=> ?
=> ? = 8
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> []
=> 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> [1]
=> 1 = 2 - 1
[2,1] => [2,1] => [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2 = 3 - 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3 = 4 - 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 4 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 4 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4 = 5 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4 = 5 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3 = 4 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4 = 5 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3 = 4 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4 = 5 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4 = 5 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 7 - 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 6 - 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 7 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 5 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 6 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 5 - 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 7 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 6 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 7 - 1
[1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? = 6 - 1
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 7 - 1
[1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 5 - 1
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 7 - 1
[1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 6 - 1
[1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 5 - 1
[1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? = 6 - 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 7 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 6 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 7 - 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 5 - 1
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 6 - 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 5 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 7 - 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? = 6 - 1
[1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 7 - 1
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 6 - 1
[1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 7 - 1
[1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? = 6 - 1
[1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 7 - 1
[1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> ? = 6 - 1
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 7 - 1
[1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? = 6 - 1
[1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 7 - 1
[1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? = 5 - 1
[1,3,2,4,7,5,6] => [1,3,2,4,6,7,5] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 6 - 1
[1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? = 5 - 1
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 7 - 1
[1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 6 - 1
[1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 7 - 1
[1,3,2,7,4,5,6] => [1,3,2,5,6,7,4] => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> ? = 6 - 1
[1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 7 - 1
[1,4,2,3,5,6,7] => [1,3,4,2,5,6,7] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 7 - 1
[1,4,2,3,5,7,6] => [1,3,4,2,5,7,6] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 6 - 1
[1,4,2,3,6,5,7] => [1,3,4,2,6,5,7] => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 7 - 1
[1,4,2,3,7,5,6] => [1,3,4,2,6,7,5] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> ? = 6 - 1
[1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 7 - 1
[1,5,2,3,4,6,7] => [1,3,4,5,2,6,7] => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 7 - 1
[2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 7 - 1
[2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 6 - 1
[2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ? = 7 - 1
Description
The largest part of an integer partition.
Matching statistic: St000069
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> ([],1)
=> 1
[1,2] => [2,1] => [1,1,0,0]
=> ([],2)
=> 2
[2,1] => [1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ([],3)
=> 3
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ([],3)
=> 3
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 3
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 3
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 3
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 3
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 3
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> 4
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> 4
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> 4
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[2,3,7,6,4,5,1] => [1,5,4,6,7,3,2] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[2,3,7,6,5,4,1] => [1,4,5,6,7,3,2] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[2,4,7,6,3,5,1] => [1,5,3,6,7,4,2] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[2,4,7,6,5,3,1] => [1,3,5,6,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[2,6,7,3,4,5,1] => [1,5,4,3,7,6,2] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[2,6,7,4,3,5,1] => [1,5,3,4,7,6,2] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[2,6,7,5,3,4,1] => [1,4,3,5,7,6,2] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[2,7,3,6,4,5,1] => [1,5,4,6,3,7,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
[2,7,4,6,3,5,1] => [1,5,3,6,4,7,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
[2,7,4,6,5,3,1] => [1,3,5,6,4,7,2] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,6),(1,2),(1,3),(1,4),(2,6),(3,6),(4,5),(6,5)],7)
