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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St001075
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001075: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> 1
[2,1] => {{1,2}}
=> 2
[1,2,3] => {{1},{2},{3}}
=> 1
[1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => {{1,2,3}}
=> 3
[3,1,2] => {{1,2,3}}
=> 3
[3,2,1] => {{1,3},{2}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,4,2,3] => {{1},{2,3,4}}
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> 4
[2,4,1,3] => {{1,2,3,4}}
=> 4
[2,4,3,1] => {{1,2,4},{3}}
=> 1
[3,1,2,4] => {{1,2,3},{4}}
=> 1
[3,1,4,2] => {{1,2,3,4}}
=> 4
[3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => {{1,2,3,4}}
=> 4
[4,1,2,3] => {{1,2,3,4}}
=> 4
[4,1,3,2] => {{1,2,4},{3}}
=> 1
[4,2,1,3] => {{1,3,4},{2}}
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> 1
[4,3,1,2] => {{1,2,3,4}}
=> 4
[4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 1
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> 1
Description
The minimal size of a block of a set partition.
Matching statistic: St000657
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> [1,1] => 1
[2,1] => {{1,2}}
=> [2] => 2
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => 1
[1,3,2] => {{1},{2,3}}
=> [1,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1] => 1
[2,3,1] => {{1,2,3}}
=> [3] => 3
[3,1,2] => {{1,2,3}}
=> [3] => 3
[3,2,1] => {{1,3},{2}}
=> [2,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => 1
[2,3,4,1] => {{1,2,3,4}}
=> [4] => 4
[2,4,1,3] => {{1,2,3,4}}
=> [4] => 4
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => 1
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => 1
[3,1,4,2] => {{1,2,3,4}}
=> [4] => 4
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => 2
[3,4,2,1] => {{1,2,3,4}}
=> [4] => 4
[4,1,2,3] => {{1,2,3,4}}
=> [4] => 4
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => 1
[4,3,1,2] => {{1,2,3,4}}
=> [4] => 4
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => 1
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> [1,4] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000655
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> [1,1] => [1,0,1,0]
=> 1
[2,1] => {{1,2}}
=> [2] => [1,1,0,0]
=> 2
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,2] => {{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => {{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 3
[3,1,2] => {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 3
[3,2,1] => {{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,3,1,2] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[4,2,3,5,6,7,8,1] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,3,1,4,5,6,7] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,5,4,7,8,6,1] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,5,4,6,7,8,1] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,5,4,8,1,6,7] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,5,4,7,1,8,6] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,6,4,7,5,8,1] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,6,4,7,1,8,3] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,5,4,6,1,3,7] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,2,1,4,3,7,8,5] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,5,4,7,1,6,3] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,1,7,4,8,6] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,1,4,8,3,5,6] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,7,4,8,1,6,3] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,2,7,4,3,5,8,1] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,3,1,8,5,6,4] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,5,4,6,7,1,3] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,7,4,1,3,5,6] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,6,7,1,8,4] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,3,1,4,5,8,6] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,6,4,8,5,1,7] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,1,4,6,3,8,5] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,3,1,7,4,6,5] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,7,4,6,1,5,3] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,1,4,3,8,6,5] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,8,4,1,5,3,6] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,6,4,7,8,1,5] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,2,1,4,3,8,5,7] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,8,4,7,1,6] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,3,5,7,1,6,4] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,2,8,4,3,7,5,1] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,8,4,3,5,6,1] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,3,7,1,5,6,4] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,2,8,4,3,5,1,7] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,2,3,7,8,5,1,4] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,3,5,6,7,1,4] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[4,2,3,5,8,1,6,7] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[4,2,3,7,1,8,6,5] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,3,8,4,1,5,6] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,2,3,5,8,1,4,6] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,2,3,8,4,7,5,1] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,6,4,1,7,8,5] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,1,6,7,8,4] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,1,4,6,7,8,3] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,2,3,8,1,4,5,7] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[8,2,3,6,4,1,5,7] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,8,4,7,5,1,6] => {{1,3,5,6,7,8},{2},{4}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[4,2,3,8,7,5,1,6] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,6,7,8,4,1] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[4,2,3,8,6,7,1,5] => {{1,4,5,6,7,8},{2},{3}}
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
Matching statistic: St000993
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [2]
=> 1
[2,1] => [2]
=> [1,1]
=> 2
[1,2,3] => [1,1,1]
=> [3]
=> 1
[1,3,2] => [2,1]
=> [2,1]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> 3
[3,1,2] => [3]
=> [1,1,1]
=> 3
[3,2,1] => [2,1]
=> [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 4
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 4
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 4
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2,2]
=> 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 4
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 4
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 4
[4,3,2,1] => [2,2]
=> [2,2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? = 8
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? = 8
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? = 8
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? = 8
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ? = 2
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ? = 8
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ? = 8
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? = 8
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? = 8
[7,8,5,6,3,1,2,4] => ?
