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Your data matches 60 different statistics following compositions of up to 3 maps.
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Matching statistic: St001392
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(load all 3 compositions to match this statistic)
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0 = 1 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 2 = 3 - 1
[2,1]
=> 0 = 1 - 1
[1,1,1]
=> 0 = 1 - 1
[4]
=> 3 = 4 - 1
[3,1]
=> 2 = 3 - 1
[2,2]
=> 1 = 2 - 1
[2,1,1]
=> 0 = 1 - 1
[1,1,1,1]
=> 0 = 1 - 1
[5]
=> 4 = 5 - 1
[4,1]
=> 3 = 4 - 1
[3,2]
=> 1 = 2 - 1
[3,1,1]
=> 2 = 3 - 1
[2,2,1]
=> 0 = 1 - 1
[2,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,1]
=> 0 = 1 - 1
[6]
=> 5 = 6 - 1
[5,1]
=> 4 = 5 - 1
[4,2]
=> 3 = 4 - 1
[4,1,1]
=> 3 = 4 - 1
[3,3]
=> 2 = 3 - 1
[3,2,1]
=> 0 = 1 - 1
[3,1,1,1]
=> 2 = 3 - 1
[2,2,2]
=> 1 = 2 - 1
[2,2,1,1]
=> 0 = 1 - 1
[2,1,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> 0 = 1 - 1
[7]
=> 6 = 7 - 1
[6,1]
=> 5 = 6 - 1
[5,2]
=> 4 = 5 - 1
[5,1,1]
=> 4 = 5 - 1
[4,3]
=> 2 = 3 - 1
[4,2,1]
=> 3 = 4 - 1
[4,1,1,1]
=> 3 = 4 - 1
[3,3,1]
=> 2 = 3 - 1
[3,2,2]
=> 1 = 2 - 1
[3,2,1,1]
=> 0 = 1 - 1
[3,1,1,1,1]
=> 2 = 3 - 1
[2,2,2,1]
=> 0 = 1 - 1
[2,2,1,1,1]
=> 0 = 1 - 1
[2,1,1,1,1,1]
=> 0 = 1 - 1
[1,1,1,1,1,1,1]
=> 0 = 1 - 1
[8]
=> 7 = 8 - 1
[7,1]
=> 6 = 7 - 1
[6,2]
=> 5 = 6 - 1
[6,1,1]
=> 5 = 6 - 1
[5,3]
=> 4 = 5 - 1
[5,2,1]
=> 4 = 5 - 1
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000733
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [[1]]
=> [[1]]
=> 1
[2]
=> [[1,2]]
=> [[1,2]]
=> [[1],[2]]
=> 2
[1,1]
=> [[1],[2]]
=> [[1],[2]]
=> [[1,2]]
=> 1
[3]
=> [[1,2,3]]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[4]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[3,1]
=> [[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 3
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 2
[2,1,1]
=> [[1,4],[2],[3]]
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[5]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5
[4,1]
=> [[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 4
[3,2]
=> [[1,2,5],[3,4]]
=> [[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 2
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 5
[4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3,4,6],[2,5]]
=> [[1,2],[3,5],[4],[6]]
=> 4
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,3,5,6],[2],[4]]
=> [[1,2,4],[3],[5],[6]]
=> 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 3
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> [[1,3,6],[2,5],[4]]
=> 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,4,6],[2],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,4],[2,5],[3,6]]
=> [[1,2,3],[4,5,6]]
=> 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [[1,2,4,5,6,7],[3]]
=> [[1,3],[2],[4],[5],[6],[7]]
=> 6
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3,4,6,7],[2,5]]
=> [[1,2],[3,5],[4],[6],[7]]
=> 5
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [[1,3,5,6,7],[2],[4]]
=> [[1,2,4],[3],[5],[6],[7]]
=> 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[1,3,5,6],[2,4,7]]
=> [[1,2],[3,4],[5,7],[6]]
=> 3
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[1,2,4,7],[3,5],[6]]
=> [[1,3,6],[2,5],[4],[7]]
=> 4
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[1,4,6,7],[2],[3],[5]]
=> [[1,2,3,5],[4],[6],[7]]
=> 4
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6,7],[4]]
=> [[1,3,4],[2,6],[5,7]]
=> 3
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,4,6],[2,5],[3,7]]
=> [[1,2,3],[4,5,7],[6]]
=> 2
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,3,5],[2,6],[4],[7]]
=> [[1,2,4,7],[3,6],[5]]
=> 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,5,7],[2],[3],[4],[6]]
=> [[1,2,3,4,6],[5],[7]]
=> 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[1,4],[2,6],[3],[5],[7]]
=> [[1,2,3,5,7],[4,6]]
=> 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,6],[2],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [[1,2,4,5,6,7,8],[3]]
=> [[1,3],[2],[4],[5],[6],[7],[8]]
=> 7
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3,4,6,7,8],[2,5]]
=> [[1,2],[3,5],[4],[6],[7],[8]]
=> 6
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [[1,3,5,6,7,8],[2],[4]]
=> [[1,2,4],[3],[5],[6],[7],[8]]
=> 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [[1,3,5,6,8],[2,4,7]]
=> [[1,2],[3,4],[5,7],[6],[8]]
=> 5
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [[1,2,4,7,8],[3,5],[6]]
=> [[1,3,6],[2,5],[4],[7],[8]]
=> 5
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001265
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001265: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 79%●distinct values known / distinct values provided: 62%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001265: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 79%●distinct values known / distinct values provided: 62%
Values
