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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000439
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(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ 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]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 3
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 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]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 3
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 3
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> 3
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 4
Description
The position of the first down step of a Dyck path.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [2] => 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,2] => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1] => 2 = 3 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [1,3] => 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [3] => 3 = 4 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [2,2] => 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [1,4] => 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,2] => 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,1] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1] => 3 = 4 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [2,3] => 2 = 3 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [1,5] => 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,3] => 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,2] => 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1] => 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [3,2] => 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [2,4] => 2 = 3 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [1,6] => 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => [2,4] => 2 = 3 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [1,2,2] => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [2,3] => 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [1,3,1] => 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4] => 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1] => 2 = 3 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [1,2,2] => 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2] => 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [3,3] => 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => [2,5] => 2 = 3 - 1
[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] => [1,7] => 1 = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => [2,5] => 2 = 3 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => [1,2,3] => 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => [2,4] => 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [1,4] => 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,2] => 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [2,3] => 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1] => 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,1] => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1] => 4 = 5 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [2,2,2] => 2 = 3 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,2] => 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [3,3] => 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,2,3,4,5] => [3,4] => 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => [2,6] => 2 = 3 - 1
[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] => [1,8] => 1 = 2 - 1
[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] => [2,6] => 2 = 3 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,6,2,3,4,7] => [1,2,4] => 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,7] => [2,5] => 2 = 3 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => [1,3,2] => 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => [3,3] => 3 = 4 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [2,9] => ? = 3 - 1
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,7,2,9,3,4,5,6,8] => ? => ? = 3 - 1
[8,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,9,2,3,4,5,6,7,10,8] => [2,7,1] => ? = 3 - 1
[9,9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,8,11,10] => [1,9,1] => ? = 2 - 1
Description
The first part of an integer composition.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 0 => 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 2 = 3 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 100 => 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 3 = 4 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 010 => 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 3 = 4 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0100 => 2 = 3 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 10000 => 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 0100 => 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 0010 => 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 01000 => 2 = 3 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => 100000 => 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,1,2,3,6] => 01000 => 2 = 3 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1010 => 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 0100 => 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1001 => 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0101 => 2 = 3 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1010 => 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0010 => 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => 00100 => 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => 010000 => 2 = 3 - 1
[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] => 1000000 => 1 = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,1,2,3,4,7] => 010000 => 2 = 3 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,5,2,3,6] => 10100 => 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,1,2,4,6] => 01000 => 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1000 => 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0010 => 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0100 => 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0101 => 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 4 = 5 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => 01010 => 2 = 3 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0010 => 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => 00100 => 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,2,3,4,5] => 001000 => 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => 0100000 => 2 = 3 - 1
[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] => 10000000 => 1 = 2 - 1