=> ?
=> ? = 2
[7,5,6,8,3,1,2,4] => ?
=> ?
=> ? = 2
[6,7,8,4,2,3,1,5] => ?
=> ?
=> ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ? = 8
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? = 8
[7,8,6,2,1,3,4,5] => ?
=> ?
=> ? = 2
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? = 8
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ? = 8
[8,5,4,3,6,1,2,7] => ?
=> ?
=> ? = 2
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ? = 1
[3,5,7,6,8,4,2,1] => ?
=> ?
=> ? = 2
[3,5,4,7,8,6,2,1] => ?
=> ?
=> ? = 1
[4,3,5,7,8,6,2,1] => ?
=> ?
=> ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ? = 8
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ? = 8
[2,3,5,7,6,8,4,1] => ?
=> ?
=> ? = 2
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ? = 8
[4,3,5,2,8,7,6,1] => ?
=> ?
=> ? = 2
[3,2,5,4,7,8,6,1] => ?
=> ?
=> ? = 1
[2,3,5,4,7,8,6,1] => ?
=> ?
=> ? = 1
[3,4,6,7,5,2,8,1] => ?
=> ?
=> ? = 1
[2,5,6,7,4,3,8,1] => ?
=> ?
=> ? = 2
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ? = 2
[3,4,2,7,6,5,8,1] => ?
=> ?
=> ? = 2
[3,5,4,2,7,6,8,1] => ?
=> ?
=> ? = 1
[3,4,2,5,7,6,8,1] => ?
=> ?
=> ? = 1
[3,5,4,6,2,7,8,1] => ?
=> ?
=> ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ? = 1
[2,5,4,3,6,7,8,1] => ?
=> ?
=> ? = 2
[1,4,3,6,8,7,5,2] => ?
=> ?
=> ? = 1
[1,4,3,7,6,8,5,2] => ?
=> ?
=> ? = 1
[1,4,6,5,3,8,7,2] => ?
=> ?
=> ? = 1
[1,3,4,5,7,6,8,2] => ?
=> ?
=> ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ? = 1
[2,6,1,7,3,8,4,5] => ?
=> ?
=> ? = 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001038
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1,1,0,0]
=> 1
[2,1] => [2]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 3
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,4,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,2,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,3,1,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? = 8
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? = 8
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? = 8
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? = 8
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ? = 2
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ? = 8
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ? = 8
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? = 8
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? = 8
[7,8,5,6,3,1,2,4] => ?
=> ?
=> ? = 2
[7,5,6,8,3,1,2,4] => ?
=> ?
=> ? = 2
[6,7,8,4,2,3,1,5] => ?
=> ?
=> ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ? = 8
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? = 8
[7,8,6,2,1,3,4,5] => ?
=> ?
=> ? = 2
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? = 8
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ? = 8
[8,5,4,3,6,1,2,7] => ?
=> ?
=> ? = 2
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ? = 1
[3,5,7,6,8,4,2,1] => ?
=> ?
=> ? = 2
[3,5,4,7,8,6,2,1] => ?
=> ?
=> ? = 1
[4,3,5,7,8,6,2,1] => ?
=> ?
=> ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ? = 8
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ? = 8
[2,3,5,7,6,8,4,1] => ?
=> ?
=> ? = 2
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ? = 8
[4,3,5,2,8,7,6,1] => ?
=> ?
=> ? = 2
[3,2,5,4,7,8,6,1] => ?
=> ?
=> ? = 1
[2,3,5,4,7,8,6,1] => ?
=> ?
=> ? = 1
[3,4,6,7,5,2,8,1] => ?
=> ?
=> ? = 1
[2,5,6,7,4,3,8,1] => ?
=> ?
=> ? = 2
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ? = 2
[3,4,2,7,6,5,8,1] => ?
=> ?
=> ? = 2
[3,5,4,2,7,6,8,1] => ?
