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Your data matches 58 different statistics following compositions of up to 3 maps.
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Matching statistic: St000439
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 4
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 8
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 6
[3,2]
=> [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]
=> 5
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [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]
=> 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 9
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 7
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 6
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 5
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 6
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 5
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [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
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> 10
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> 8
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 7
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 6
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 6
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 5
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 5
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [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
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [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
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 11
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> 9
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> 8
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> 7
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 7
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 6
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> 5
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 7
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 5
Description
The position of the first down step of a Dyck path.
Matching statistic: St000382
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 90% ●values known / values provided: 100%●distinct values known / distinct values provided: 90%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 90% ●values known / values provided: 100%●distinct values known / distinct values provided: 90%
Values
[1]
=> [1,0,1,0]
=> [1,1] => 1 = 4 - 3
[2]
=> [1,1,0,0,1,0]
=> [2,1] => 2 = 5 - 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 1 = 4 - 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 6 - 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 4 - 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 4 - 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 7 - 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2 = 5 - 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 5 - 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1 = 4 - 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1 = 4 - 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => 5 = 8 - 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3 = 6 - 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 5 - 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1 = 4 - 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1 = 4 - 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1 = 4 - 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => 1 = 4 - 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => 6 = 9 - 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => 4 = 7 - 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3 = 6 - 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2 = 5 - 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3 = 6 - 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 4 - 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1 = 4 - 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2 = 5 - 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 1 = 4 - 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => 1 = 4 - 3
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => 7 = 10 - 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => 5 = 8 - 3
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => 4 = 7 - 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => 3 = 6 - 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3 = 6 - 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2 = 5 - 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1 = 4 - 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2 = 5 - 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2 = 5 - 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1 = 4 - 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => 1 = 4 - 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1 = 4 - 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => 1 = 4 - 3
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => 8 = 11 - 3
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => 6 = 9 - 3
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => 5 = 8 - 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => 4 = 7 - 3
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => 4 = 7 - 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => 3 = 6 - 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => 2 = 5 - 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => 4 = 7 - 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2 = 5 - 3
[]
=> []
=> [] => ? = 2 - 3
Description
The first part of an integer composition.
Matching statistic: St000297
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 99%●distinct values known / distinct values provided: 90%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 99%●distinct values known / distinct values provided: 90%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 0 => 0 = 4 - 4
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 5 - 4
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 0 = 4 - 4
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 2 = 6 - 4
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0 = 4 - 4
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 0 = 4 - 4
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 3 = 7 - 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 100 => 1 = 5 - 4
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 5 - 4
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 010 => 0 = 4 - 4
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 0 = 4 - 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 11110 => 4 = 8 - 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 1100 => 2 = 6 - 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 5 - 4
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 0 = 4 - 4
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 0 = 4 - 4
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0110 => 0 = 4 - 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 01111 => 0 = 4 - 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => 111110 => 5 = 9 - 4
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 11100 => 3 = 7 - 4
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 1100 => 2 = 6 - 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1010 => 1 = 5 - 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1101 => 2 = 6 - 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0 = 4 - 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0110 => 0 = 4 - 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 1 = 5 - 4
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0101 => 0 = 4 - 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 01110 => 0 = 4 - 4
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => 1111110 => 6 = 10 - 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => 111100 => 4 = 8 - 4
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => 11100 => 3 = 7 - 4
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => 11010 => 2 = 6 - 4
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 2 = 6 - 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1000 => 1 = 5 - 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 0 = 4 - 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1001 => 1 = 5 - 4
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1010 => 1 = 5 - 4
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0100 => 0 = 4 - 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 01110 => 0 = 4 - 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 0 = 4 - 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 01101 => 0 = 4 - 4
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => 11111110 => 7 = 11 - 4
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => 1111100 => 5 = 9 - 4
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => 111100 => 4 = 8 - 4
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => 111010 => 3 = 7 - 4
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => 11100 => 3 = 7 - 4
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => 11000 => 2 = 6 - 4
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => 10110 => 1 = 5 - 4
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 11101 => 3 = 7 - 4
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1000 => 1 = 5 - 4
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,7,8,6,4,3,2,1,9] => ? => ? = 8 - 4
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [3,6,7,5,4,2,1,8] => ? => ? = 6 - 4
[]
=> []
=> [] => => ? = 2 - 4
Description
The number of leading ones in a binary word.