[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] => 0100000 => 2 = 3 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,6,2,3,4,7] => 101000 => 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,1,2,3,5,7] => 010000 => 2 = 3 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => 10010 => 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,1,2,6] => 00100 => 3 = 4 - 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [5,1,7,2,3,4,6,8] => ? => ? = 2 - 1
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [5,1,6,7,2,3,4,8] => ? => ? = 2 - 1
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [4,5,7,1,2,3,6,8] => ? => ? = 4 - 1
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,7,2,9,3,4,5,6,8] => ? => ? = 3 - 1
[7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [5,6,1,2,3,7,4,8] => ? => ? = 3 - 1
[7,3,1,1,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [3,6,1,7,2,4,5,8] => ? => ? = 3 - 1
[7,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [5,6,1,2,3,4,7,8] => ? => ? = 3 - 1
[7,4,4]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6,8] => ? => ? = 2 - 1
[7,4,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,6,1,3,7,4,5,8] => ? => ? = 3 - 1
[8,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,9,2,3,4,5,6,7,10,8] => ? => ? = 3 - 1
[7,7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,1,0,0]
=> [5,7,1,2,3,4,6,9,8] => ? => ? = 3 - 1
[6,6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [5,1,2,3,6,4,8,7] => ? => ? = 2 - 1
[7,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [5,1,2,3,4,7,6,8] => ? => ? = 2 - 1
[7,5,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3,8] => ? => ? = 2 - 1
[6,4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ? => ? = 3 - 1
[7,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,2,7,4,5,6,8] => ? => ? = 2 - 1
[9,9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,1,2,3,4,5,6,7,8,11,10] => ? => ? = 2 - 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: St000745
Mp00043: Integer partitions —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: 93% ●values known / values provided: 93%●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: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[1,2],[3]]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 2 = 3 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3 = 4 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[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 = 3 - 1
[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]]
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[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 = 3 - 1
[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]]
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [[1,3,4,5],[2],[6]]
=> 2 = 3 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[1,2,4],[3],[5]]
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,5],[3,4]]
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [[1,4,5,6],[2],[3]]
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [[1,3,4,5,6,7],[2]]
=> 2 = 3 - 1
[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]]
=> 1 = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [[1,3,4,5,6],[2],[7]]
=> 2 = 3 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [[1,2,4,5],[3],[6]]
=> 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [[1,3,4,5],[2],[6]]
=> 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[1,4],[2],[3],[5]]
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[1,3],[2,5],[4]]
=> 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[1,2],[3,5],[4]]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[1,5],[2],[3],[4]]
=> 4 = 5 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [[1,3,5,6],[2],[4]]
=> 2 = 3 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,4,5],[2],[3]]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [[1,4,5,6],[2],[3]]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [[1,4,5,6,7],[2],[3]]
=> 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [[1,3,4,5,6,7,8],[2]]
=> 2 = 3 - 1
[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]]
=> 1 = 2 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [[1,3,4,5,6,7],[2],[8]]
=> 2 = 3 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [[1,2,4,5,6],[3],[7]]
=> 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [[1,3,4,5,6],[2],[7]]
=> 2 = 3 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[1,2,3,5],[4],[6]]
=> 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [[1,4,5],[2],[3],[6]]
=> 3 = 4 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 2 - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 3 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 3 - 1
[7,4,1]
=> [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
[7,3,2]
=> [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
[7,5,1]
=> [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
[5,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [6,4,2,3,5,7,8,1] => ?
=> ? = 3 - 1
[7,5,2]
=> [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
[7,3,2,2]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,1,2,6,7] => ?
=> ? = 2 - 1
[7,3,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [8,4,2,3,5,6,1,7] => ?
=> ? = 3 - 1
[6,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,3,5,6,8,1] => ?
=> ? = 3 - 1
[4,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,2,3,7,8,1] => ?
=> ? = 3 - 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,2,8,1] => ?
=> ? = 4 - 1
[7,7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [8,9,2,1,3,4,5,6,7] => [[1,3,4,5,6,7],[2,9],[8]]
=> ? = 3 - 1
[7,6,2]
=> [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
[7,3,3,2]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,1,2,6,7] => ?
=> ? = 2 - 1
[6,6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [7,8,4,1,2,3,5,6] => ?
=> ? = 2 - 1
[8,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [9,2,3,4,5,6,7,8,10,1] => [[1,3,4,5,6,7,8,10],[2],[9]]
=> ? = 3 - 1
[7,7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,1,0,0]
=> [8,9,2,3,1,4,5,6,7] => ?
=> ? = 3 - 1
[7,4,3,2]
=> [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
[7,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [8,5,2,3,4,6,7,1] => ?
=> ? = 3 - 1
[9,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,5,6,7,8,9,1] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 3 - 1
[7,4,3,3]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,1,2,3,7] => ?
=> ? = 2 - 1
[7,3,3,2,2]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,1,2,7] => ?
=> ? = 2 - 1
[6,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,8,1,2] => ?
=> ? = 2 - 1
[5,4,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [6,5,7,2,3,4,8,1] => ?
=> ? = 3 - 1
[5,4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,7,8,1,2] => ?
=> ? = 2 - 1
[5,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,1,2,3] => ?
=> ? = 2 - 1
[5,3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,2,3,8,1] => ?
=> ? = 3 - 1
[5,3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,1,2] => ?
=> ? = 2 - 1
[4,3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,8,1,2] => ?
