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Your data matches 65 different statistics following compositions of up to 3 maps.
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Matching statistic: St000439
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding 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]
=> 2
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 3
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 4
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 3
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 3
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 5
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 4
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
Description
The position of the first down step of a Dyck path.
Matching statistic: St000382
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 2 - 1
[1,2] => [2] => 2 = 3 - 1
[2,1] => [1,1] => 1 = 2 - 1
[1,2,3] => [3] => 3 = 4 - 1
[1,3,2] => [2,1] => 2 = 3 - 1
[2,1,3] => [1,2] => 1 = 2 - 1
[2,3,1] => [2,1] => 2 = 3 - 1
[3,1,2] => [1,2] => 1 = 2 - 1
[3,2,1] => [1,1,1] => 1 = 2 - 1
[1,2,3,4] => [4] => 4 = 5 - 1
[1,2,4,3] => [3,1] => 3 = 4 - 1
[1,3,2,4] => [2,2] => 2 = 3 - 1
[1,3,4,2] => [3,1] => 3 = 4 - 1
[1,4,2,3] => [2,2] => 2 = 3 - 1
[1,4,3,2] => [2,1,1] => 2 = 3 - 1
[2,1,3,4] => [1,3] => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => 1 = 2 - 1
[2,3,1,4] => [2,2] => 2 = 3 - 1
[2,3,4,1] => [3,1] => 3 = 4 - 1
[2,4,1,3] => [2,2] => 2 = 3 - 1
[2,4,3,1] => [2,1,1] => 2 = 3 - 1
[3,1,2,4] => [1,3] => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => 1 = 2 - 1
[3,2,1,4] => [1,1,2] => 1 = 2 - 1
[3,2,4,1] => [1,2,1] => 1 = 2 - 1
[3,4,1,2] => [2,2] => 2 = 3 - 1
[3,4,2,1] => [2,1,1] => 2 = 3 - 1
[4,1,2,3] => [1,3] => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => 1 = 2 - 1
[4,2,1,3] => [1,1,2] => 1 = 2 - 1
[4,2,3,1] => [1,2,1] => 1 = 2 - 1
[4,3,1,2] => [1,1,2] => 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => 1 = 2 - 1
[1,2,3,4,5] => [5] => 5 = 6 - 1
[1,2,3,5,4] => [4,1] => 4 = 5 - 1
[1,2,4,3,5] => [3,2] => 3 = 4 - 1
[1,2,4,5,3] => [4,1] => 4 = 5 - 1
[1,2,5,3,4] => [3,2] => 3 = 4 - 1
[1,2,5,4,3] => [3,1,1] => 3 = 4 - 1
[1,3,2,4,5] => [2,3] => 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => 2 = 3 - 1
[1,3,4,2,5] => [3,2] => 3 = 4 - 1
[1,3,4,5,2] => [4,1] => 4 = 5 - 1
[1,3,5,2,4] => [3,2] => 3 = 4 - 1
[1,3,5,4,2] => [3,1,1] => 3 = 4 - 1
[1,4,2,3,5] => [2,3] => 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => 2 = 3 - 1
[1,4,3,5,2] => [2,2,1] => 2 = 3 - 1
[1,4,5,2,3] => [3,2] => 3 = 4 - 1
[7,6,5,3,2,4,8,1] => ? => ? = 2 - 1
[5,6,4,3,2,7,1,8] => ? => ? = 3 - 1
[6,5,7,3,4,1,2,8] => ? => ? = 2 - 1
[6,7,3,4,2,1,5,8] => ? => ? = 3 - 1
[7,6,4,2,3,1,5,8] => ? => ? = 2 - 1
[6,7,4,2,3,1,5,8] => ? => ? = 3 - 1
[7,5,4,3,1,2,6,8] => ? => ? = 2 - 1
[4,3,5,6,1,2,7,8] => ? => ? = 2 - 1
[5,6,2,1,3,4,7,8] => ? => ? = 3 - 1
[4,3,5,1,2,6,7,8] => ? => ? = 2 - 1
[4,5,2,1,3,6,7,8] => ? => ? = 3 - 1
[4,3,5,2,8,7,6,1] => ? => ? = 2 - 1
[2,1,3,6,5,4,8,7] => ? => ? = 2 - 1
[2,1,4,3,6,7,5,8] => ? => ? = 2 - 1
[3,2,6,4,5,1,7,8] => ? => ? = 2 - 1
[8,6,3,2,5,1,4,7] => ? => ? = 2 - 1
[8,7,3,2,5,1,4,6] => ? => ? = 2 - 1
[4,5,2,3,7,1,6,8] => ? => ? = 3 - 1
[4,5,1,3,7,2,6,8] => ? => ? = 3 - 1
[6,3,4,1,2,7,5,8] => ? => ? = 2 - 1
[3,6,2,1,4,7,5,8] => ? => ? = 3 - 1
[3,6,2,1,5,7,4,8] => ? => ? = 3 - 1
[5,8,4,6,7,1,3,2] => ? => ? = 3 - 1
[6,3,7,8,5,2,4,1] => ? => ? = 2 - 1
[8,6,1,7,3,2,5,4] => ? => ? = 2 - 1
[8,7,2,1,4,6,3,5] => ? => ? = 2 - 1
[8,5,7,4,1,2,6,3] => ? => ? = 2 - 1
[8,1,5,4,2,7,6,3] => ? => ? = 2 - 1
[2,1,8,5,7,6,3,4] => ? => ? = 2 - 1
[7,5,1,4,6,8,3,2] => ? => ? = 2 - 1
[6,3,8,1,5,7,4,2] => ? => ? = 2 - 1
[7,4,2,3,1,8,6,5] => ? => ? = 2 - 1
[5,7,1,8,3,6,2,4] => ? => ? = 3 - 1
[6,5,7,1,3,8,2,4] => ? => ? = 2 - 1
[7,3,5,8,1,6,2,4] => ? => ? = 2 - 1
[4,3,2,5,7,1,8,6] => ? => ? = 2 - 1
[8,2,4,7,5,1,6,3] => ? => ? = 2 - 1
[3,1,2,6,5,8,4,7] => ? => ? = 2 - 1
[2,3,1,7,5,4,8,6] => ? => ? = 3 - 1
[2,3,8,6,1,4,7,5] => ? => ? = 4 - 1
[2,1,7,3,6,4,5,8] => ? => ? = 2 - 1
[2,1,6,3,5,8,4,7] => ? => ? = 2 - 1
[2,6,1,8,3,5,4,7] => ? => ? = 3 - 1
[8,2,4,1,7,5,3,6] => ? => ? = 2 - 1
[2,7,1,8,3,5,6,4] => ? => ? = 3 - 1
[3,2,1,7,4,5,6,8] => ? => ? = 2 - 1
[8,3,5,7,1,6,4,2] => ? => ? = 2 - 1
[6,3,5,8,7,1,4,2] => ? => ? = 2 - 1
[8,7,4,6,1,2,5,3] => ? => ? = 2 - 1
[5,7,1,2,3,6,4,8] => ? => ? = 3 - 1
Description
The first part of an integer composition.