Matching statistic: St000011
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 4 - 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 5 - 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 4 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 6 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 7 - 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 5 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 8 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 6 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 9 - 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 7 - 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 6 - 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 5 - 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 6 - 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 4 - 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 5 - 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 4 - 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 10 - 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 8 - 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5 = 7 - 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 6 - 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 6 - 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 4 - 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 5 - 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 4 - 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 2 = 4 - 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 2 = 4 - 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 9 = 11 - 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 9 - 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 6 = 8 - 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 7 - 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 7 - 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 7 - 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 5 - 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 10 - 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 11 - 2
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 8 - 2
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 6 - 2
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: St000738
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 98%●distinct values known / distinct values provided: 90%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 98%●distinct values known / distinct values provided: 90%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2 = 4 - 2
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[1,2],[3]]
=> 3 = 5 - 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 2 = 4 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 4 = 6 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 2 = 4 - 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2 = 4 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 5 = 7 - 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 3 = 5 - 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3 = 5 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 2 = 4 - 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 2 = 4 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[1,2,3,4,5],[6]]
=> 6 = 8 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 4 = 6 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 3 = 5 - 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2 = 4 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2 = 4 - 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 2 = 4 - 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,3,4,5,6],[2]]
=> 2 = 4 - 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7]]
=> 7 = 9 - 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [[1,2,3,4,6],[5]]
=> 5 = 7 - 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [[1,2,3],[4,5]]
=> 4 = 6 - 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 3 = 5 - 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 4 = 6 - 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 4 - 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 2 = 4 - 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,5],[3,4]]
=> 3 = 5 - 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 2 = 4 - 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [[1,3,4,5,6],[2]]
=> 2 = 4 - 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [[1,2,3,4,5,6,7],[8]]
=> 8 = 10 - 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [[1,2,3,4,5,7],[6]]
=> 6 = 8 - 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [[1,2,3,4],[5,6]]
=> 5 = 7 - 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [[1,2,3,5,6],[4]]
=> 4 = 6 - 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[1,2,3],[4,5]]
=> 4 = 6 - 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 3 = 5 - 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 2 = 4 - 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[1,2,4],[3,5]]
=> 3 = 5 - 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[1,2,5],[3,4]]
=> 3 = 5 - 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> 2 = 4 - 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [[1,3,4,5,6],[2]]
=> 2 = 4 - 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[1,3,5],[2,4]]
=> 2 = 4 - 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [[1,3,4,5],[2,6]]
=> 2 = 4 - 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [[1,2,3,4,5,6,7,8],[9]]
=> 9 = 11 - 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [[1,2,3,4,5,6,8],[7]]
=> 7 = 9 - 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [[1,2,3,4,5],[6,7]]
=> 6 = 8 - 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [[1,2,3,4,6,7],[5]]
=> 5 = 7 - 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [[1,2,3,4],[5,6]]
=> 5 = 7 - 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [[1,2,3,5],[4,6]]
=> 4 = 6 - 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,5,6] => [[1,2,4,5,6],[3]]
=> 3 = 5 - 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [[1,2,3,4],[5,6]]
=> 5 = 7 - 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[1,2,4],[3,5]]
=> 3 = 5 - 2
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => [[1,2,3,4,5,6,7,8,10],[9]]
=> ? = 11 - 2
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ?