[1]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0 = 1 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {1,6} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,6} - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,4,6,7} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,4,6,7} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,4,6,7} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,4,6,7} - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,3,4,5,6,6,7,8} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,3,4,5,6,6,7,8} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,3,4,5,6,6,7,8} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {1,3,4,5,6,6,7,8} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,3,4,5,6,6,7,8} - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? ∊ {1,3,4,5,6,6,7,8} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {1,3,4,5,6,6,7,8} - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,3,4,5,6,6,7,8} - 1
Description
The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St001232
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 75%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,2}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,1,2}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,2,7}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,2,7}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,2,7}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,2,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,2,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 3
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,2,4,7,8}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000462
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000462: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 75%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000462: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 0 = 1 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => 3 = 4 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 0 = 1 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => 0 = 1 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => 4 = 5 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => 3 = 4 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1 = 2 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => 0 = 1 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => ? = 1 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,8,6,7] => ? ∊ {1,1,4,6} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,1,2,3,7,4,5] => ? ∊ {1,1,4,6} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => 4 = 5 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => 3 = 4 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => 2 = 3 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => 2 = 3 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => 1 = 2 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => 0 = 1 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? ∊ {1,1,4,6} - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8] => ? ∊ {1,1,4,6} - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,6,9,7,8] => ? ∊ {1,1,1,3,5,5,6,7} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,8,5,6] => ? ∊ {1,1,1,3,5,5,6,7} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,7,4,5] => ? ∊ {1,1,1,3,5,5,6,7} - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,7,4,6] => ? ∊ {1,1,1,3,5,5,6,7} - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => 2 = 3 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => 3 = 4 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => 2 = 3 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => 1 = 2 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => 0 = 1 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,7,2,4,5,6] => ? ∊ {1,1,1,3,5,5,6,7} - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => 0 = 1 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5,7] => ? ∊ {1,1,1,3,5,5,6,7} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,8,1,2,4,5,6,7] => ? ∊ {1,1,1,3,5,5,6,7} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8,9] => ? ∊ {1,1,1,3,5,5,6,7} - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,6,7,10,8,9] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,9,6,7] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,7,2,3,4,8,5,6] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,8,5,7] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,2,6,3,7,4,5] => 6 = 7 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [4,1,2,3,7,5,6] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,2,3,6,4,5,7] => 3 = 4 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => 2 = 3 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => 4 = 5 - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => 3 = 4 - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,2,7,4,5,6] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => 1 = 2 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => 2 = 3 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => 0 = 1 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,8,2,4,5,6,7] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,2,3,5,6,7] => 1 = 2 - 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,5,1,2,4,6,7] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6,8] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,9,1,2,4,5,6,7,8] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,3,4,4,5,5,6,6,8} - 1
Description
The major index minus the number of excedences of a permutation.
This occurs in the context of Eulerian polynomials [1].