=> ?
=> ? = 1
[3,4,2,5,7,6,8,1] => ?
=> ?
=> ? = 1
[3,5,4,6,2,7,8,1] => ?
=> ?
=> ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ? = 1
[2,5,4,3,6,7,8,1] => ?
=> ?
=> ? = 2
[1,4,3,6,8,7,5,2] => ?
=> ?
=> ? = 1
[1,4,3,7,6,8,5,2] => ?
=> ?
=> ? = 1
[1,4,6,5,3,8,7,2] => ?
=> ?
=> ? = 1
[1,3,4,5,7,6,8,2] => ?
=> ?
=> ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ? = 1
[2,6,1,7,3,8,4,5] => ?
=> ?
=> ? = 2
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000297
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [2]
=> 100 => 1
[2,1] => [2]
=> [1,1]
=> 110 => 2
[1,2,3] => [1,1,1]
=> [3]
=> 1000 => 1
[1,3,2] => [2,1]
=> [2,1]
=> 1010 => 1
[2,1,3] => [2,1]
=> [2,1]
=> 1010 => 1
[2,3,1] => [3]
=> [1,1,1]
=> 1110 => 3
[3,1,2] => [3]
=> [1,1,1]
=> 1110 => 3
[3,2,1] => [2,1]
=> [2,1]
=> 1010 => 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 10000 => 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 10010 => 1
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 10010 => 1
[1,3,4,2] => [3,1]
=> [2,1,1]
=> 10110 => 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> 10110 => 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 10010 => 1
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 10010 => 1
[2,1,4,3] => [2,2]
=> [2,2]
=> 1100 => 2
[2,3,1,4] => [3,1]
=> [2,1,1]
=> 10110 => 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> 11110 => 4
[2,4,1,3] => [4]
=> [1,1,1,1]
=> 11110 => 4
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 10110 => 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> 10110 => 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> 11110 => 4
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 10010 => 1
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 10110 => 1
[3,4,1,2] => [2,2]
=> [2,2]
=> 1100 => 2
[3,4,2,1] => [4]
=> [1,1,1,1]
=> 11110 => 4
[4,1,2,3] => [4]
=> [1,1,1,1]
=> 11110 => 4
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 10110 => 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> 10110 => 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 10010 => 1
[4,3,1,2] => [4]
=> [1,1,1,1]
=> 11110 => 4
[4,3,2,1] => [2,2]
=> [2,2]
=> 1100 => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 100000 => 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 10100 => 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> 10100 => 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? => ? = 8
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? => ? = 8
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? => ? = 8
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ? => ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? => ? = 8
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ? => ? = 2
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ? => ? = 8
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ? => ? = 8
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? => ? = 8
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? => ? = 8
[7,8,5,6,3,1,2,4] => ?
=> ?
=> ? => ? = 2
[7,5,6,8,3,1,2,4] => ?
=> ?
=> ? => ? = 2
[6,7,8,4,2,3,1,5] => ?
=> ?
=> ? => ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ? => ? = 8
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? => ? = 8
[7,8,6,2,1,3,4,5] => ?
=> ?
=> ? => ? = 2
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? => ? = 8
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ? => ? = 8
[8,5,4,3,6,1,2,7] => ?
=> ?
=> ? => ? = 2
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ? => ? = 1
[3,5,7,6,8,4,2,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,7,8,6,2,1] => ?
=> ?
=> ? => ? = 1
[4,3,5,7,8,6,2,1] => ?
=> ?
=> ? => ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ? => ? = 8
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ? => ? = 8
[2,3,5,7,6,8,4,1] => ?
=> ?
=> ? => ? = 2
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ? => ? = 8
[4,3,5,2,8,7,6,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,4,7,8,6,1] => ?
=> ?
=> ? => ? = 1
[2,3,5,4,7,8,6,1] => ?
=> ?
=> ? => ? = 1
[3,4,6,7,5,2,8,1] => ?
=> ?
=> ? => ? = 1
[2,5,6,7,4,3,8,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ? => ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ? => ? = 2
[3,4,2,7,6,5,8,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,2,7,6,8,1] => ?
=> ?
=> ? => ? = 1
[3,4,2,5,7,6,8,1] => ?
=> ?
=> ? => ? = 1
[3,5,4,6,2,7,8,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ? => ? = 1
[2,5,4,3,6,7,8,1] => ?
=> ?