=> ? = 8 - 2
[]
=> []
=> [] => []
=> ? = 2 - 2
[8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7,9] => [[1,3,5,7,9],[2,4,6,8]]
=> ? = 4 - 2
[7,6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6,9,8] => [[1,3,4,6,8],[2,5,7,9]]
=> ? = 4 - 2
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000054
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [1] => 1 = 4 - 3
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 2 = 5 - 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1 = 4 - 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2] => 3 = 6 - 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 1 = 4 - 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,2,3] => 1 = 4 - 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3] => 4 = 7 - 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 2 = 5 - 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 2 = 5 - 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 1 = 4 - 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 1 = 4 - 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [5,1,2,3,4] => 5 = 8 - 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [3,4,1,2] => 3 = 6 - 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 2 = 5 - 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 1 = 4 - 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 1 = 4 - 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 1 = 4 - 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [1,2,3,4,5] => 1 = 4 - 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [6,1,2,3,4,5] => 6 = 9 - 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => [4,5,1,2,3] => 4 = 7 - 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,1,4,2] => 3 = 6 - 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2 = 5 - 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,2,4] => 3 = 6 - 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 1 = 4 - 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,3] => 1 = 4 - 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 2 = 5 - 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 1 = 4 - 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [1,5,2,3,4] => 1 = 4 - 3
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => 7 = 10 - 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => [5,6,1,2,3,4] => 5 = 8 - 3
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => [4,1,5,2,3] => 4 = 7 - 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [3,5,1,2,4] => 3 = 6 - 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4] => 3 = 6 - 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => 2 = 5 - 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3] => 1 = 4 - 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 2 = 5 - 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,4,3] => 2 = 5 - 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 1 = 4 - 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [1,5,2,3,4] => 1 = 4 - 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 1 = 4 - 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [1,4,2,3,5] => 1 = 4 - 3
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [8,1,2,3,4,5,6,7] => 8 = 11 - 3
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,1,2,3,4,5,8] => [6,7,1,2,3,4,5] => 6 = 9 - 3
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,6,2,3,4,7] => [5,1,6,2,3,4] => 5 = 8 - 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,7] => [4,6,1,2,3,5] => 4 = 7 - 3
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => [4,1,2,5,3] => 4 = 7 - 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [3,4,5,1,2] => 3 = 6 - 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,6] => [2,5,1,3,4] => 2 = 5 - 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => [4,1,2,3,5] => 4 = 7 - 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2 = 5 - 3
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,7,2,3,4,5,8] => [6,1,7,2,3,4,5] => ? = 9 - 3
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,7,1,2,3,4,6,8] => [5,7,1,2,3,4,6] => ? = 8 - 3
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,8,2,3,4,5,6,9] => [7,1,8,2,3,4,5,6] => ? = 10 - 3
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,9,2,3,4,5,6,7,10] => ? => ? = 11 - 3
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [4,6,7,1,2,3,5,8] => ? => ? = 7 - 3
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [5,7,8,1,2,3,4,6,9] => ? => ? = 8 - 3
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [4,6,1,7,2,3,5,8] => [4,6,1,7,2,3,5] => ? = 7 - 3
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [3,6,7,1,2,4,5,8] => ? => ? = 6 - 3
[6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4,8,7] => ? => ? = 8 - 3
[6,6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [5,1,6,2,3,4,8,7] => ? => ? = 8 - 3
[6,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [4,6,1,2,3,5,8,7] => ? => ? = 7 - 3
[7,7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5,9,8] => ? => ? = 9 - 3
[8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,1,2,3,4,5,6,7,10,9] => ? => ? = 11 - 3
[]
=> []
=> [] => ? => ? = 2 - 3
[6,5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [1,2,4,5,6,7,3] => ? = 4 - 3
[6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => [1,3,4,5,6,7,2] => ? = 4 - 3
[7,6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => [1,3,4,5,6,7,8,2] => ? = 4 - 3
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: St000971
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 92%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 2 = 4 - 2
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 5 - 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 4 - 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 6 - 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2 = 4 - 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 4 - 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5 = 7 - 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3 = 5 - 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 5 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 4 - 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 4 - 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 6 = 8 - 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 4 = 6 - 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3 = 5 - 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2 = 4 - 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2 = 4 - 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2 = 4 - 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 2 = 4 - 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7}}
=> 7 = 9 - 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,6},{5}}
=> 5 = 7 - 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 4 = 6 - 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3 = 5 - 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4 = 6 - 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 2 = 4 - 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2 = 4 - 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3 = 5 - 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2 = 4 - 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> {{1,2},{3,4,6},{5}}
=> 2 = 4 - 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8}}
=> 8 = 10 - 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,7},{6}}
=> 6 = 8 - 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> {{1,2,3,6},{4,5}}
=> 5 = 7 - 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1,2,3,5,6},{4}}
=> 4 = 6 - 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 4 = 6 - 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 3 = 5 - 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2 = 4 - 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 5 - 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3 = 5 - 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2 = 4 - 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> {{1,2},{3,6},{4,5}}
=> 2 = 4 - 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 2 = 4 - 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> {{1,2},{3,5,6},{4}}
=> 2 = 4 - 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 11 - 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,8},{7}}
=> 7 = 9 - 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,7},{5,6}}
=> 6 = 8 - 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,6,7},{5}}
=> 5 = 7 - 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> {{1,2,6},{3,4,5}}
=> 5 = 7 - 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,6},{4},{5}}
=> 4 = 6 - 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> {{1,2,4,5,6},{3}}