Matching statistic: St000326
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
St000326: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => => ? = 2 - 1
[1,2] => 0 => 2 = 3 - 1
[2,1] => 1 => 1 = 2 - 1
[1,2,3] => 00 => 3 = 4 - 1
[1,3,2] => 01 => 2 = 3 - 1
[2,1,3] => 10 => 1 = 2 - 1
[2,3,1] => 01 => 2 = 3 - 1
[3,1,2] => 10 => 1 = 2 - 1
[3,2,1] => 11 => 1 = 2 - 1
[1,2,3,4] => 000 => 4 = 5 - 1
[1,2,4,3] => 001 => 3 = 4 - 1
[1,3,2,4] => 010 => 2 = 3 - 1
[1,3,4,2] => 001 => 3 = 4 - 1
[1,4,2,3] => 010 => 2 = 3 - 1
[1,4,3,2] => 011 => 2 = 3 - 1
[2,1,3,4] => 100 => 1 = 2 - 1
[2,1,4,3] => 101 => 1 = 2 - 1
[2,3,1,4] => 010 => 2 = 3 - 1
[2,3,4,1] => 001 => 3 = 4 - 1
[2,4,1,3] => 010 => 2 = 3 - 1
[2,4,3,1] => 011 => 2 = 3 - 1
[3,1,2,4] => 100 => 1 = 2 - 1
[3,1,4,2] => 101 => 1 = 2 - 1
[3,2,1,4] => 110 => 1 = 2 - 1
[3,2,4,1] => 101 => 1 = 2 - 1
[3,4,1,2] => 010 => 2 = 3 - 1
[3,4,2,1] => 011 => 2 = 3 - 1
[4,1,2,3] => 100 => 1 = 2 - 1
[4,1,3,2] => 101 => 1 = 2 - 1
[4,2,1,3] => 110 => 1 = 2 - 1
[4,2,3,1] => 101 => 1 = 2 - 1
[4,3,1,2] => 110 => 1 = 2 - 1
[4,3,2,1] => 111 => 1 = 2 - 1
[1,2,3,4,5] => 0000 => 5 = 6 - 1
[1,2,3,5,4] => 0001 => 4 = 5 - 1
[1,2,4,3,5] => 0010 => 3 = 4 - 1
[1,2,4,5,3] => 0001 => 4 = 5 - 1
[1,2,5,3,4] => 0010 => 3 = 4 - 1
[1,2,5,4,3] => 0011 => 3 = 4 - 1
[1,3,2,4,5] => 0100 => 2 = 3 - 1
[1,3,2,5,4] => 0101 => 2 = 3 - 1
[1,3,4,2,5] => 0010 => 3 = 4 - 1
[1,3,4,5,2] => 0001 => 4 = 5 - 1
[1,3,5,2,4] => 0010 => 3 = 4 - 1
[1,3,5,4,2] => 0011 => 3 = 4 - 1
[1,4,2,3,5] => 0100 => 2 = 3 - 1
[1,4,2,5,3] => 0101 => 2 = 3 - 1
[1,4,3,2,5] => 0110 => 2 = 3 - 1
[1,4,3,5,2] => 0101 => 2 = 3 - 1
[1,4,5,2,3] => 0010 => 3 = 4 - 1
[1,4,5,3,2] => 0011 => 3 = 4 - 1
[8,5,4,6,3,7,2,1] => ? => ? = 2 - 1
[7,6,4,5,3,8,2,1] => ? => ? = 2 - 1
[7,6,4,5,8,2,3,1] => ? => ? = 2 - 1
[7,8,4,3,5,2,6,1] => ? => ? = 3 - 1
[8,7,5,4,2,3,6,1] => ? => ? = 2 - 1
[7,8,4,5,6,3,1,2] => ? => ? = 3 - 1
[7,8,6,3,4,5,1,2] => ? => ? = 3 - 1
[6,4,5,7,3,8,1,2] => ? => ? = 2 - 1
[7,5,4,6,8,2,1,3] => ? => ? = 2 - 1
[8,6,5,7,3,1,2,4] => ? => ? = 2 - 1
[8,6,3,2,1,4,5,7] => ? => ? = 2 - 1
[6,5,7,3,4,1,2,8] => ? => ? = 2 - 1
[5,6,4,7,1,2,3,8] => ? => ? = 3 - 1
[6,7,3,4,2,1,5,8] => ? => ? = 3 - 1
[7,6,4,2,3,1,5,8] => ? => ? = 2 - 1
[6,7,4,2,3,1,5,8] => ? => ? = 3 - 1
[6,3,4,5,2,1,7,8] => ? => ? = 2 - 1
[5,6,4,3,1,2,7,8] => ? => ? = 3 - 1
[6,4,3,5,1,2,7,8] => ? => ? = 2 - 1
[5,4,6,2,1,3,7,8] => ? => ? = 2 - 1
[6,5,1,2,3,4,7,8] => ? => ? = 2 - 1
[4,5,3,7,8,6,2,1] => ? => ? = 3 - 1
[4,3,5,7,8,6,2,1] => ? => ? = 2 - 1
[4,3,5,6,8,7,2,1] => ? => ? = 2 - 1
[3,2,6,7,5,8,4,1] => ? => ? = 2 - 1
[4,3,2,6,8,7,5,1] => ? => ? = 2 - 1
[4,5,3,2,8,7,6,1] => ? => ? = 3 - 1
[4,3,5,2,8,7,6,1] => ? => ? = 2 - 1
[3,4,2,6,5,8,7,1] => ? => ? = 3 - 1
[2,1,3,6,5,4,8,7] => ? => ? = 2 - 1
[2,1,5,4,3,6,8,7] => ? => ? = 2 - 1
[2,4,3,5,6,7,1,8] => ? => ? = 3 - 1
[2,1,5,4,6,7,3,8] => ? => ? = 2 - 1
[3,2,1,6,5,7,4,8] => ? => ? = 2 - 1
[2,1,4,3,6,7,5,8] => ? => ? = 2 - 1
[3,2,1,4,5,7,6,8] => ? => ? = 2 - 1
[2,1,3,5,4,6,7,8] => ? => ? = 2 - 1
[3,2,1,7,5,6,4,8] => ? => ? = 2 - 1
[3,2,6,4,5,1,7,8] => ? => ? = 2 - 1
[4,2,3,1,5,8,7,6] => ? => ? = 2 - 1
[4,2,3,1,6,5,7,8] => ? => ? = 2 - 1
[4,2,3,1,7,6,5,8] => ? => ? = 2 - 1
[2,1,3,6,4,8,5,7] => ? => ? = 2 - 1
[3,6,1,2,8,4,5,7] => ? => ? = 3 - 1
[3,1,2,7,4,5,6,8] => ? => ? = 2 - 1
[4,5,1,2,7,3,8,6] => ? => ? = 3 - 1
[4,1,7,2,3,8,5,6] => ? => ? = 2 - 1
[1,2,5,6,8,3,4,7] => ? => ? = 6 - 1
[1,2,5,3,7,8,4,6] => ? => ? = 4 - 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 => 1 = 2 - 1
[1,2] => [2,1] => [1,1] => 11 => 2 = 3 - 1
[2,1] => [1,2] => [2] => 10 => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1] => 111 => 3 = 4 - 1
[1,3,2] => [3,1,2] => [1,2] => 110 => 2 = 3 - 1
[2,1,3] => [2,3,1] => [2,1] => 101 => 1 = 2 - 1
[2,3,1] => [2,1,3] => [1,2] => 110 => 2 = 3 - 1
