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Your data matches 74 different statistics following compositions of up to 3 maps.
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Matching statistic: St000157
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => [[1],[2]]
=> 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [[1,3,5],[2,4,6]]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [[1,4],[2,5],[3],[6]]
=> 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [[1,3],[2,4],[5],[6]]
=> 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [[1,4],[2,5],[3],[6]]
=> 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [[1,3,5],[2,4,6]]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [[1,5],[2,6],[3],[4]]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 7
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [[1,3,6],[2,4,7],[5],[8]]
=> 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [[1,4,7],[2,5,8],[3],[6]]
=> 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [[1,5],[2,6],[3],[4],[7],[8]]
=> 6
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [[1,3],[2,4],[5],[6],[7],[8]]
=> 6
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [[1,5],[2,6],[3],[4],[7],[8]]
=> 6
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [[1,5],[2,6],[3],[4],[7],[8]]
=> 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [[1,3,6],[2,4,7],[5],[8]]
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [[1,4,7],[2,5,8],[3],[6]]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [[1,7],[2,8],[3],[4],[5],[6]]
=> 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [[1,4,7],[2,5,8],[3],[6]]
=> 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [[1,3,6],[2,4,7],[5],[8]]
=> 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [[1,5],[2,6],[3,7],[4,8]]
=> 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 9
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
=> 8
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [[1,9],[2,10],[3],[4],[5],[6],[7],[8]]
=> 8
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [[1,5],[2,6],[3,7],[4,8],[9],[10]]
=> 8
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [[1,7],[2,8],[3,9],[4,10],[5],[6]]
=> 8
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000245
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => [1,2] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [1,2,3,4] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [1,4,2,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,4,2,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,2,5,3,4,6] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [1,6,2,3,4,5] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,2,5,3,4,6] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,4,2,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [1,6,2,3,4,5] => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 7
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 6
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [1,8,2,3,4,5,6,7] => 6
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 6
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [1,8,2,3,4,5,6,7] => 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [1,8,2,3,4,5,6,7] => 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 9
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => 8
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => 8
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => 8
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => 8
Description
The number of ascents of a permutation.
Matching statistic: St000393
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1 => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 111 => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 101 => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 11111 => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 10101 => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 11011 => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 10111 => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 11011 => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 10101 => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 11101 => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => 1111111 => 7
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 1011011 => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 1101101 => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => 1110111 => 6
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => 1011111 => 6
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => 1110111 => 6
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => 1110111 => 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 1011011 => 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 1101101 => 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => 1111101 => 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 1101101 => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 1011011 => 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => 1110111 => 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => 111111111 => 9
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => 101111111 => 8
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => 111111101 => 8
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => 111011111 => 8
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => 111110111 => 8
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000507
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [1,0]
=> [[1],[2]]
=> 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 7
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 6
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 6
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 6
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> 9
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 8
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> 8
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> 8
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> 8
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000672
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => [1,2] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [1,2,3,4] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [1,4,2,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,4,2,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,2,5,3,4,6] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [1,6,2,3,4,5] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,2,5,3,4,6] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,4,2,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [1,6,2,3,4,5] => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 7
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 6
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [1,8,2,3,4,5,6,7] => 6
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 6
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [1,8,2,3,4,5,6,7] => 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [1,8,2,3,4,5,6,7] => 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 9
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => 8
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => 8
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => 8
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => 8
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St001176
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => [1,1]
=> 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [1,1,1,1]
=> 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,2]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1]
=> 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,3]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [2,2,1,1]
=> 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,2,1,1]
=> 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [2,2,1,1]
=> 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,3]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [2,2,1,1]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1]
=> 7
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,3,1,1]
=> 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,3,1,1]
=> 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [2,2,1,1,1,1]
=> 6
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [2,2,1,1,1,1]
=> 6
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [2,2,1,1,1,1]
=> 6
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [2,2,1,1,1,1]
=> 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,3,1,1]
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,3,1,1]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [2,2,1,1,1,1]
=> 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,3,1,1]
=> 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,3,1,1]
=> 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [2,2,2,2]
=> 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1]
=> 9
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [2,2,1,1,1,1,1,1]
=> 8
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [2,2,1,1,1,1,1,1]
=> 8
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [2,2,2,2,1,1]
=> 8
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [2,2,2,2,1,1]
=> 8
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001087
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => [1,2] => 0 = 1 - 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [1,2,3,4] => 2 = 3 - 1