=> 3 = 5 - 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 5 = 7 - 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 3 = 5 - 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3 = 5 - 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 12 - 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,9},{8}}
=> ? = 10 - 2
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,10},{9}}
=> ? = 11 - 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,9},{7,8}}
=> ? = 10 - 2
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,10},{8,9}}
=> ? = 11 - 2
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 7 - 2
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,7,9},{6},{8}}
=> ? = 8 - 2
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6,7}}
=> ? = 7 - 2
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,5,6,8},{4},{7}}
=> ? = 6 - 2
[]
=> []
=> []
=> {}
=> ? = 2 - 2
[7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 4 - 2
[6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 4 - 2
[6,5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 4 - 2
[6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 4 - 2
[8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 4 - 2
[7,6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 4 - 2
[9,8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 4 - 2
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: St000675
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000675: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 92%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 4 - 3
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 5 - 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 4 - 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 6 - 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 4 - 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 7 - 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 5 - 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 5 - 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 4 - 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 4 - 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 8 - 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 6 - 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 5 - 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 4 - 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 4 - 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 4 - 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 9 - 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 4 = 7 - 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3 = 6 - 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 5 - 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 6 - 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 4 - 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 4 - 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 5 - 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 4 - 3
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 7 = 10 - 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> 5 = 8 - 3
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 4 = 7 - 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 3 = 6 - 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 6 - 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 5 - 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 4 - 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 5 - 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 5 - 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 4 - 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 1 = 4 - 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1 = 4 - 3
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 11 - 3
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 6 = 9 - 3
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> 5 = 8 - 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> 4 = 7 - 3
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 4 = 7 - 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 3 = 6 - 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 5 - 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 4 = 7 - 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 5 - 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 5 - 3
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 12 - 3
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 10 - 3
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 11 - 3
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 10 - 3
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ?
=> ? = 11 - 3
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ?
=> ? = 8 - 3
[7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 9 - 3
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 10 - 3
[7,7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 9 - 3
[8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 11 - 3
[]
=> []
=> []
=> []
=> ? = 2 - 3
[7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 4 - 3
[6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 3
[6,5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 3
[8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 4 - 3
[7,6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 3
[9,8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 4 - 3
Description
The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.
Matching statistic: St000069
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Values
[1]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 4 - 3
[2]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 2 = 5 - 3
[1,1]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1 = 4 - 3
[3]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 3 = 6 - 3
[2,1]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 4 - 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 4 - 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4 = 7 - 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 5 - 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 5 - 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 4 - 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 4 - 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 5 = 8 - 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3 = 6 - 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 2 = 5 - 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 4 - 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 4 - 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 4 - 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 4 - 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> 6 = 9 - 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4 = 7 - 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3 = 6 - 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2 = 5 - 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3 = 6 - 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 4 - 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 4 - 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 5 - 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1 = 4 - 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 4 - 3
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> 7 = 10 - 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(6,1)],7)
=> 5 = 8 - 3
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 4 = 7 - 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> 3 = 6 - 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3 = 6 - 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 5 - 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1 = 4 - 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2 = 5 - 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2 = 5 - 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 4 - 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 4 - 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1 = 4 - 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 4 - 3
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> 8 = 11 - 3
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(7,1)],8)
=> 6 = 9 - 3
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
=> 5 = 8 - 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
=> 4 = 7 - 3
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> 4 = 7 - 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> 3 = 6 - 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> 2 = 5 - 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> 4 = 7 - 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 5 - 3
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 11 - 3
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(4,7),(5,7),(6,1),(6,7)],8)
=> ? = 7 - 3
[5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(6,2)],7)
=> ? = 7 - 3
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> ?