[3,1,2] => [1,3,2] => [2,1] => 101 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [3] => 100 => 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1111 => 4 = 5 - 1
[1,2,4,3] => [4,3,1,2] => [1,1,2] => 1110 => 3 = 4 - 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => 1101 => 2 = 3 - 1
[1,3,4,2] => [4,2,1,3] => [1,1,2] => 1110 => 3 = 4 - 1
[1,4,2,3] => [4,1,3,2] => [1,2,1] => 1101 => 2 = 3 - 1
[1,4,3,2] => [4,1,2,3] => [1,3] => 1100 => 2 = 3 - 1
[2,1,3,4] => [3,4,2,1] => [2,1,1] => 1011 => 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [2,2] => 1010 => 1 = 2 - 1
[2,3,1,4] => [3,2,4,1] => [1,2,1] => 1101 => 2 = 3 - 1
[2,3,4,1] => [3,2,1,4] => [1,1,2] => 1110 => 3 = 4 - 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1101 => 2 = 3 - 1
[2,4,3,1] => [3,1,2,4] => [1,3] => 1100 => 2 = 3 - 1
[3,1,2,4] => [2,4,3,1] => [2,1,1] => 1011 => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [2,2] => 1010 => 1 = 2 - 1
[3,2,1,4] => [2,3,4,1] => [3,1] => 1001 => 1 = 2 - 1
[3,2,4,1] => [2,3,1,4] => [2,2] => 1010 => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => 1101 => 2 = 3 - 1
[3,4,2,1] => [2,1,3,4] => [1,3] => 1100 => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => 1011 => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [2,2] => 1010 => 1 = 2 - 1
[4,2,1,3] => [1,3,4,2] => [3,1] => 1001 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [2,2] => 1010 => 1 = 2 - 1
[4,3,1,2] => [1,2,4,3] => [3,1] => 1001 => 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => [4] => 1000 => 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 11111 => 5 = 6 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => 11110 => 4 = 5 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => 11101 => 3 = 4 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => 11110 => 4 = 5 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 11101 => 3 = 4 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 11100 => 3 = 4 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => 11011 => 2 = 3 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => 11010 => 2 = 3 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => 11101 => 3 = 4 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => 11110 => 4 = 5 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 11101 => 3 = 4 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => 11100 => 3 = 4 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 11011 => 2 = 3 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 11010 => 2 = 3 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => 11001 => 2 = 3 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,2,2] => 11010 => 2 = 3 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 11101 => 3 = 4 - 1
[8,5,6,4,7,3,2,1] => [1,4,3,5,2,6,7,8] => ? => ? => ? = 2 - 1
[7,8,3,4,5,6,2,1] => [2,1,6,5,4,3,7,8] => ? => ? => ? = 3 - 1
[7,5,4,6,3,8,2,1] => [2,4,5,3,6,1,7,8] => ? => ? => ? = 2 - 1
[7,3,4,5,6,8,2,1] => [2,6,5,4,3,1,7,8] => ? => ? => ? = 2 - 1
[6,5,3,4,7,8,2,1] => [3,4,6,5,2,1,7,8] => ? => ? => ? = 2 - 1
[5,3,4,6,7,8,2,1] => [4,6,5,3,2,1,7,8] => ? => ? => ? = 2 - 1
[8,6,7,5,3,2,4,1] => [1,3,2,4,6,7,5,8] => ? => ? => ? = 2 - 1
[7,6,8,5,2,3,4,1] => [2,3,1,4,7,6,5,8] => ? => ? => ? = 2 - 1
[7,8,5,6,2,3,4,1] => [2,1,4,3,7,6,5,8] => ? => ? => ? = 3 - 1
[6,5,7,8,2,3,4,1] => [3,4,2,1,7,6,5,8] => ? => ? => ? = 2 - 1
[7,8,5,3,4,2,6,1] => [2,1,4,6,5,7,3,8] => ? => ? => ? = 3 - 1
[7,8,4,3,5,2,6,1] => [2,1,5,6,4,7,3,8] => ? => ? => ? = 3 - 1
[7,8,3,4,2,5,6,1] => [2,1,6,5,7,4,3,8] => ? => ? => ? = 3 - 1
[7,8,4,2,3,5,6,1] => [2,1,5,7,6,4,3,8] => ? => ? => ? = 3 - 1
[8,3,4,5,2,6,7,1] => [1,6,5,4,7,3,2,8] => ? => ? => ? = 2 - 1
[7,6,3,4,5,2,8,1] => [2,3,6,5,4,7,1,8] => ? => ? => ? = 2 - 1
[6,5,3,4,2,7,8,1] => [3,4,6,5,7,2,1,8] => ? => ? => ? = 2 - 1
[6,5,4,2,3,7,8,1] => [3,4,5,7,6,2,1,8] => ? => ? => ? = 2 - 1