=> ? = 2 - 1
[7,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0]
=> [8,7,9,1,2,3,4,5,6] => [[1,2,3,4,5,6],[7,9],[8]]
=> ? = 2 - 1
[8,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,8,1,2] => [[1,2,5,6,7,8],[3,4],[9]]
=> ? = 2 - 1
[8,7,6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,2,1,3] => [[1,3],[2],[4],[5],[6],[7],[8],[9]]
=> ? = 3 - 1
[8,6,5,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,7,6,5,4,3,2,1,8] => [[1,8],[2],[3],[4],[5],[6],[7],[9]]
=> ? = 8 - 1
[9,7,6,5,4,3,2,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [10,8,7,6,5,4,3,2,1,9] => [[1,9],[2],[3],[4],[5],[6],[7],[8],[10]]
=> ? = 9 - 1
[9,9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,11,1,2,3,4,5,6,7,8,9] => [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 2 - 1
[7,7,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [8,9,2,3,4,5,6,7,1] => [[1,3,4,5,6,7],[2,9],[8]]
=> ? = 3 - 1
[8,6,5,4,3,2]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,7,6,5,4,3,1,2,8] => [[1,2,8],[3],[4],[5],[6],[7],[9]]
=> ? = 2 - 1
[7,6,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,7,9,6,5,4,2,3,1] => [[1,3],[2,9],[4],[5],[6],[7],[8]]
=> ? = 3 - 1
[7,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,3,1,2] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? = 2 - 1
[7,6,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,7,9,6,5,4,3,1,2] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? = 2 - 1
[9,8,8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [10,9,11,8,7,6,5,4,3,2,1] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 11 - 1
[8,7,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [9,8,10,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? = 2 - 1
[7,7,6,5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,2,1,3] => [[1,3],[2,9],[4],[5],[6],[7],[8]]
=> ? = 3 - 1
[7,7,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,4,2,3,1] => [[1,3],[2,9],[4],[5],[6],[7],[8]]
=> ? = 3 - 1
[8,8,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? = 2 - 1
[9,9,8,7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [10,11,9,8,7,6,5,4,3,2,1] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 11 - 1
[8,7,6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6],[7],[8],[9]]
=> ? = 4 - 1
[10,9,8,7,6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,10,9,8,7,6,5,4,3,1,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2 - 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000011
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 3 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 2 = 3 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> 1 = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 3 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> 2 = 3 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> 1 = 2 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 2 = 3 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 3 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[7,3,1,1,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 3 - 1
[6,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 4 - 1
[5,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[8,1,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
[7,3,2,2]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[7,3,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 3 - 1
[6,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[5,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[4,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[2,2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[7,5,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 - 1
[7,4,2,2]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[7,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[6,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[5,5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[5,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[8,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[7,5,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[7,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[7,3,3,2,1]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5 - 1
[6,6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[6,6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2 - 1
[6,6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 4 - 1
[6,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[5,4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[5,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[4,4,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[4,3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 3 - 1
[4,3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 - 1
[7,6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[7,6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 3 - 1
[7,5,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3 - 1
[7,5,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[7,5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[7,4,4,2]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[6,6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 - 1
[6,4,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[6,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[5,4,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[5,4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 2 - 1
[5,3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 3 - 1
[4,3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 2 - 1
[6,5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 5 - 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: St000678
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 80%
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 3 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[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 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 2 = 3 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 3 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 2 = 3 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 3 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 4 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 4 - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 4 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 3 - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 4 - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 4 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 4 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[6,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
[6,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 4 - 1
[5,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 4 - 1