=> ? = 1
[2,7,5,6,3,4,1] => [1,4,3,6,5,7,2] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[2,7,5,6,4,3,1] => [1,3,4,6,5,7,2] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 1
[2,7,6,4,5,3,1] => [1,3,5,4,6,7,2] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(4,3),(5,4),(6,3)],7)
=> ? = 1
[3,2,7,6,4,5,1] => [1,5,4,6,7,2,3] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,2,7,6,5,4,1] => [1,4,5,6,7,2,3] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,4,7,6,2,5,1] => [1,5,2,6,7,4,3] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,4,7,6,5,1,2] => [2,1,5,6,7,4,3] => [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[3,6,7,2,4,5,1] => [1,5,4,2,7,6,3] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,6,7,4,2,5,1] => [1,5,2,4,7,6,3] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,6,7,5,2,4,1] => [1,4,2,5,7,6,3] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,7,2,6,4,5,1] => [1,5,4,6,2,7,3] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,7,4,5,6,1,2] => [2,1,6,5,4,7,3] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[3,7,4,6,2,5,1] => [1,5,2,6,4,7,3] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,7,5,4,6,1,2] => [2,1,6,4,5,7,3] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[3,7,5,6,2,4,1] => [1,4,2,6,5,7,3] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 1
[3,7,5,6,4,1,2] => [2,1,4,6,5,7,3] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(1,6),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ? = 2
[3,7,5,6,4,2,1] => [1,2,4,6,5,7,3] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> ? = 1
[3,7,6,4,5,1,2] => [2,1,5,4,6,7,3] => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[3,7,6,5,4,1,2] => [2,1,4,5,6,7,3] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,6),(1,5),(1,6),(2,5),(5,3),(5,4),(6,3),(6,4)],7)
=> ? = 2
[4,2,7,6,3,5,1] => [1,5,3,6,7,2,4] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[4,2,7,6,5,3,1] => [1,3,5,6,7,2,4] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,3,7,6,2,5,1] => [1,5,2,6,7,3,4] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[4,3,7,6,5,1,2] => [2,1,5,6,7,3,4] => [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[4,5,7,6,1,2,3] => [3,2,1,6,7,5,4] => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 3
[4,5,7,6,2,1,3] => [3,1,2,6,7,5,4] => [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 3
[4,6,7,2,3,5,1] => [1,5,3,2,7,6,4] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[4,6,7,3,2,5,1] => [1,5,2,3,7,6,4] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[4,7,2,6,3,5,1] => [1,5,3,6,2,7,4] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
[4,7,2,6,5,3,1] => [1,3,5,6,2,7,4] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,6),(1,2),(1,3),(1,4),(2,6),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,7,3,5,6,1,2] => [2,1,6,5,3,7,4] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[4,7,3,6,2,5,1] => [1,5,2,6,3,7,4] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
[4,7,5,3,6,1,2] => [2,1,6,3,5,7,4] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[4,7,5,6,1,2,3] => [3,2,1,6,5,7,4] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 3
[4,7,5,6,1,3,2] => [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,5),(0,6),(1,3),(1,4),(3,5),(3,6),(4,5),(4,6),(6,2)],7)
=> ? = 2
[4,7,5,6,2,1,3] => [3,1,2,6,5,7,4] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 3
[4,7,5,6,2,3,1] => [1,3,2,6,5,7,4] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(2,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 1
[4,7,5,6,3,1,2] => [2,1,3,6,5,7,4] => [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,6),(1,2),(1,3),(2,6),(3,6),(6,4),(6,5)],7)
=> ? = 2
[4,7,6,2,5,3,1] => [1,3,5,2,6,7,4] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(4,3),(5,4),(6,3)],7)
=> ? = 1
[4,7,6,3,5,1,2] => [2,1,5,3,6,7,4] => [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[4,7,6,5,1,3,2] => [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,3),(1,6),(3,4),(3,5),(5,2),(6,4),(6,5)],7)
=> ? = 2
Description
The number of maximal elements of a poset.
Matching statistic: St000476
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 80%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,5,3,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[7,6,5,4,3,8,2,1] => [8,7,5,4,3,2,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,5,6,4,3,8,2,1] => [8,7,5,4,2,3,1,6] => ?
=> ?
=> ? = 1 - 1
[6,5,7,4,3,8,2,1] => [8,7,5,4,2,1,3,6] => ?
=> ?
=> ? = 1 - 1
[5,6,7,4,3,8,2,1] => [8,7,5,4,1,2,3,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,6,4,5,3,8,2,1] => [8,7,5,3,4,2,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,7,4,5,3,8,2,1] => [8,7,5,3,4,1,2,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,5,4,6,3,8,2,1] => [8,7,5,3,2,4,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,4,5,6,3,8,2,1] => [8,7,5,2,3,4,1,6] => ?
=> ?
=> ? = 1 - 1
[6,5,4,7,3,8,2,1] => [8,7,5,3,2,1,4,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,6,4,7,3,8,2,1] => [8,7,5,3,1,2,4,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,4,5,7,3,8,2,1] => [8,7,5,2,3,1,4,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,4,6,7,3,8,2,1] => [8,7,5,2,1,3,4,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[4,5,6,7,3,8,2,1] => [8,7,5,1,2,3,4,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,6,5,3,4,8,2,1] => [8,7,4,5,3,2,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,5,6,3,4,8,2,1] => [8,7,4,5,2,3,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,5,7,3,4,8,2,1] => [8,7,4,5,2,1,3,6] => ?
=> ?
=> ? = 1 - 1
[5,6,7,3,4,8,2,1] => [8,7,4,5,1,2,3,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,6,4,3,5,8,2,1] => [8,7,4,3,5,2,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,7,4,3,5,8,2,1] => [8,7,4,3,5,1,2,6] => ?