[6,5,3,2,4,7,8,1] => [3,4,6,7,5,2,1,8] => ? => ? => ? = 2 - 1
[7,8,6,3,4,5,1,2] => [2,1,3,6,5,4,8,7] => ? => ? => ? = 3 - 1
[8,5,3,4,6,7,1,2] => [1,4,6,5,3,2,8,7] => ? => ? => ? = 2 - 1
[6,5,7,8,4,2,1,3] => [3,4,2,1,5,7,8,6] => ? => ? => ? = 2 - 1
[8,4,3,5,6,2,1,7] => [1,5,6,4,3,7,8,2] => ? => ? => ? = 2 - 1
[8,5,6,2,3,4,1,7] => [1,4,3,7,6,5,8,2] => ? => ? => ? = 2 - 1
[8,6,3,2,4,5,1,7] => [1,3,6,7,5,4,8,2] => ? => ? => ? = 2 - 1
[8,4,2,3,5,1,6,7] => [1,5,7,6,4,8,3,2] => ? => ? => ? = 2 - 1
[6,7,5,3,4,2,1,8] => [3,2,4,6,5,7,8,1] => ? => ? => ? = 3 - 1
[5,6,4,3,2,7,1,8] => ? => ? => ? => ? = 3 - 1
[7,6,4,5,3,1,2,8] => [2,3,5,4,6,8,7,1] => ? => ? => ? = 2 - 1
[5,6,4,7,1,2,3,8] => [4,3,5,2,8,7,6,1] => ? => ? => ? = 3 - 1
[6,7,3,2,1,4,5,8] => [3,2,6,7,8,5,4,1] => ? => ? => ? = 3 - 1
[7,5,4,2,3,1,6,8] => [2,4,5,7,6,8,3,1] => ? => ? => ? = 2 - 1
[7,5,4,3,1,2,6,8] => ? => ? => ? => ? = 2 - 1
[7,5,2,3,1,4,6,8] => [2,4,7,6,8,5,3,1] => ? => ? => ? = 2 - 1
[5,6,4,3,8,7,2,1] => [4,3,5,6,1,2,7,8] => ? => ? => ? = 3 - 1
[5,4,7,6,3,8,2,1] => [4,5,2,3,6,1,7,8] => ? => ? => ? = 2 - 1
[3,2,5,7,8,6,4,1] => [6,7,4,2,1,3,5,8] => ? => ? => ? = 2 - 1
[3,2,5,7,6,8,4,1] => [6,7,4,2,3,1,5,8] => ? => ? => ? = 2 - 1
[3,4,2,7,8,6,5,1] => [6,5,7,2,1,3,4,8] => ? => ? => ? = 3 - 1
[5,4,3,6,2,8,7,1] => [4,5,6,3,7,1,2,8] => ? => ? => ? = 2 - 1
[3,4,2,6,5,8,7,1] => [6,5,7,3,4,1,2,8] => ? => ? => ? = 3 - 1
[3,2,5,4,6,8,7,1] => [6,7,4,5,3,1,2,8] => ? => ? => ? = 2 - 1
[3,4,2,7,6,5,8,1] => ? => ? => ? => ? = 3 - 1
[3,4,2,5,7,6,8,1] => ? => ? => ? => ? = 3 - 1
[2,1,5,4,6,8,7,3] => [7,8,4,5,3,1,2,6] => ? => ? => ? = 2 - 1
[2,1,5,7,6,4,8,3] => [7,8,4,2,3,5,1,6] => ? => ? => ? = 2 - 1
[3,5,4,2,1,7,8,6] => [6,4,5,7,8,2,1,3] => ? => ? => ? = 3 - 1
[2,1,4,5,3,6,8,7] => [7,8,5,4,6,3,1,2] => ? => ? => ? = 2 - 1
[4,3,7,6,5,2,1,8] => [5,6,2,3,4,7,8,1] => ? => ? => ? = 2 - 1
[3,2,6,5,7,4,1,8] => [6,7,3,4,2,5,8,1] => ? => ? => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000383
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 2 - 1
[1,2] => [2] => [2] => 2 = 3 - 1
[2,1] => [1,1] => [1,1] => 1 = 2 - 1
[1,2,3] => [3] => [3] => 3 = 4 - 1
[1,3,2] => [2,1] => [1,2] => 2 = 3 - 1
[2,1,3] => [1,2] => [2,1] => 1 = 2 - 1
[2,3,1] => [2,1] => [1,2] => 2 = 3 - 1
[3,1,2] => [1,2] => [2,1] => 1 = 2 - 1
[3,2,1] => [1,1,1] => [1,1,1] => 1 = 2 - 1
[1,2,3,4] => [4] => [4] => 4 = 5 - 1
[1,2,4,3] => [3,1] => [1,3] => 3 = 4 - 1
[1,3,2,4] => [2,2] => [2,2] => 2 = 3 - 1
[1,3,4,2] => [3,1] => [1,3] => 3 = 4 - 1
[1,4,2,3] => [2,2] => [2,2] => 2 = 3 - 1
[1,4,3,2] => [2,1,1] => [1,1,2] => 2 = 3 - 1
[2,1,3,4] => [1,3] => [3,1] => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,2,1] => 1 = 2 - 1
[2,3,1,4] => [2,2] => [2,2] => 2 = 3 - 1
[2,3,4,1] => [3,1] => [1,3] => 3 = 4 - 1
[2,4,1,3] => [2,2] => [2,2] => 2 = 3 - 1
[2,4,3,1] => [2,1,1] => [1,1,2] => 2 = 3 - 1
[3,1,2,4] => [1,3] => [3,1] => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,2,1] => 1 = 2 - 1
[3,2,1,4] => [1,1,2] => [2,1,1] => 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,2,1] => 1 = 2 - 1
[3,4,1,2] => [2,2] => [2,2] => 2 = 3 - 1
[3,4,2,1] => [2,1,1] => [1,1,2] => 2 = 3 - 1
[4,1,2,3] => [1,3] => [3,1] => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,2,1] => 1 = 2 - 1
[4,2,1,3] => [1,1,2] => [2,1,1] => 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,2,1] => 1 = 2 - 1
[4,3,1,2] => [1,1,2] => [2,1,1] => 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => 1 = 2 - 1
[1,2,3,4,5] => [5] => [5] => 5 = 6 - 1
[1,2,3,5,4] => [4,1] => [1,4] => 4 = 5 - 1
[1,2,4,3,5] => [3,2] => [2,3] => 3 = 4 - 1
[1,2,4,5,3] => [4,1] => [1,4] => 4 = 5 - 1
[1,2,5,3,4] => [3,2] => [2,3] => 3 = 4 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,3] => 3 = 4 - 1
[1,3,2,4,5] => [2,3] => [3,2] => 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => [1,2,2] => 2 = 3 - 1