[8,6]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[8,1,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 - 1
[7,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[6,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[5,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[4,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[4,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
[3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
[3,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[2,2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
[8,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[7,7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[6,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[5,5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 - 1
[5,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 3 - 1
[5,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 - 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001050
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 70%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 3 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3 = 4 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 4 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2 = 3 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 2 = 3 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7}}
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2,3,4,6},{5}}
=> 2 = 3 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 2 = 3 - 1
[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}}
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6},{7}}
=> 2 = 3 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8}}
=> 1 = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,7},{6}}
=> 2 = 3 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> {{1},{2,3,6},{4,5}}
=> 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 5 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> {{1,2,4,5},{3},{6}}
=> 2 = 3 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> {{1,2,5},{3,4},{6}}
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,6},{5},{7}}
=> 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7},{8}}
=> 2 = 3 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9}}
=> 1 = 2 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8},{7}}
=> 2 = 3 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,6,7},{5}}
=> 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> {{1},{2,3,4,7},{5,6}}
=> 2 = 3 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> {{1},{2,4,5,6},{3}}
=> 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> {{1},{2,3,6},{4},{5}}
=> 3 = 4 - 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,6,8},{5},{7}}
=> ? = 3 - 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5,6}}
=> ? = 2 - 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,4,5,6,7},{3},{8}}
=> ? = 3 - 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4,5},{8}}
=> ? = 3 - 1
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,5,6,8},{4},{7}}
=> ? = 3 - 1
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5},{6}}
=> ? = 2 - 1
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5,6},{7}}
=> ? = 4 - 1
[7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> {{1},{2,8},{3,4,5,6,7}}
=> ? = 3 - 1
[6,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 3 - 1
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4},{5},{8}}
=> ? = 3 - 1
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,7,8},{5,6},{9}}
=> ? = 3 - 1
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> {{1,7},{2,3,4,5,6},{8}}
=> ? = 4 - 1
[7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,4,5,6,8},{3},{7}}
=> ? = 3 - 1
[7,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,5,7,8},{4},{6}}
=> ? = 2 - 1
[7,3,3]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> {{1},{2,3,6,7,8},{4,5}}
=> ? = 2 - 1
[7,3,1,1,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> {{1},{2,3,8},{4,6,7},{5}}
=> ? = 3 - 1
[7,2,2,2]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> {{1},{2,3,7,8},{4,5,6}}
=> ? = 2 - 1
[6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1,2},{3,4,5,6,8},{7}}
=> ? = 3 - 1
[6,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,3,4,5,7},{2},{6},{8}}
=> ? = 4 - 1
[5,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,4,6,7},{3},{5},{8}}
=> ? = 3 - 1
[3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,6},{5},{7,8}}
=> ? = 2 - 1
[2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 4 - 1
[8,6]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8,9},{3}}
=> ? = 2 - 1
[8,1,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> {{1},{2,9},{3,4,5,6,7,8}}
=> ? = 3 - 1
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8,9}}
=> ? = 2 - 1
[7,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,8},{7}}
=> ? = 3 - 1
[7,5,2]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2,4,5,7,8},{3},{6}}
=> ? = 2 - 1
[7,4,3]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> {{1},{2,3,6,7,8},{4},{5}}
=> ? = 2 - 1
[7,3,2,2]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> {{1},{2,3,7,8},{4,6},{5}}
=> ? = 2 - 1
[7,3,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> {{1},{2,8},{3,4,6,7},{5}}
=> ? = 3 - 1
[7,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7,8},{2},{9}}
=> ? = 3 - 1
[6,6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1,2},{3,4,5,7,8},{6}}
=> ? = 2 - 1
[6,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> {{1,2},{3,4,5,8},{6,7}}
=> ? = 3 - 1
[6,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> {{1,3,4,6,7},{2},{5},{8}}
=> ? = 3 - 1
[5,4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,5,6,7},{3},{4},{8}}
=> ? = 3 - 1
[4,3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0]
=> {{1,2,6,7},{3,5},{4},{8}}
=> ? = 3 - 1
[4,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,5,6},{4},{7,8}}
=> ? = 2 - 1
[4,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> {{1,7},{2,3,5,6},{4},{8}}
=> ? = 4 - 1
[3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,6},{4,5},{7,8}}
=> ? = 2 - 1
[3,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,6},{5},{7},{8}}
=> ? = 5 - 1
[2,2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> {{1,8},{2,3,4,5,6,7},{9}}
=> ? = 4 - 1
[8,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1},{2,3,4,5,6,7,8},{9}}
=> ? = 3 - 1
[7,7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,9},{8}}
=> ? = 3 - 1
[7,6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,7,8},{6}}
=> ? = 2 - 1
[7,5,3]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> {{1},{2,4,6,7,8},{3},{5}}
=> ? = 2 - 1
