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Your data matches 24 different statistics following compositions of up to 3 maps.
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Matching statistic: St000250
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(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000250: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000250: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 5
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 4
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 4
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6}}
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6}}
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6}}
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4,6},{5}}
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5,6}}
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6}}
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> {{1},{2},{3,4},{5,6}}
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> {{1},{2},{3,5},{4},{6}}
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3,6},{4},{5}}
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> {{1},{2},{3,5,6},{4}}
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2},{3,4,5},{6}}
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> {{1},{2},{3,6},{4,5}}
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3,4,6},{5}}
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4,5,6}}
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6}}
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5,6}}
=> 6
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2,3},{4,5},{6}}
=> 6
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> {{1},{2,3},{4,6},{5}}
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5,6}}
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> {{1},{2,4},{3},{5},{6}}
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> {{1},{2,4},{3},{5,6}}
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> {{1},{2,5},{3},{4},{6}}
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2,6},{3},{4},{5}}
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> {{1},{2,5,6},{3},{4}}
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> {{1},{2,4,5},{3},{6}}
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> {{1},{2,6},{3},{4,5}}
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> {{1},{2,4,6},{3},{5}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> {{1},{2,4,5,6},{3}}
=> 5
Description
The number of blocks ([[St000105]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000507
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [1,2] => [[1,2]]
=> 2 = 3 - 1
[1,1,0,0]
=> [1,2] => [1,2] => [[1,2]]
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [[1,2,3]]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [[1,2,3]]
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [[1,2],[3]]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [[1,2,3],[4]]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [[1,2,3],[4]]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [[1,2,4],[3]]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [[1,2,3],[4],[5]]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 4 = 5 - 1
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: St000245
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 2 - 2
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1 = 3 - 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [1,3,2] => 1 = 3 - 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 5 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 3 = 5 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 3 = 5 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 3 = 5 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 3 = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [1,3,5,4,2] => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 3 = 5 - 2
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)
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [.,.]
=> [1] => => ? = 2 - 2
[1,0,1,0]
=> [[.,.],.]
=> [1,2] => 0 => 1 = 3 - 2
[1,1,0,0]
=> [.,[.,.]]
=> [2,1] => 1 => 1 = 3 - 2
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> [1,2,3] => 00 => 2 = 4 - 2
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => 10 => 2 = 4 - 2
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => 10 => 2 = 4 - 2
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => 01 => 1 = 3 - 2
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => 11 => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 100 => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 100 => 3 = 5 - 2
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 010 => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 110 => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 3 = 5 - 2
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 110 => 3 = 5 - 2
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 010 => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 001 => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 101 => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 3 = 5 - 2
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 101 => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 011 => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0000 => 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1000 => 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 1000 => 4 = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0100 => 3 = 5 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1100 => 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 1000 => 4 = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1100 => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 0100 => 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0010 => 3 = 5 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 1010 => 3 = 5 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1100 => 4 = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1010 => 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0110 => 3 = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1110 => 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1000 => 4 = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1100 => 4 = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 1100 => 4 = 6 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 0110 => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1110 => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 0100 => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1010 => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0010 => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0001 => 3 = 5 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 1001 => 3 = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 1010 => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1001 => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0101 => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1101 => 3 = 5 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1100 => 4 = 6 - 2
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000672
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 2 - 2
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1 = 3 - 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [3,1,2] => 1 = 3 - 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [3,1,4,2] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 3 = 5 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [4,1,2,3,5] => 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 3 = 5 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 3 = 5 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [4,1,2,5,3] => 3 = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 3 = 5 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 3 = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => 3 = 5 - 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [1,7,2,3,4,5,6] => [1,2,3,4,7,5,6] => ? = 7 - 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => [1,5,6,7,2,3,4] => [1,2,5,3,6,7,4] => ? = 7 - 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [1,4,5,6,7,2,3] => [1,4,2,5,6,7,3] => ? = 7 - 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? = 7 - 2
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: St001237
(load all 99 compositions to match this statistic)
(load all 99 compositions to match this statistic)
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
St001237: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> 2
[1,0,1,0]
=> [1,1,0,0]
=> 3
[1,1,0,0]
=> [1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 4
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 4
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 5
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 6
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 6
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 7
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
Matching statistic: St001211
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 3
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 6
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 6
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001492
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 6
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 6
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 6
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 6
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 8
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 7
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St000083
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000083: Binary trees ⟶ ℤResult quality: 71% ●values known / values provided: 96%●distinct values known / distinct values provided: 71%
Mp00223: Permutations —runsort⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000083: Binary trees ⟶ ℤResult quality: 71% ●values known / values provided: 96%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [1] => [.,.]