[1,3,4,2,5] => [3,2] => [2,3] => 3 = 4 - 1
[1,3,4,5,2] => [4,1] => [1,4] => 4 = 5 - 1
[1,3,5,2,4] => [3,2] => [2,3] => 3 = 4 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,3] => 3 = 4 - 1
[1,4,2,3,5] => [2,3] => [3,2] => 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => [1,2,2] => 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => [2,1,2] => 2 = 3 - 1
[1,4,3,5,2] => [2,2,1] => [1,2,2] => 2 = 3 - 1
[1,4,5,2,3] => [3,2] => [2,3] => 3 = 4 - 1
[7,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 3 - 1
[6,7,8,5,4,3,2,1] => [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 4 - 1
[8,4,5,6,7,3,2,1] => [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[7,4,5,6,8,3,2,1] => [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[6,4,5,7,8,3,2,1] => [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[5,4,6,7,8,3,2,1] => [1,4,1,1,1] => [1,1,1,4,1] => ? = 2 - 1
[8,7,3,4,5,6,2,1] => [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[7,8,3,4,5,6,2,1] => [2,4,1,1] => [1,1,4,2] => ? = 3 - 1
[8,5,3,4,6,7,2,1] => [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[8,4,3,5,6,7,2,1] => [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[7,6,3,4,5,8,2,1] => [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[7,5,3,4,6,8,2,1] => [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[6,5,3,4,7,8,2,1] => [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[5,4,3,6,7,8,2,1] => [1,1,4,1,1] => [1,1,4,1,1] => ? = 2 - 1
[8,7,6,2,3,4,5,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[8,6,4,2,3,5,7,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[8,5,4,2,3,6,7,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[8,4,3,2,5,6,7,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[7,6,5,3,2,4,8,1] => ? => ? => ? = 2 - 1
[7,6,5,2,3,4,8,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[7,6,4,2,3,5,8,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[7,6,3,2,4,5,8,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[7,5,4,2,3,6,8,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[7,5,3,2,4,6,8,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[7,4,3,2,5,6,8,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[6,5,4,2,3,7,8,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[6,5,3,2,4,7,8,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[5,4,3,2,6,7,8,1] => [1,1,1,4,1] => [1,4,1,1,1] => ? = 2 - 1
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => ? = 2 - 1
[6,7,8,5,4,3,1,2] => [3,1,1,1,2] => [2,1,1,1,3] => ? = 4 - 1
[5,6,7,8,3,4,1,2] => [4,2,2] => [2,2,4] => ? = 5 - 1
[8,7,3,4,5,6,1,2] => [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[8,5,3,4,6,7,1,2] => [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[8,4,3,5,6,7,1,2] => [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[8,3,4,5,6,7,1,2] => [1,5,2] => [2,5,1] => ? = 2 - 1
[4,5,6,7,3,8,1,2] => [4,2,2] => [2,2,4] => ? = 5 - 1
[7,6,3,4,5,8,1,2] => [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[7,5,3,4,6,8,1,2] => [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[7,3,4,5,6,8,1,2] => [1,5,2] => [2,5,1] => ? = 2 - 1
[6,4,3,5,7,8,1,2] => [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[6,3,4,5,7,8,1,2] => [1,5,2] => [2,5,1] => ? = 2 - 1
[5,4,3,6,7,8,1,2] => [1,1,4,2] => [2,4,1,1] => ? = 2 - 1
[5,3,4,6,7,8,1,2] => [1,5,2] => [2,5,1] => ? = 2 - 1
[4,3,5,6,7,8,1,2] => [1,5,2] => [2,5,1] => ? = 2 - 1
[3,4,5,6,7,8,1,2] => [6,2] => [2,6] => ? = 7 - 1
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => ? = 2 - 1
[6,7,8,5,4,2,1,3] => [3,1,1,1,2] => [2,1,1,1,3] => ? = 4 - 1
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [3,1,1,1,1,1] => ? = 2 - 1
[7,8,6,5,4,1,2,3] => [2,1,1,1,3] => [3,1,1,1,2] => ? = 3 - 1
[7,6,8,5,4,1,2,3] => [1,2,1,1,3] => [3,1,1,2,1] => ? = 2 - 1
Description
The last part of an integer composition.