[7,5,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> {{1},{2,4,8},{3},{5,6,7}}
=> ? = 3 - 1
[7,4,4]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2,5,6,7,8},{3,4}}
=> ? = 2 - 1
[7,4,3,1]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,3,6,8},{4},{5},{7}}
=> ? = 3 - 1
[7,4,2,2]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> {{1},{2,3,7,8},{4},{5,6}}
=> ? = 2 - 1
[7,4,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> {{1},{2,8},{3,5,6,7},{4}}
=> ? = 3 - 1
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000971
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 3 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 4 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3 = 4 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2 = 3 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 2 = 3 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6,7}}
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,5,6},{2},{3},{4}}
=> 2 = 3 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2 = 3 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> {{1,6},{2,3},{4},{5}}
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7}}
=> 2 = 3 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,6,7},{2},{3},{4},{5}}
=> 2 = 3 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> {{1},{2,5,6},{3},{4}}
=> 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> {{1,4},{2},{3},{5,6}}
=> 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 4 = 5 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> {{1,2},{3,6},{4},{5}}
=> 2 = 3 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> {{1,5},{2,3},{4},{6}}
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,7},{2,3},{4},{5},{6}}
=> 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 3 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> ? = 2 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 3 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> {{1},{2,6,7},{3},{4},{5}}
=> 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> {{1,5},{2},{3},{4},{6,7}}
=> 2 = 3 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> {{1},{2},{3,5,6},{4}}
=> 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> {{1,2,5,6},{3},{4}}
=> 3 = 4 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> {{1,3},{2},{4},{5,6}}
=> 2 = 3 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6}}
=> 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2 = 3 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ? = 4 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [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}}
=> ? = 3 - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> ? = 2 - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? = 3 - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1},{2,7,8},{3},{4},{5},{6}}
=> ? = 2 - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> {{1,6},{2},{3},{4},{5},{7,8}}
=> ? = 3 - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,2},{3,8},{4},{5},{6},{7}}
=> ? = 3 - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,7},{2,3},{4},{5},{6},{8}}
=> ? = 4 - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2,3},{4},{5},{6},{7},{8}}
=> ? = 4 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 3 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> ? = 2 - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,9,10},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 3 - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1},{2,8,9},{3},{4},{5},{6},{7}}
=> ? = 2 - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> {{1,7},{2},{3},{4},{5},{6},{8,9}}
=> ? = 3 - 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> {{1},{2},{3,7,8},{4},{5},{6}}
=> ? = 2 - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> {{1,2,7,8},{3},{4},{5},{6}}
=> ? = 4 - 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> {{1,5},{2},{3},{4},{6},{7,8}}
=> ? = 3 - 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1,2},{3},{4,8},{5},{6},{7}}
=> ? = 3 - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,2},{3,9},{4},{5},{6},{7},{8}}
=> ? = 3 - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> {{1,6},{2,3},{4},{5},{7},{8}}
=> ? = 4 - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,8},{2,3},{4},{5},{6},{7},{9}}
=> ? = 4 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,10},{2,3},{4},{5},{6},{7},{8},{9}}
=> ? = 4 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> ? = 3 - 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> {{1},{2},{3},{4,7,8},{5},{6}}
=> ? = 2 - 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> {{1,3,7,8},{2},{4},{5},{6}}
=> ? = 3 - 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> {{1},{2,6},{3},{4},{5},{7,8}}
=> ? = 2 - 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1,2},{3},{4},{5,8},{6},{7}}
=> ? = 3 - 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,2},{3,7},{4},{5},{6},{8}}
=> ? = 3 - 1
[7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> {{1},{2},{3},{4},{5,7,8},{6}}
=> ? = 2 - 1
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> {{1,4,7,8},{2},{3},{5},{6}}
=> ? = 3 - 1
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> {{1},{2,3,7,8},{4},{5},{6}}
=> ? = 2 - 1
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> {{1,2,6},{3},{4},{5},{7,8}}
=> ? = 4 - 1
[7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,3},{2},{4},{5},{6},{7,8}}
=> ? = 3 - 1
[6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 2 - 1
[6,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1,2},{3},{4},{5},{6,8},{7}}
=> ? = 3 - 1
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> {{1,2},{3,4,8},{5},{6},{7}}
=> ? = 3 - 1
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> {{1,2},{3,8},{4},{5},{6},{7},{9}}
=> ? = 3 - 1
[2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 2 - 1
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,4},{2,3},{5},{6},{7},{8}}
=> ? = 4 - 1
[7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 2 - 1
[7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> {{1,5,7,8},{2},{3},{4},{6}}
=> ? = 3 - 1
[7,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> {{1},{2,4,7,8},{3},{5},{6}}
=> ? = 2 - 1
[7,3,3]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> {{1},{2},{3,6},{4},{5},{7,8}}
=> ? = 2 - 1
[7,3,1,1,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> {{1,7,8},{2},{3,5},{4},{6}}
=> ? = 3 - 1
[7,2,2,2]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> {{1},{2,5},{3},{4},{6},{7,8}}
=> ? = 2 - 1
[7,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 3 - 1
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: St000759
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1]
