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Your data matches 115 different statistics following compositions of up to 3 maps.
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Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [[1,2]]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,6],[3,4],[5]]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,5],[3],[4]]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [[1,2]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,2,8],[3,4],[5,6],[7]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3],[2,5],[4,7],[6]]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,2,5],[3,4,8],[6],[7]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,2,3,8],[4,5],[6],[7]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,7],[3,4],[5],[6]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3],[2,6],[4],[5]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,2,3,7],[4],[5],[6]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,6],[3],[4],[5]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,2,5],[3,4,8],[6,7]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,2,3,8],[4,5],[6,7]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,7],[3,4],[5,6]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,2,3,8],[4,5,6],[7]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,2,5],[3,4,7],[6]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,2,3,7],[4,5],[6]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,6],[3,4],[5]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,6],[4],[5]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,5],[3],[4]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,2,3,7],[4,5,6]]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,2,5],[3,4,6]]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,2,3,6],[4,5]]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,2,3,5],[4]]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000382
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> 10 => [1,1] => 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => [2,1] => 2
[1,1,0,0,1,0]
=> [2]
=> 100 => [1,2] => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => [1,1] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => [2,1,1,1] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => [3,1] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => [1,1,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => [1,2,1,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => [2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => [1,3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => [1,2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => [1,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => [1,1,2,1,1,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => [3,1,1,1] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => [2,2,2,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => [1,2,1,1,2,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => [1,1,1,1,2,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => [2,1,2,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => [1,3,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => [4,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => [2,1,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => [1,2,2,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => [1,1,2,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => [3,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => [1,1,1,2,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => [2,2,1,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => [1,2,1,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => [2,1,1,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => [3,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => [1,1,1,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => [2,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => [1,2,1,2] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => [1,1,1,2] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => [2,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => [1,3,1,1] => 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => [1,2,1,1] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => [1,1,1,1] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 110 => [2,1] => 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> 11000000000 => [2,9] => ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> 110000000000 => [2,10] => ? = 2
Description
The first part of an integer composition.
Matching statistic: St000297
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,0]
=> [2]
=> 100 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 110 => 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> 1100000000 => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> 11000000000 => ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> 110000000000 => ? = 2
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> 10 => 01 => 2 = 1 + 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 001 => 3 = 2 + 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 011 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 01 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 010101 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 00101 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 01001 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 0001 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 01011 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0011 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 01101 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 001 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 0111 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 011 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 01 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1011010 => 0100101 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 000101 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 0011001 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 01101001 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1010110 => 0101001 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 110110 => 001001 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 01110001 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1001110 => 0110001 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 010001 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 00001 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 001011 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 0110011 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 101100 => 010011 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 00011 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 0101101 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 110010 => 001101 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 0110101 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101010 => 010101 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 00101 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 0111001 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 01001 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 0001 => 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 010111 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => 00111 => 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 011011 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 01011 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => 0011 => 3 = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 011101 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => 01101 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 110 => 001 => 3 = 2 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> 1100000000 => 0011111111 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> 11000000000 => 00111111111 => ? = 2 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> 110000000000 => 001111111111 => ? = 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: St000383
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => ? = 6
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,2] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,2] => ? = 2
[1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,2] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,2] => ? = 2
[1,1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => ? = 6
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 2
Description
The last part of an integer composition.
Matching statistic: St000993
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
St000993: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1]
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,7]
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [7,7]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> ? = 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 3
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 3
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 3
[1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> [8,9,10,11,12,13,14,1,2,3,4,5,6,7] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]
=> [9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8] => ? = 2
[1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0,0]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16,17,18]]
=> [10,11,12,13,14,15,16,17,18,1,2,3,4,5,6,7,8,9] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> [8,9,10,11,12,13,14,1,2,3,4,5,6,7] => ? = 2
[1,1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17,18,19,20]]
=> [11,12,13,14,15,16,17,18,19,20,1,2,3,4,5,6,7,8,9,10] => ? = 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000883
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 3
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 3
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 3
[1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> [8,9,10,11,12,13,14,1,2,3,4,5,6,7] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]
=> [9,10,11,12,13,14,15,16,1,2,3,4,5,6,7,8] => ? = 2
[1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16,17,18]]
=> [10,11,12,13,14,15,16,17,18,1,2,3,4,5,6,7,8,9] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> [8,9,10,11,12,13,14,1,2,3,4,5,6,7] => ? = 2
[1,1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 6
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17,18,19,20]]
=> [11,12,13,14,15,16,17,18,19,20,1,2,3,4,5,6,7,8,9,10] => ? = 2
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St000617
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 92%●distinct values known / distinct values provided: 60%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 92%●distinct values known / distinct values provided: 60%
Values
[1,0,1,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0,0]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0,0]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
Description
The number of global maxima of a Dyck path.