Matching statistic: St000011
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[5,6,4,7,8,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[5,4,6,7,3,8,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[6,7,5,8,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,4,5,8,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,3,2,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,2,3,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[5,6,3,4,7,2,8,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 - 1
[4,5,3,6,7,2,8,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[5,6,4,7,2,3,8,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 - 1
[5,6,7,2,3,4,8,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 4 - 1
[4,3,5,6,2,7,8,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[4,5,3,2,6,7,8,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[4,5,2,3,6,7,8,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,4,3,1,2] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,4,5,8,3,1,2] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[5,6,4,7,8,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,3,4,1,2] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[5,6,4,7,3,8,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[4,5,6,7,3,8,1,2] => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 5 - 1
[5,6,4,3,7,8,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[5,4,6,3,7,8,1,2] => [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[5,6,3,4,7,8,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[4,5,3,6,7,8,1,2] => [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,4,2,1,3] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,4,8,2,1,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,4,1,2,3] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,4,8,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,4,5,8,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[5,6,4,7,8,1,2,3] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,3,2,1,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,2,3,1,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,3,1,2,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,2,1,3,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,7,5,8,1,2,3,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[5,6,4,7,3,2,1,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[5,6,4,3,7,2,1,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[5,6,4,7,2,3,1,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[5,6,4,3,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[5,6,3,4,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[3,4,5,6,2,7,1,8] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 5 - 1
[3,4,2,5,6,7,1,8] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 - 1
[5,6,3,4,7,1,2,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[5,6,4,7,1,2,3,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[4,5,3,2,6,1,7,8] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[3,4,2,5,6,1,7,8] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
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: St000745
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 1 = 2 - 1
[1,2] => [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2 = 3 - 1
[2,1] => [1,1,0,0]
=> [1,2] => [[1,2]]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3 = 4 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [[1,2],[3]]
=> 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => [[1,3],[2]]
=> 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1 = 2 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1 = 2 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4 = 5 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 2 = 3 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 2 = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2 = 3 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2 = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 2 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 2 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5 = 6 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,5],[2],[3],[4]]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,4],[2],[3],[5]]
=> 3 = 4 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[1,5],[2],[3],[4]]
=> 4 = 5 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 2 = 3 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,3],[2,5],[4]]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[1,5],[2],[3],[4]]
=> 4 = 5 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [5,2,1,3,4,6,7,8] => ?
=> ? = 3 - 1
[6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [5,2,1,3,4,6,7,8] => ?
=> ? = 3 - 1
[6,5,4,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [5,4,1,2,3,6,7,8] => ?
=> ? = 2 - 1
[6,4,5,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [5,4,1,2,3,6,7,8] => ?
=> ? = 2 - 1
[6,7,5,4,3,8,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [6,2,1,3,4,5,7,8] => ?
=> ? = 3 - 1
[6,7,4,5,3,8,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [6,2,1,3,4,5,7,8] => ?
=> ? = 3 - 1
[6,5,4,3,7,8,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 2 - 1
[6,5,3,4,7,8,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 2 - 1
[6,3,4,5,7,8,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 2 - 1
[5,4,3,6,7,8,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3,7,8] => ?
=> ? = 2 - 1
[5,3,4,6,7,8,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3,7,8] => ?
=> ? = 2 - 1
[6,7,4,5,8,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [5,2,1,3,4,6,7,8] => ?
=> ? = 3 - 1
[5,6,3,4,7,2,8,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [7,5,2,1,3,4,6,8] => ?
=> ? = 3 - 1
[4,5,3,6,7,2,8,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [7,5,4,2,1,3,6,8] => ?
=> ? = 3 - 1
[5,6,4,7,2,3,8,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,1,3,5,6,8] => ?
=> ? = 3 - 1
[5,4,3,6,2,7,8,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [7,6,4,1,2,3,5,8] => ?
=> ? = 2 - 1
[4,3,5,2,6,7,8,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [7,6,5,3,1,2,4,8] => ?
=> ? = 2 - 1
[6,7,4,5,8,3,1,2] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [5,2,1,3,4,6,7,8] => ?
=> ? = 3 - 1
[6,5,4,7,8,3,1,2] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [5,4,1,2,3,6,7,8] => ?
=> ? = 2 - 1
[6,7,4,5,3,8,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [6,2,1,3,4,5,7,8] => ?
=> ? = 3 - 1
[6,5,4,3,7,8,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 2 - 1
[5,6,4,3,7,8,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [6,5,2,1,3,4,7,8] => ?
=> ? = 3 - 1
[5,4,6,3,7,8,1,2] => [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [6,5,3,1,2,4,7,8] => ?
=> ? = 2 - 1
[5,6,3,4,7,8,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [6,5,2,1,3,4,7,8] => ?
=> ? = 3 - 1
[6,4,3,5,7,8,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 2 - 1
[6,3,4,5,7,8,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 2 - 1
[5,4,3,6,7,8,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3,7,8] => ?
=> ? = 2 - 1
[5,3,4,6,7,8,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3,7,8] => ?
=> ? = 2 - 1
[6,7,5,4,8,2,1,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [5,2,1,3,4,6,7,8] => ?
=> ? = 3 - 1
[6,7,5,4,8,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [5,2,1,3,4,6,7,8] => ?
=> ? = 3 - 1
[6,7,4,5,8,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [5,2,1,3,4,6,7,8] => ?
=> ? = 3 - 1
[6,5,4,7,8,1,2,3] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [5,4,1,2,3,6,7,8] => ?
=> ? = 2 - 1
[6,4,5,7,8,1,2,3] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> [5,4,1,2,3,6,7,8] => ?
=> ? = 2 - 1
[5,4,6,7,3,2,1,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 2 - 1
[5,6,4,3,7,2,1,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [8,5,2,1,3,4,6,7] => ?
=> ? = 3 - 1
[5,4,6,7,2,3,1,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 2 - 1
[5,6,4,3,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [8,6,2,1,3,4,5,7] => ?
=> ? = 3 - 1
[5,6,3,4,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [8,6,2,1,3,4,5,7] => ?