=> ?
=> ? = 1 - 1
[7,6,3,4,5,8,2,1] => [8,7,3,4,5,2,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,5,4,3,6,8,2,1] => [8,7,4,3,2,5,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,5,3,4,6,8,2,1] => [8,7,3,4,2,5,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[7,3,4,5,6,8,2,1] => [8,7,2,3,4,5,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,5,4,3,7,8,2,1] => [8,7,4,3,2,1,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,6,4,3,7,8,2,1] => [8,7,4,3,1,2,5,6] => ?
=> ?
=> ? = 1 - 1
[6,4,5,3,7,8,2,1] => [8,7,4,2,3,1,5,6] => ?
=> ?
=> ? = 1 - 1
[4,5,6,3,7,8,2,1] => [8,7,4,1,2,3,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,5,3,4,7,8,2,1] => [8,7,3,4,2,1,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,6,3,4,7,8,2,1] => [8,7,3,4,1,2,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,4,3,5,7,8,2,1] => [8,7,3,2,4,1,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[6,3,4,5,7,8,2,1] => [8,7,2,3,4,1,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[5,4,3,6,7,8,2,1] => [8,7,3,2,1,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[4,5,3,6,7,8,2,1] => [8,7,3,1,2,4,5,6] => ?
=> ?
=> ? = 1 - 1
[4,3,5,6,7,8,2,1] => [8,7,2,1,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[3,4,5,6,7,8,2,1] => [8,7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,5,4,3,2,7,1] => [8,6,5,4,3,2,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,5,6,4,3,2,7,1] => [8,6,5,4,2,3,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,4,5,3,2,7,1] => [8,6,5,3,4,2,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,5,4,6,3,2,7,1] => [8,6,5,3,2,4,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,5,3,4,2,7,1] => [8,6,4,5,3,2,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,5,6,3,4,2,7,1] => [8,6,4,5,2,3,7,1] => ?
=> ?
=> ? = 1 - 1
[8,6,4,3,5,2,7,1] => [8,6,4,3,5,2,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,3,4,5,2,7,1] => [8,6,3,4,5,2,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,5,4,3,6,2,7,1] => [8,6,4,3,2,5,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,4,5,3,6,2,7,1] => [8,6,4,2,3,5,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,5,4,2,3,7,1] => [8,5,6,4,3,2,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,5,6,4,2,3,7,1] => [8,5,6,4,2,3,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,6,4,5,2,3,7,1] => [8,5,6,3,4,2,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[8,5,4,6,2,3,7,1] => [8,5,6,3,2,4,7,1] => ?
=> ?
=> ? = 1 - 1
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000054
(load all 66 compositions to match this statistic)
(load all 66 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 1
[1,2] => [2,1] => [1,1,0,0]
=> [2,1] => 2
[2,1] => [1,2] => [1,0,1,0]
=> [1,2] => 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 3
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 3
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 4
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 4
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 3
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 4
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 4
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 3
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 6
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => ? = 5
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 6
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 5
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 6
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => ? = 4
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 6
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => ? = 4
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => ? = 5
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => ? = 4
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,4,7,6,3,2,1] => ? = 5
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,6,3,2,1] => ? = 4
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 6
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 5
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 6
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => ? = 4
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 5
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 4
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 6
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => ? = 5
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 6
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 5
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 6
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => ? = 3
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 6
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => ? = 3
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,6,4,3,2,1] => ? = 5
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => ? = 3
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [5,4,7,6,3,2,1] => ? = 5
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,6,4,2,1] => ? = 3
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 6
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 5