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [1,4,2,3] => 1 = 2 - 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 4 = 5 - 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,4,2,3,6,5] => 2 = 3 - 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,2,5,3,4,6] => 3 = 4 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [1,6,2,3,4,5] => 3 = 4 - 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,2,5,3,4,6] => 3 = 4 - 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,4,2,3,6,5] => 2 = 3 - 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [1,6,2,3,4,5] => 3 = 4 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 6 = 7 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 4 = 5 - 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 5 = 6 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [1,8,2,3,4,5,6,7] => 5 = 6 - 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 5 = 6 - 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,3,6,4,5,7,8] => 5 = 6 - 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 4 = 5 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 4 = 5 - 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [1,8,2,3,4,5,6,7] => 5 = 6 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,8,2,5,3,4,6,7] => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,2,3,4,7,5,6] => 4 = 5 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [1,8,2,3,4,5,6,7] => 5 = 6 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 8 = 9 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [1,10,2,3,4,5,6,7,8,9] => 7 = 8 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [1,10,2,3,4,5,6,7,8,9] => 7 = 8 - 1
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [1,10,2,3,4,5,6,7,8,9] => 7 = 8 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [1,10,2,3,4,5,6,7,8,9] => 7 = 8 - 1
Description
The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern $123$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => 1 => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => 111 => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 101 => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 11111 => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 10101 => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 11011 => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 10111 => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 11011 => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 10101 => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 11101 => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => 1111111 => 7
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 1011011 => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 1101101 => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => 1110111 => 6
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => 1011111 => 6
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => 1110111 => 6
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => 1110111 => 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 1011011 => 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 1101101 => 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => 1111101 => 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 1101101 => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 1011011 => 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => 1110111 => 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => 111111111 => 9
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => 101111111 => ? = 8
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => 111111101 => 8
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => 111011111 => ? = 8
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => 111110111 => ? = 8
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000662
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 89%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 89%
Values
[[1]]
=> [(1,2)]
=> [2,1] => [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [4,3,2,1] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [3,2,4,1] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [4,3,5,2,6,1] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [6,4,2,5,3,1] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [5,4,6,3,2,1] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [6,4,2,5,3,1] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [4,3,5,2,6,1] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,4,3,2,6,1] => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => 7
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [6,5,8,4,2,7,3,1] => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,5,3,6,4,2,8,1] => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [8,7,5,2,6,4,3,1] => 6
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [7,6,8,5,4,3,2,1] => 6
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [8,7,5,2,6,4,3,1] => 6
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [8,7,5,2,6,4,3,1] => 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [6,5,8,4,2,7,3,1] => 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,5,3,6,4,2,8,1] => 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,8,1] => 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,5,3,6,4,2,8,1] => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [6,5,8,4,2,7,3,1] => 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [7,6,5,4,8,3,2,1] => 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 9
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [9,8,10,7,6,5,4,3,2,1] => ? = 8
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [9,8,7,6,5,4,3,2,10,1] => ? = 8
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [9,8,7,6,10,5,4,3,2,1] => ? = 8
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [9,8,7,6,5,4,10,3,2,1] => ? = 8
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000703
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 89%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 89%
Values
[[1]]
=> [(1,2)]
=> [2,1] => [2,1] => 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [4,1,2,3] => 3
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [6,1,2,3,4,5] => 5
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [6,5,2,1,4,3] => 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,6,3,4,5] => 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [6,5,2,1,4,3] => 4
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [4,1,2,3,6,5] => 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [8,1,2,3,4,5,6,7] => 7
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,8,7,4,3,6,5] => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [6,5,2,1,4,3,8,7] => 5
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [8,1,6,3,2,5,4,7] => 6
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [2,1,8,3,4,5,6,7] => 6
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [8,1,6,3,2,5,4,7] => 6
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [8,1,6,3,2,5,4,7] => 6
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,8,7,4,3,6,5] => 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [6,5,2,1,4,3,8,7] => 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [6,1,2,3,4,5,8,7] => 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [6,5,2,1,4,3,8,7] => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,8,7,4,3,6,5] => 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [4,1,2,3,8,5,6,7] => 6
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [10,1,2,3,4,5,6,7,8,9] => 9
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [2,1,10,3,4,5,6,7,8,9] => ? = 8
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [8,1,2,3,4,5,6,7,10,9] => ? = 8
[[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [4,1,2,3,10,5,6,7,8,9] => ? = 8
[[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [6,1,2,3,4,5,10,7,8,9] => ? = 8
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000632The jump number of the poset. St001427The number of descents of a signed permutation. St001554The number of distinct nonempty subtrees of a binary tree. St000021The number of descents of a permutation. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000362The size of a minimal vertex cover of a graph. St000482The (zero)-forcing number of a graph. St000495The number of inversions of distance at most 2 of a permutation. St000619The number of cyclic descents of a permutation. St000702The number of weak deficiencies of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000991The number of right-to-left minima of a permutation. St001388The number of non-attacking neighbors of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001960The number of descents of a permutation minus one if its first entry is not one. St001812The biclique partition number of a graph. St001684The reduced word complexity of a permutation. St000077The number of boxed and circled entries. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001769The reflection length of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001209The pmaj statistic of a parking function. St001557The number of inversions of the second entry of a permutation. St001645The pebbling number of a connected graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001894The depth of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
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