=> ? = 3 - 1
[5,4,6,2,3,7,1,8] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [8,6,3,1,2,4,5,7] => ?
=> ? = 2 - 1
[5,4,6,7,3,1,2,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 2 - 1
[5,6,3,4,7,1,2,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [8,5,2,1,3,4,6,7] => ?
=> ? = 3 - 1
[4,3,5,6,7,1,2,8] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,1,2,6,7] => ?
=> ? = 2 - 1
[5,4,6,7,2,1,3,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 2 - 1
[5,4,6,7,1,2,3,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 2 - 1
[5,4,6,3,2,1,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,1,2,4,5,6] => ?
=> ? = 2 - 1
[5,4,6,2,3,1,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,1,2,4,5,6] => ?
=> ? = 2 - 1
[3,4,2,5,6,1,7,8] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,2,1,3,6] => ?
=> ? = 3 - 1
[5,4,6,3,1,2,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,1,2,4,5,6] => ?
=> ? = 2 - 1
[4,3,5,6,1,2,7,8] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,1,2,5,6] => ?
=> ? = 2 - 1
[5,4,6,2,1,3,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,1,2,4,5,6] => ?
=> ? = 2 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000759
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> 1 = 2 - 1
[1,2] => [1,0,1,0]
=> [1]
=> 2 = 3 - 1
[2,1] => [1,1,0,0]
=> []
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 3 = 4 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
=> []
=> 1 = 2 - 1
[3,2,1] => [1,1,1,0,0,0]
=> []
=> 1 = 2 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 4 = 5 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3 = 4 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2 = 3 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3 = 4 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2 = 3 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2 = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 2 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 2 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 2 = 3 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 2 = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 1 = 2 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> 1 = 2 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 1 = 2 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 1 = 2 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 1 = 2 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 5 = 6 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 3 = 4 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 4 = 5 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3 = 4 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2 = 3 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 3 = 4 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 4 = 5 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 3 = 4 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 3 = 4 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 3 = 4 - 1
[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]
=> ? = 6 - 1
[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]
=> ? = 6 - 1
[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]
=> ? = 6 - 1
[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]
=> ? = 4 - 1
[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]
=> ? = 5 - 1
[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]
=> ? = 5 - 1
[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]
=> ? = 6 - 1
[1,2,4,5,7,3,6] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> ? = 6 - 1
[1,2,4,5,7,6,3] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> ? = 6 - 1
[1,2,4,6,3,5,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> ? = 5 - 1
[1,2,4,6,3,7,5] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> ? = 5 - 1
[1,2,4,6,5,3,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> ? = 5 - 1
[1,2,4,6,5,7,3] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> ? = 5 - 1
[1,2,4,6,7,3,5] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> ? = 6 - 1
[1,2,4,6,7,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> ? = 6 - 1
[1,2,4,7,3,5,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 5 - 1
[1,2,4,7,3,6,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 5 - 1
[1,2,4,7,5,3,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 5 - 1
[1,2,4,7,5,6,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 5 - 1
[1,2,4,7,6,3,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 5 - 1
[1,2,4,7,6,5,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 5 - 1
[1,2,5,3,6,4,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> ? = 4 - 1
[1,2,5,3,7,4,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> ? = 4 - 1
[1,2,5,3,7,6,4] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> ? = 4 - 1
[1,2,5,4,6,3,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> ? = 4 - 1
[1,2,5,4,7,3,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> ? = 4 - 1
[1,2,5,4,7,6,3] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> ? = 4 - 1
[1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> ? = 5 - 1
[1,2,5,6,4,3,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> ? = 5 - 1
[1,2,6,3,7,4,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> ? = 4 - 1
[1,2,6,3,7,5,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> ? = 4 - 1
[1,2,6,4,7,3,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> ? = 4 - 1
[1,2,6,4,7,5,3] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> ? = 4 - 1
[1,2,6,5,7,3,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> ? = 4 - 1
[1,2,6,5,7,4,3] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> ? = 4 - 1
[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]
=> ? = 3 - 1
[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]
=> ? = 3 - 1
[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]
=> ? = 3 - 1
[1,3,2,5,7,4,6] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,1,1]
=> ? = 3 - 1
[1,3,2,5,7,6,4] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,1,1]
=> ? = 3 - 1
[1,3,2,6,4,7,5] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1,1]
=> ? = 3 - 1
[1,3,2,6,5,7,4] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1,1]
=> ? = 3 - 1
[1,3,2,6,7,4,5] => [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,1,1]
=> ? = 3 - 1
[1,3,2,6,7,5,4] => [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,1,1]
=> ? = 3 - 1
[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]
=> ? = 4 - 1
[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]
=> ? = 4 - 1
[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]
=> ? = 4 - 1
[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]
=> ? = 4 - 1
[1,3,4,2,7,5,6] => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,1,1]
=> ? = 4 - 1
[1,3,4,2,7,6,5] => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,1,1]
=> ? = 4 - 1
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000678
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 83%●distinct values known / distinct values provided: 80%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 83%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1,0]
=> ? = 2 - 1
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,4,5,3,2] => [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[8,6,7,5,4,3,2,1] => [1,2,3,4,5,7,6,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[6,7,8,5,4,3,2,1] => [1,2,3,4,5,8,7,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 4 - 1
[8,7,5,6,4,3,2,1] => [1,2,3,4,6,5,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[8,6,5,7,4,3,2,1] => [1,2,3,4,7,5,6,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[8,5,6,7,4,3,2,1] => [1,2,3,4,7,6,5,8] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[7,6,5,8,4,3,2,1] => [1,2,3,4,8,5,6,7] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[6,7,5,8,4,3,2,1] => [1,2,3,4,8,5,7,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[7,5,6,8,4,3,2,1] => [1,2,3,4,8,6,5,7] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[6,5,7,8,4,3,2,1] => [1,2,3,4,8,7,5,6] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[8,7,6,4,5,3,2,1] => [1,2,3,5,4,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 - 1
[8,7,5,4,6,3,2,1] => [1,2,3,6,4,5,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[8,7,4,5,6,3,2,1] => [1,2,3,6,5,4,7,8] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[8,5,6,4,7,3,2,1] => [1,2,3,7,4,6,5,8] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 - 1