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 6
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => ? = 3
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 5
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => ? = 3
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 6
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => ? = 4
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 6
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => ? = 4
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 6
[1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => ? = 3
[1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 6
[1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => ? = 3
[1,2,5,6,3,7,4] => [4,7,3,6,5,2,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,7,6,5,3,2,1] => ? = 4
[1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,7,6,5,4,2,1] => ? = 3
[1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3,7,6,5,2,1] => ? = 4
[1,2,5,6,7,4,3] => [3,4,7,6,5,2,1] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,6,5,2,1] => ? = 3
[1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 6
[1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => ? = 4
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000141
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 2 = 3 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 3 - 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 4 - 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3 = 4 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 4 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2 = 3 - 1
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4 = 5 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 4 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 4 = 5 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2 = 3 - 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 6 - 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 7 - 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 5 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 6 - 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 5 - 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ? = 7 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => ? = 6 - 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 7 - 1
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 4 - 1
[1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 6 - 1
[1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 4 - 1
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => ? = 7 - 1
[1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 5 - 1
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 7 - 1
[1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 4 - 1
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 5 - 1
[1,2,3,6,7,5,4] => [1,2,3,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 4 - 1
[1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ? = 6 - 1
[1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 5 - 1
[1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 6 - 1
[1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 4 - 1
[1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 5 - 1
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 4 - 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => ? = 7 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => ? = 6 - 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => ? = 7 - 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 5 - 1
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => ? = 6 - 1
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 5 - 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => ? = 7 - 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => ? = 6 - 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 7 - 1
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 3 - 1
[1,2,4,5,7,3,6] => [1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => ? = 6 - 1
[1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 3 - 1
[1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => ? = 7 - 1
[1,2,4,6,3,7,5] => [1,2,5,3,7,4,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => ? = 5 - 1
[1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 7 - 1
[1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 3 - 1
[1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => ? = 5 - 1
[1,2,4,6,7,5,3] => [1,2,7,3,6,4,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 3 - 1
[1,2,4,7,3,5,6] => [1,2,5,3,6,7,4] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => ? = 6 - 1
[1,2,4,7,3,6,5] => [1,2,5,3,7,6,4] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => ? = 5 - 1
[1,2,4,7,5,3,6] => [1,2,6,3,5,7,4] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => ? = 6 - 1
[1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 3 - 1
[1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => ? = 5 - 1
[1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 3 - 1
[1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => ? = 7 - 1
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => ? = 6 - 1