Matching statistic: St000054
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 58%●distinct values known / distinct values provided: 50%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 58%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 4 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 4 = 3 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? ∊ {3,5} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 5 = 4 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? ∊ {3,5} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,2] => ? ∊ {1,3,5,6} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ? ∊ {1,3,5,6} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 5 = 4 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 5 = 4 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 4 = 3 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 3 = 2 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? ∊ {1,3,5,6} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,8,4,5,6,7,1,3] => ? ∊ {1,3,5,6} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? ∊ {1,1,3,4,5,5,6,7} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,6,4,5,1,2,8,3] => ? ∊ {1,1,3,4,5,5,6,7} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ? ∊ {1,1,3,4,5,5,6,7} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? ∊ {1,1,3,4,5,5,6,7} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 4 = 3 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 5 = 4 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 4 = 3 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 3 = 2 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? ∊ {1,1,3,4,5,5,6,7} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? ∊ {1,1,3,4,5,5,6,7} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,8,7,5,6,1,3,4] => ? ∊ {1,1,3,4,5,5,6,7} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,9,4,5,6,7,8,1,3] => ? ∊ {1,1,3,4,5,5,6,7} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,10,2] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [8,7,4,5,6,1,2,9,3] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [7,3,6,5,1,2,8,4] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,5,4,1,6,2,8,3] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => 5 = 4 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 4 = 3 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 3 = 2 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 3 = 2 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => 2 = 1 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [2,8,4,7,6,1,3,5] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [2,8,6,5,1,7,3,4] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [2,9,8,5,6,7,1,3,4] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [2,10,4,5,6,7,8,9,1,3] => ? ∊ {1,1,3,3,4,4,4,5,5,5,6,6,7,8} + 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000060
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000060: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000060: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,4,6] => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => ? ∊ {3,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,1,6] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? ∊ {3,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,6,8] => ? ∊ {1,3,5,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? ∊ {1,3,5,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [5,3,1,2,4,6] => 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,3,1,4,6] => 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,1,6] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,1,6] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => ? ∊ {1,3,5,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => ? ∊ {1,3,5,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,7,9] => ? ∊ {1,1,3,4,5,5,6,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [7,2,1,3,4,5,6,8] => ? ∊ {1,1,3,4,5,5,6,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,2,4,5,7] => ? ∊ {1,1,3,4,5,5,6,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [6,2,3,1,4,5,7] => ? ∊ {1,1,3,4,5,5,6,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,1,6] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [4,3,5,1,2,6] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,1,6] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => ? ∊ {1,1,3,4,5,5,6,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => ? ∊ {1,1,3,4,5,5,6,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,1,8] => ? ∊ {1,1,3,4,5,5,6,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1,9] => ? ∊ {1,1,3,4,5,5,6,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [9,1,2,3,4,5,6,7,8,10] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> [8,2,1,3,4,5,6,7,9] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [7,3,1,2,4,5,6,8] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [7,2,3,1,4,5,6,8] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,1,2,3,5,7] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,2,1,4,5,7] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,1,5,7] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,6,1,2,3,4,7] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [5,3,4,1,2,6] => 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,1,6] => 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,1,7] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,1,6] => 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,1,8] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,1,2,7] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,1,8] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1,9] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1,10] => ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
Description
The greater neighbor of the maximum.
Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the smallest path leaf label of the binary tree associated to a permutation ([[St000724]]), see also [3].
Matching statistic: St000154
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 62%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 62%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? ∊ {1,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,1,2,6,3,4] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => ? ∊ {1,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,8,6,7] => ? ∊ {1,1,4,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,1,2,3,7,4,5] => ? ∊ {1,1,4,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,6,3,4] => 5
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,7,1,2,4,5,6] => ? ∊ {1,1,4,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8] => ? ∊ {1,1,4,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,6,9,7,8] => ? ∊ {1,1,1,4,4,5,6,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,8,5,6] => ? ∊ {1,1,1,4,4,5,6,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,7,4,5] => ? ∊ {1,1,1,4,4,5,6,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,7,4,6] => ? ∊ {1,1,1,4,4,5,6,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => 5
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,7,2,4,5,6] => ? ∊ {1,1,1,4,4,5,6,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5,7] => ? ∊ {1,1,1,4,4,5,6,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,8,1,2,4,5,6,7] => ? ∊ {1,1,1,4,4,5,6,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8,9] => ? ∊ {1,1,1,4,4,5,6,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,6,7,10,8,9] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,9,6,7] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,7,2,3,4,8,5,6] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,8,5,7] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,2,6,3,7,4,5] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [4,1,2,3,7,5,6] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,2,3,6,4,5,7] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => 5
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,2,7,4,5,6] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,8,2,4,5,6,7] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,2,3,5,6,7] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,5,1,2,4,6,7] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6,8] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,9,1,2,4,5,6,7,8] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8,9,10] => ? ∊ {1,1,1,1,1,2,4,4,4,4,5,5,6,6,7,8}
Description
The sum of the descent bottoms of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$
For the descent tops, see [[St000111]].
Matching statistic: St000199
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [[0,1],[1,0]]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {3,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {3,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? ∊ {1,3,5,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,3,5,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,3,5,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {1,3,5,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 3
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => [[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => [[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
Description
The column of the unique '1' in the last row of the alternating sign matrix.
Matching statistic: St000200
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000200: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000200: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 55%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [[0,1],[1,0]]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {3,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {3,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? ∊ {1,3,5,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,3,5,6}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,3,5,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {1,3,5,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? ∊ {1,1,3,4,5,5,6,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => [[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => [[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {1,1,1,3,3,4,4,4,4,5,5,5,6,6,7,8}
Description
The row of the unique '1' in the last column of the alternating sign matrix.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001438The number of missing boxes of a skew partition. St001557The number of inversions of the second entry of a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001463The number of distinct columns in the nullspace of a graph. St000456The monochromatic index of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000454The largest eigenvalue of a graph if it is integral. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St000259The diameter of a connected graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000654The first descent of a permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000406The number of occurrences of the pattern 3241 in a permutation. St000650The number of 3-rises of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001556The number of inversions of the third entry of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000383The last part of an integer composition. St001330The hat guessing number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001060The distinguishing index of a graph. St001645The pebbling number of a connected graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000352The Elizalde-Pak rank of a permutation. 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 $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000007The number of saliances of the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing.
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