[8,6,4,5,7,3,2,1] => [1,2,3,7,5,4,6,8] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[8,4,5,6,7,3,2,1] => [1,2,3,7,6,5,4,8] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[6,7,5,4,8,3,2,1] => [1,2,3,8,4,5,7,6] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[7,5,6,4,8,3,2,1] => [1,2,3,8,4,6,5,7] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 - 1
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 - 1
[7,6,4,5,8,3,2,1] => [1,2,3,8,5,4,6,7] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[6,7,4,5,8,3,2,1] => [1,2,3,8,5,4,7,6] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[7,5,4,6,8,3,2,1] => [1,2,3,8,6,4,5,7] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[7,4,5,6,8,3,2,1] => [1,2,3,8,6,5,4,7] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[6,5,4,7,8,3,2,1] => [1,2,3,8,7,4,5,6] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[5,6,4,7,8,3,2,1] => [1,2,3,8,7,4,6,5] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[6,4,5,7,8,3,2,1] => [1,2,3,8,7,5,4,6] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[5,4,6,7,8,3,2,1] => [1,2,3,8,7,6,4,5] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[8,7,6,5,3,4,2,1] => [1,2,4,3,5,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[7,6,8,5,3,4,2,1] => [1,2,4,3,5,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[7,8,5,6,3,4,2,1] => [1,2,4,3,6,5,8,7] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[8,5,6,7,3,4,2,1] => [1,2,4,3,7,6,5,8] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[7,6,5,8,3,4,2,1] => [1,2,4,3,8,5,6,7] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[6,7,5,8,3,4,2,1] => [1,2,4,3,8,5,7,6] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[6,5,7,8,3,4,2,1] => [1,2,4,3,8,7,5,6] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[8,7,6,4,3,5,2,1] => [1,2,5,3,4,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[7,8,6,4,3,5,2,1] => [1,2,5,3,4,6,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[7,6,8,4,3,5,2,1] => [1,2,5,3,4,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[8,7,6,3,4,5,2,1] => [1,2,5,4,3,6,7,8] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 - 1
[7,8,6,3,4,5,2,1] => [1,2,5,4,3,6,8,7] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[8,6,7,3,4,5,2,1] => [1,2,5,4,3,7,6,8] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2 - 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000054
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1] => 1 = 2 - 1
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 2 = 3 - 1
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 3 = 4 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2 = 3 - 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,3,2] => 1 = 2 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 4 = 5 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 3 = 4 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 3 = 4 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 3 = 4 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => 2 = 3 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,4,3,2] => 1 = 2 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,4,3,2] => 1 = 2 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => 2 = 3 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 1 = 2 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 1 = 2 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,4,3,2] => 1 = 2 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,4,3,2] => 1 = 2 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,4,3,2] => 1 = 2 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => 1 = 2 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 5 = 6 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => 4 = 5 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,5,4,2,1] => 3 = 4 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,5,4,2,1] => 3 = 4 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => 2 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,4,3,1] => 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => 3 = 4 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => 3 = 4 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => 3 = 4 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,4,3,1] => 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,5,4,3,1] => 2 = 3 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,5,4,3,1] => 2 = 3 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => 3 = 4 - 1
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 7 - 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 6 - 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 7 - 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 6 - 1
[1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 6 - 1
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 6 - 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 7 - 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 6 - 1
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [5,7,4,6,3,2,1] => ? = 6 - 1
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [5,4,7,6,3,2,1] => ? = 6 - 1
[1,2,3,6,7,5,4] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [5,4,7,6,3,2,1] => ? = 6 - 1
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,7,6,4,5] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [4,7,6,5,3,2,1] => ? = 5 - 1
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 5 - 1
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [4,3,7,6,5,2,1] => ? = 5 - 1
[1,2,4,5,6,3,7] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [5,4,3,7,6,2,1] => ? = 6 - 1
[1,2,4,5,6,7,3] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [6,5,4,3,7,2,1] => ? = 7 - 1
[1,2,4,5,7,3,6] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [5,4,3,7,6,2,1] => ? = 6 - 1
[1,2,4,5,7,6,3] => [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [5,7,4,3,6,2,1] => ? = 6 - 1
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 5 - 1
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [4,3,7,6,5,2,1] => ? = 5 - 1
[1,2,4,6,5,3,7] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [4,7,3,6,5,2,1] => ? = 5 - 1
[1,2,4,6,5,7,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 5 - 1
[1,2,4,6,7,3,5] => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [5,4,3,7,6,2,1] => ? = 6 - 1
[1,2,4,6,7,5,3] => [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => [5,4,7,3,6,2,1] => ? = 6 - 1
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [4,3,7,6,5,2,1] => ? = 5 - 1
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [4,3,7,6,5,2,1] => ? = 5 - 1
[1,2,4,7,5,3,6] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [4,7,3,6,5,2,1] => ? = 5 - 1
[1,2,4,7,5,6,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 5 - 1
[1,2,4,7,6,3,5] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => [4,7,3,6,5,2,1] => ? = 5 - 1
[1,2,4,7,6,5,3] => [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [4,7,6,3,5,2,1] => ? = 5 - 1
[1,2,5,3,4,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,5,3,4,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [3,7,6,5,4,2,1] => ? = 4 - 1
[1,2,5,3,6,4,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [3,7,6,5,4,2,1] => ? = 4 - 1
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
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001050The number of terminal closers of a set partition. St000069The number of maximal elements of a poset. St000068The number of minimal elements in a poset. St000007The number of saliances of the permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000617The number of global maxima of a Dyck path. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000363The number of minimal vertex covers of a graph. St001322The size of a minimal independent dominating set in a graph. St001316The domatic number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. 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. St000501The size of the first part in the decomposition of a permutation. St000990The first ascent of a permutation. St000989The number of final rises of a permutation. St000546The number of global descents of a permutation. St000654The first descent of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000740The last entry of a permutation. 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. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St000084The number of subtrees. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function.
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