[1,2,5,3,6,4,7] => [1,2,4,6,3,5,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => ? = 7 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000505
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00091: Set partitions —rotate increasing⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 80%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00091: Set partitions —rotate increasing⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => {{1}}
=> {{1}}
=> 1
[1,2] => [2,1] => {{1,2}}
=> {{1,2}}
=> 2
[2,1] => [1,2] => {{1},{2}}
=> {{1},{2}}
=> 1
[1,2,3] => [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 3
[1,3,2] => [3,2,1] => {{1,3},{2}}
=> {{1,2},{3}}
=> 2
[2,1,3] => [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 3
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1
[3,1,2] => [3,1,2] => {{1,3},{2}}
=> {{1,2},{3}}
=> 2
[3,2,1] => [2,1,3] => {{1,2},{3}}
=> {{1},{2,3}}
=> 1
[1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 4
[1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
=> {{1,2,3},{4}}
=> 3
[1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 4
[1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
=> {{1,2},{3},{4}}
=> 2
[1,4,2,3] => [3,4,2,1] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 3
[1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
=> {{1,2},{3,4}}
=> 2
[2,1,3,4] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 4
[2,1,4,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 3
[2,3,1,4] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 4
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1
[2,4,1,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 3
[2,4,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1},{2},{3,4}}
=> 1
[3,1,2,4] => [3,1,4,2] => {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 4
[3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> {{1,2},{3},{4}}
=> 2
[3,2,1,4] => [2,1,4,3] => {{1,2},{3,4}}
=> {{1,4},{2,3}}
=> 4
[3,2,4,1] => [2,1,3,4] => {{1,2},{3},{4}}
=> {{1},{2,3},{4}}
=> 1
[3,4,1,2] => [4,1,2,3] => {{1,4},{2},{3}}
=> {{1,2},{3},{4}}
=> 2
[3,4,2,1] => [3,1,2,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 3
[4,1,3,2] => [4,3,1,2] => {{1,4},{2,3}}
=> {{1,2},{3,4}}
=> 2
[4,2,1,3] => [2,4,1,3] => {{1,2,4},{3}}
=> {{1,2,3},{4}}
=> 3
[4,2,3,1] => [2,3,1,4] => {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1
[4,3,1,2] => [4,2,1,3] => {{1,4},{2},{3}}
=> {{1,2},{3},{4}}
=> 2
[4,3,2,1] => [3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 5
[1,2,3,5,4] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> {{1,2,3,4},{5}}
=> 4
[1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> {{1,2,3,5},{4}}
=> 5
[1,2,4,5,3] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> {{1,2,3},{4},{5}}
=> 3
[1,2,5,3,4] => [2,4,5,3,1] => {{1,2,4},{3,5}}
=> {{1,4},{2,3,5}}
=> 4
[1,2,5,4,3] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> {{1,2,3},{4,5}}
=> 3
[1,3,2,4,5] => [3,2,4,5,1] => {{1,3,4,5},{2}}
=> {{1,2,4,5},{3}}
=> 5
[1,3,2,5,4] => [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> {{1,2,4},{3},{5}}
=> 4
[1,3,4,2,5] => [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> 5
[1,3,4,5,2] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> {{1,2},{3},{4},{5}}
=> 2
[1,3,5,2,4] => [4,2,5,3,1] => {{1,4},{2},{3,5}}
=> {{1,4},{2,5},{3}}
=> 4
[1,3,5,4,2] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> {{1,2},{3},{4,5}}
=> 2
[1,4,2,3,5] => [3,4,2,5,1] => {{1,3},{2,4,5}}
=> {{1,3,5},{2,4}}
=> 5
[1,4,2,5,3] => [3,5,2,4,1] => {{1,3},{2,5},{4}}
=> {{1,3},{2,4},{5}}
=> 3
[1,4,3,2,5] => [4,3,2,5,1] => {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 5
[1,4,3,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> {{1,2},{3,4},{5}}
=> 2
[1,4,5,2,3] => [4,5,2,3,1] => {{1,4},{2,5},{3}}
=> {{1,3},{2,5},{4}}
=> 3
[7,6,5,4,3,8,2,1] => [7,5,4,3,2,1,6,8] => ?
=> ?
=> ? = 1
[7,5,6,4,3,8,2,1] => [7,5,4,2,3,1,6,8] => ?
=> ?
=> ? = 1
[6,5,7,4,3,8,2,1] => [7,5,4,2,1,3,6,8] => ?
=> ?
=> ? = 1
[5,6,7,4,3,8,2,1] => [7,5,4,1,2,3,6,8] => ?
=> ?
=> ? = 1
[7,6,4,5,3,8,2,1] => [7,5,3,4,2,1,6,8] => ?
=> ?
=> ? = 1
[6,7,4,5,3,8,2,1] => [7,5,3,4,1,2,6,8] => ?
=> ?
=> ? = 1
[7,5,4,6,3,8,2,1] => [7,5,3,2,4,1,6,8] => ?
=> ?
=> ? = 1
[7,4,5,6,3,8,2,1] => [7,5,2,3,4,1,6,8] => ?
=> ?
=> ? = 1
[6,5,4,7,3,8,2,1] => [7,5,3,2,1,4,6,8] => ?
=> ?
=> ? = 1
[5,6,4,7,3,8,2,1] => [7,5,3,1,2,4,6,8] => {{1,7},{2,5},{3},{4},{6},{8}}
=> {{1},{2,8},{3,6},{4},{5},{7}}
=> ? = 1
[6,4,5,7,3,8,2,1] => [7,5,2,3,1,4,6,8] => ?
=> ?
=> ? = 1
[5,4,6,7,3,8,2,1] => [7,5,2,1,3,4,6,8] => ?
=> ?
=> ? = 1
[4,5,6,7,3,8,2,1] => [7,5,1,2,3,4,6,8] => {{1,7},{2,5},{3},{4},{6},{8}}
=> {{1},{2,8},{3,6},{4},{5},{7}}
=> ? = 1
[7,6,5,3,4,8,2,1] => [7,4,5,3,2,1,6,8] => ?
=> ?
=> ? = 1
[7,5,6,3,4,8,2,1] => ? => ?
=> ?
=> ? = 1
[6,5,7,3,4,8,2,1] => [7,4,5,2,1,3,6,8] => ?
=> ?
=> ? = 1
[5,6,7,3,4,8,2,1] => [7,4,5,1,2,3,6,8] => ?
=> ?
=> ? = 1
[7,6,4,3,5,8,2,1] => [7,4,3,5,2,1,6,8] => ?
=> ?
=> ? = 1
[6,7,4,3,5,8,2,1] => ? => ?
=> ?
=> ? = 1
[7,6,3,4,5,8,2,1] => [7,3,4,5,2,1,6,8] => ?
=> ?
=> ? = 1
[7,5,4,3,6,8,2,1] => [7,4,3,2,5,1,6,8] => ?
=> ?
=> ? = 1
[7,5,3,4,6,8,2,1] => [7,3,4,2,5,1,6,8] => ?
=> ?