Matching statistic: St000765
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000765: Integer compositions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000765: Integer compositions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 2
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [2] => 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 2
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [2] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [1,1] => 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => ? = 3
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [4,4,3] => ? = 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 5
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => ? = 3
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [4,4,3] => ? = 2
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [5,5] => ? = 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [2,2,2,2,2,2] => ? = 6
[1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 5
[1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [3,3,3,2] => ? = 3
[1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [4,4,3] => ? = 2
[1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [6,6] => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [5,5] => ? = 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [7,3] => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [9,1] => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [3,1,1,1,1,1,1,1] => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [2,1,1,1,1,1,1,1,1] => ? = 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]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => ? = 10
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> [7,7] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,8]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]
=> [8,8] => ? = 2
[1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [2,2,2,2,2,2] => ? = 6
[1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => ? = 5
[1,1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0,0]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [4,4,3] => ? = 2
[1,1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0,0]
=> [4,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10]]
=> [4,3,3] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [6,6] => ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [5,5] => ? = 2
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,0]
=> [7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [7,3] => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0]
=> [9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [9,1] => ? = 1
[1,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [2,1,1,1,1,1,1,1,1] => ? = 1
[1,1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [3,1,1,1,1,1,1,1] => ? = 1
[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => ? = 10
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [9,9]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16,17,18]]
=> [9,9] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0,0]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [6,6] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> [7,7] => ? = 2
[1,1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [2,2,2,2,2,2] => ? = 6
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,10]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17,18,19,20]]
=> [10,10] => ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0,0]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [5,5] => ? = 2
Description
The number of weak records in an integer composition.
A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$.
The following 105 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000439The position of the first down step of a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000654The first descent of a permutation. St000392The length of the longest run of ones in a binary word. St000288The number of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000678The number of up steps after the last double rise of a Dyck path. St000974The length of the trunk of an ordered tree. St000674The number of hills of a Dyck path. St000010The length of the partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000475The number of parts equal to 1 in a partition. St001176The size of a partition minus its first part. St000759The smallest missing part in an integer partition. St000546The number of global descents of a permutation. St001050The number of terminal closers of a set partition. St000054The first entry of the permutation. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001829The common independence number of a graph. St000918The 2-limited packing number of a graph. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000363The number of minimal vertex covers of a graph. St000729The minimal arc length of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000312The number of leaves in a graph. St000234The number of global ascents of a permutation. St001316The domatic number of a graph. St000989The number of final rises of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000022The number of fixed points of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000717The number of ordinal summands of a poset. St000990The first ascent 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. St000504The cardinality of the first block of a set partition. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000056The decomposition (or block) number of a permutation. St000287The number of connected components of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000221The number of strong fixed points of a permutation. St000315The number of isolated vertices of a graph. St000286The number of connected components of the complement of a graph. St000335The difference of lower and upper interactions. St000553The number of blocks of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001672The restrained domination number of a graph. St000090The variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000338The number of pixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001479The number of bridges of a graph. St001826The maximal number of leaves on a vertex of a graph. St000756The sum of the positions of the left to right maxima of a permutation. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000461The rix statistic of a permutation. St001330The hat guessing number of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001545The second Elser number of a connected graph. St000237The number of small exceedances. St000241The number of cyclical small excedances. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000264The girth of a graph, which is not a tree. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001846The number of elements which do not have a complement in the lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000711The number of big exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001498The normalised height of a Nakayama algebra with magnitude 1.
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