=> ? = 2 - 2
[1,0,1,0]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1 = 3 - 2
[1,1,0,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 2 = 4 - 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 2 = 4 - 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> 2 = 4 - 2
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> 1 = 3 - 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 2 = 4 - 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 2 = 4 - 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 3 = 5 - 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 3 = 5 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 3 = 5 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 3 = 5 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 3 = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 3 = 5 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 3 = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 3 = 5 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 4 = 6 - 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 8 - 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> ? = 7 - 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => [1,6,7,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> ? = 7 - 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => [1,5,6,7,2,3,4] => [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ? = 7 - 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [1,4,5,6,7,2,3] => [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ? = 7 - 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 8 - 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 7 - 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 8 - 2
Description
The number of left oriented leafs of a binary tree except the first one.
In other other words, this is the sum of canopee vector of the tree.
The canopee of a non empty binary tree T with n internal nodes is the list l of 0 and 1 of length n-1 obtained by going along the leaves of T from left to right except the two extremal ones, writing 0 if the leaf is a right leaf and 1 if the leaf is a left leaf.
This is also the number of nodes having a right child. Indeed each of said right children will give exactly one left oriented leaf.
Matching statistic: St000354
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 96%●distinct values known / distinct values provided: 71%
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 96%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [1] => [1] => ? = 2 - 2
[1,0,1,0]
=> [2,1] => [1,2] => [2,1] => 1 = 3 - 2
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => [3,2,1] => 2 = 4 - 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [3,2,1] => 2 = 4 - 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,2,3] => [3,2,1] => 2 = 4 - 2
[1,1,0,1,0,0]
=> [2,1,3] => [1,3,2] => [3,1,2] => 1 = 3 - 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 3 = 5 - 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,2,4,3] => [4,3,1,2] => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 3 = 5 - 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 3 = 5 - 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 2 = 4 - 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,4,2,3] => [4,1,3,2] => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 3 = 5 - 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,2,3,5,4] => [5,4,3,1,2] => 3 = 5 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,2,4,3,5] => [5,4,2,3,1] => 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,2,5,3,4] => [5,4,1,3,2] => 3 = 5 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,2,5,3,4] => [5,4,1,3,2] => 3 = 5 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,2,3,5,4] => [5,4,3,1,2] => 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,2,4,5,3] => [5,4,2,1,3] => 3 = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,2,3,5,4] => [5,4,3,1,2] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,4,2,3,5] => [5,2,4,3,1] => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 3 = 5 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 3 = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,2,3,5] => [5,2,4,3,1] => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 3 = 5 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 6 - 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8 - 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [1,7,2,3,4,5,6] => [7,1,6,5,4,3,2] => ? = 7 - 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => [1,6,7,2,3,4,5] => [7,2,1,6,5,4,3] => ? = 7 - 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => [1,5,6,7,2,3,4] => [7,3,2,1,6,5,4] => ? = 7 - 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [1,4,5,6,7,2,3] => [7,4,3,2,1,6,5] => ? = 7 - 2
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8 - 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => [7,5,4,3,2,1,6] => ? = 7 - 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 8 - 2
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001298The number of repeated entries in the Lehmer code of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001668The number of points of the poset minus the width of the 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. St000216The absolute length of a permutation. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001432The order dimension of the partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001773The number of minimal elements in Bruhat order not less than the signed permutation.
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