=> ? = 1
[7,3,4,5,6,8,2,1] => [7,2,3,4,5,1,6,8] => ?
=> ?
=> ? = 1
[6,5,4,3,7,8,2,1] => [7,4,3,2,1,5,6,8] => {{1,7},{2,4},{3},{5},{6},{8}}
=> {{1},{2,8},{3,5},{4},{6},{7}}
=> ? = 1
[5,6,4,3,7,8,2,1] => [7,4,3,1,2,5,6,8] => ?
=> ?
=> ? = 1
[6,4,5,3,7,8,2,1] => ? => ?
=> ?
=> ? = 1
[4,5,6,3,7,8,2,1] => [7,4,1,2,3,5,6,8] => ?
=> ?
=> ? = 1
[6,5,3,4,7,8,2,1] => [7,3,4,2,1,5,6,8] => ?
=> ?
=> ? = 1
[5,6,3,4,7,8,2,1] => ? => ?
=> ?
=> ? = 1
[6,4,3,5,7,8,2,1] => [7,3,2,4,1,5,6,8] => ?
=> ?
=> ? = 1
[6,3,4,5,7,8,2,1] => [7,2,3,4,1,5,6,8] => ?
=> ?
=> ? = 1
[5,4,3,6,7,8,2,1] => [7,3,2,1,4,5,6,8] => ?
=> ?
=> ? = 1
[4,5,3,6,7,8,2,1] => [7,3,1,2,4,5,6,8] => {{1,7},{2,3},{4},{5},{6},{8}}
=> {{1},{2,8},{3,4},{5},{6},{7}}
=> ? = 1
[4,3,5,6,7,8,2,1] => [7,2,1,3,4,5,6,8] => {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 1
[3,4,5,6,7,8,2,1] => [7,1,2,3,4,5,6,8] => {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 1
[8,6,5,4,3,2,7,1] => [6,5,4,3,2,7,1,8] => ?
=> ?
=> ? = 1
[8,5,6,4,3,2,7,1] => [6,5,4,2,3,7,1,8] => ?
=> ?
=> ? = 1
[8,6,4,5,3,2,7,1] => [6,5,3,4,2,7,1,8] => ?
=> ?
=> ? = 1
[8,5,4,6,3,2,7,1] => [6,5,3,2,4,7,1,8] => ?
=> ?
=> ? = 1
[8,6,5,3,4,2,7,1] => [6,4,5,3,2,7,1,8] => ?
=> ?
=> ? = 1
[8,5,6,3,4,2,7,1] => [6,4,5,2,3,7,1,8] => ?
=> ?
=> ? = 1
[8,6,4,3,5,2,7,1] => [6,4,3,5,2,7,1,8] => ?
=> ?
=> ? = 1
[8,6,3,4,5,2,7,1] => [6,3,4,5,2,7,1,8] => ?
=> ?
=> ? = 1
[8,5,4,3,6,2,7,1] => [6,4,3,2,5,7,1,8] => ?
=> ?
=> ? = 1
[8,4,5,3,6,2,7,1] => [6,4,2,3,5,7,1,8] => ?
=> ?
=> ? = 1
[8,6,5,4,2,3,7,1] => [5,6,4,3,2,7,1,8] => ?
=> ?
=> ? = 1
[8,5,6,4,2,3,7,1] => [5,6,4,2,3,7,1,8] => ?
=> ?
=> ? = 1
[8,6,4,5,2,3,7,1] => [5,6,3,4,2,7,1,8] => {{1,5},{2,6,7},{3},{4},{8}}
=> ?
=> ? = 1
[8,5,4,6,2,3,7,1] => [5,6,3,2,4,7,1,8] => ?
=> ?
=> ? = 1
[8,6,5,3,2,4,7,1] => [5,4,6,3,2,7,1,8] => ?
=> ?
=> ? = 1
Description
The biggest entry in the block containing the 1.
The following 34 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000740The last entry of a permutation. St000653The last descent of a permutation. St000297The number of leading ones in a binary word. St000738The first entry in the last row of a standard tableau. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000989The number of final rises of a permutation. St001497The position of the largest weak excedence of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000051The size of the left subtree of a binary tree. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000133The "bounce" of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001480The number of simple summands of the module J^2/J^3. St000326The position of the first one in a binary word after appending a 1 at the end. St000391The sum of the positions of the ones in a binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St000682The Grundy value of Welter's game on a binary word. St000840The number of closers smaller than the largest opener in a perfect matching. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001430The number of positive entries in a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation.
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