=> ? = 8 - 3
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(0,5),(0,6),(4,7),(5,7),(6,1),(6,2),(6,7)],8)
=> ? = 7 - 3
[7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(3,7),(4,7),(5,7),(6,1),(6,7)],8)
=> ? = 6 - 3
[6,4,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ([(0,4),(0,5),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> ? = 6 - 3
[5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(6,2),(6,3)],7)
=> ? = 7 - 3
[5,5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(5,2),(6,2)],7)
=> ? = 6 - 3
[7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ([(0,7),(7,1),(7,2),(7,3),(7,4),(7,5),(7,6)],8)
=> ? = 9 - 3
[7,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4 - 3
[6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(7,2)],8)
=> ? = 8 - 3
[5,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(5,3),(6,2),(6,3)],7)
=> ? = 6 - 3
[5,5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(3,6),(4,6),(5,6)],7)
=> ? = 5 - 3
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8)],9)
=> ? = 10 - 3
[6,6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(7,2),(7,3)],8)
=> ? = 8 - 3
[6,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(6,2),(7,2)],8)
=> ? = 7 - 3
[6,3,2,2,1]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,1),(4,6)],7)
=> ? = 5 - 3
[7,7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(8,2)],9)
=> ? = 9 - 3
[8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9)],10)
=> ? = 11 - 3
[]
=> []
=> ?
=> ? = 2 - 3
Description
The number of maximal elements of a poset.
Matching statistic: St000678
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 90%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 4 - 3
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 5 - 3
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 4 - 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 6 - 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1 = 4 - 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 4 - 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 7 - 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 5 - 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 5 - 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 4 - 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 8 - 3
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 6 - 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 5 - 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 4 - 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 4 - 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 4 - 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 9 - 3
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 7 - 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 6 - 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 5 - 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 6 - 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 4 - 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 4 - 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 5 - 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 4 - 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 10 - 3
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 5 = 8 - 3
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 4 = 7 - 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 3 = 6 - 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 6 - 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 5 - 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 4 - 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 5 - 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 5 - 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 4 - 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 4 - 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 4 - 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 1 = 4 - 3
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 3
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 9 - 3
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> 5 = 8 - 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> 4 = 7 - 3
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 4 = 7 - 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 3 = 6 - 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 2 = 5 - 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4 = 7 - 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 5 - 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 5 - 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 4 - 3
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 12 - 3
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 10 - 3
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 9 - 3
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 11 - 3
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 10 - 3
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 11 - 3
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ?
=> ? = 8 - 3
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 3
[7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9 - 3
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10 - 3
[7,7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 9 - 3
[8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 11 - 3
[]
=> []
=> []
=> []
=> ? = 2 - 3
[7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> ? = 4 - 3
[6,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 4 - 3
[6,5,4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4 - 3
[8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 4 - 3
[7,6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 4 - 3
[9,8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 4 - 3
Description
The number of up steps after the last double rise of a Dyck path.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000383The last part of an integer composition. 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. St000025The number of initial rises of a Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000026The position of the first return of a Dyck path. St000991The number of right-to-left minima of a permutation. St000989The number of final rises of a permutation. St000877The depth of the binary word interpreted as a path. St000234The number of global ascents of a permutation. St000068The number of minimal elements in a poset. St000717The number of ordinal summands of a poset. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000654The first descent of a permutation. St000990The first ascent of a permutation. St000734The last entry in the first row of a standard tableau. St000843The decomposition number of a perfect matching. St000061The number of nodes on the left branch of a binary tree. 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. St000326The position of the first one in a binary word after appending a 1 at the end. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001346The number of parking functions that give the same permutation. St000051The size of the left subtree of a binary tree. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000314The number of left-to-right-maxima of a permutation. St000352The Elizalde-Pak rank of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000241The number of cyclical small excedances. St000338The number of pixed points of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. 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. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001557The number of inversions of the second entry of a permutation.
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