=> 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 2 = 3 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3 = 4 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3 = 4 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 2 = 3 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 2 = 3 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> 2 = 3 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2 = 3 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> 2 = 3 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> 1 = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1,1]
=> 2 = 3 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 4 = 5 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> 2 = 3 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,1,1]
=> 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1]
=> 2 = 3 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> 1 = 2 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1]
=> 2 = 3 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,2,2]
=> 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1]
=> 2 = 3 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> 3 = 4 - 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [7,3,3]
=> ? = 2 - 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1]
=> ? = 3 - 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [7,4,4]
=> ? = 2 - 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,3,1]
=> ? = 3 - 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2]
=> ? = 2 - 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1]
=> ? = 3 - 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2]
=> ? = 2 - 1
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1]
=> ? = 3 - 1
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1]
=> ? = 4 - 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1,1]
=> ? = 4 - 1
[7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [7,5,5]
=> ? = 2 - 1
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> [7,4,4,1]
=> ? = 3 - 1
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,3,2]
=> ? = 2 - 1
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,1]
=> ? = 4 - 1
[7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1,1]
=> ? = 3 - 1
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1]
=> ? = 3 - 1
[6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2]
=> ? = 2 - 1
[6,4,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1]
=> ? = 3 - 1
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1]
=> ? = 5 - 1
[6,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> ? = 3 - 1
[6,2,2,1,1]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1,1]
=> ? = 4 - 1
[7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> ? = 2 - 1
[7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,5,1]
=> ? = 3 - 1
[7,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2]
=> ? = 2 - 1
[7,3,3]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3]
=> ? = 2 - 1
[7,3,1,1,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,3,1,1,1]
=> ? = 3 - 1
[7,2,2,2]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2]
=> ? = 2 - 1
[6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2]
=> ? = 2 - 1
[6,5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1]
=> ? = 3 - 1
[6,4,2,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1]
=> ? = 4 - 1
[6,3,3,1]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1]
=> ? = 3 - 1
[6,3,2,2]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2]
=> ? = 2 - 1
[6,3,2,1,1]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1,1]
=> ? = 5 - 1
[5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3]
=> ? = 2 - 1
[5,4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [5,4,4,1]
=> ? = 3 - 1
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1,1]
=> ? = 6 - 1
[8,6]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [8,6,6]
=> ? = 2 - 1
[8,1,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,1,1,1,1,1,1]
=> ? = 3 - 1
[7,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,1]
=> ? = 3 - 1
[7,5,2]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> [7,5,5,2]
=> ? = 2 - 1
[7,4,3]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3]
=> ? = 2 - 1
[7,3,2,2]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,3,2,2]
=> ? = 2 - 1
[7,3,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,3,1,1,1,1]
=> ? = 3 - 1
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1]
=> ? = 4 - 1
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ? = 3 - 1
[6,4,2,1,1]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 4 - 1
[6,3,2,2,1]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> ? = 5 - 1
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,1]
=> ? = 5 - 1
[6,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [6,3,3,1]
=> ? = 3 - 1
[5,5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1]
=> ? = 3 - 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: St000025
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 70%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 5 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 3 = 4 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4 - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 4 - 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 2 - 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 - 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 2 - 1
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[6,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2 - 1
[7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3 - 1
[7,4,2]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2 - 1
[7,3,3]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 - 1
[7,3,1,1,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[7,2,2,2]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[7,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000069The number of maximal elements of a poset. St000654The first descent of a permutation. St000068The number of minimal elements in a poset. St000054The first entry of the permutation. St000989The number of final rises of a permutation. St000007The number of saliances of the permutation. St000740The last entry of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000990The first ascent of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. 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. St000061The number of nodes on the left branch of a binary tree. St000314The number of left-to-right-maxima of a permutation. 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. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation.
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