=> ? => ? = 2
[1,4,3,6,8,7,5,2] => ?
=> ?
=> ? => ? = 1
[1,4,3,7,6,8,5,2] => ?
=> ?
=> ? => ? = 1
[1,4,6,5,3,8,7,2] => ?
=> ?
=> ? => ? = 1
[1,3,4,5,7,6,8,2] => ?
=> ?
=> ? => ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ? => ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ? => ? = 1
[2,6,1,7,3,8,4,5] => ?
=> ?
=> ? => ? = 2
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> 1 => 1
[2,1] => [2]
=> [[1,2]]
=> 0 => 2
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 11 => 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> 10 => 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> 10 => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> 00 => 3
[3,1,2] => [3]
=> [[1,2,3]]
=> 00 => 3
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> 10 => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 010 => 2
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> 000 => 4
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> 000 => 4
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> 000 => 4
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 010 => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> 000 => 4
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> 000 => 4
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> 000 => 4
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> 010 => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 1
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? => ? = 8
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? => ? = 8
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? => ? = 8
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ? => ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? => ? = 8
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ? => ? = 2
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ? => ? = 8
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ? => ? = 8
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? => ? = 8
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? => ? = 8
[7,8,5,6,3,1,2,4] => ?
=> ?
=> ? => ? = 2
[7,5,6,8,3,1,2,4] => ?
=> ?
=> ? => ? = 2
[6,7,8,4,2,3,1,5] => ?
=> ?
=> ? => ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ? => ? = 8
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? => ? = 8
[7,8,6,2,1,3,4,5] => ?
=> ?
=> ? => ? = 2
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? => ? = 8
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ? => ? = 8
[8,5,4,3,6,1,2,7] => ?
=> ?
=> ? => ? = 2
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ? => ? = 1
[3,5,7,6,8,4,2,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,7,8,6,2,1] => ?
=> ?
=> ? => ? = 1
[4,3,5,7,8,6,2,1] => ?
=> ?
=> ? => ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ? => ? = 8
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ? => ? = 8
[2,3,5,7,6,8,4,1] => ?
=> ?
=> ? => ? = 2
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ? => ? = 8
[4,3,5,2,8,7,6,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,4,7,8,6,1] => ?
=> ?
=> ? => ? = 1
[2,3,5,4,7,8,6,1] => ?
=> ?
=> ? => ? = 1
[3,4,6,7,5,2,8,1] => ?
=> ?
=> ? => ? = 1
[2,5,6,7,4,3,8,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ? => ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ? => ? = 2
[3,4,2,7,6,5,8,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,2,7,6,8,1] => ?
=> ?
=> ? => ? = 1
[3,4,2,5,7,6,8,1] => ?
=> ?
=> ? => ? = 1
[3,5,4,6,2,7,8,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ? => ? = 1
[2,5,4,3,6,7,8,1] => ?
=> ?
=> ? => ? = 2
[1,4,3,6,8,7,5,2] => ?
=> ?
=> ? => ? = 1
[1,4,3,7,6,8,5,2] => ?
=> ?
=> ? => ? = 1
[1,4,6,5,3,8,7,2] => ?
=> ?
=> ? => ? = 1
[1,3,4,5,7,6,8,2] => ?
=> ?
=> ? => ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ? => ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ? => ? = 1
[2,6,1,7,3,8,4,5] => ?
=> ?
=> ? => ? = 2
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of {1,…,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000382
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2,1] => [2]
=> [[1,2]]
=> [2] => 2
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [3] => 3
[3,1,2] => [3]
=> [[1,2,3]]
=> [3] => 3
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ? => ? = 8
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ? => ? = 8
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ? => ? = 8
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ? => ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ? => ? = 8
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ? => ? = 2
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ? => ? = 8
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ? => ? = 8
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ? => ? = 8
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ? => ? = 8
[7,8,5,6,3,1,2,4] => ?
=> ?
=> ? => ? = 2
[7,5,6,8,3,1,2,4] => ?
=> ?
=> ? => ? = 2
[6,7,8,4,2,3,1,5] => ?
=> ?
=> ? => ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ? => ? = 8
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ? => ? = 8
[7,8,6,2,1,3,4,5] => ?
=> ?
=> ? => ? = 2
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ? => ? = 8
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ? => ? = 8
[8,5,4,3,6,1,2,7] => ?
=> ?
=> ? => ? = 2
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ? => ? = 1
[3,5,7,6,8,4,2,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,7,8,6,2,1] => ?
=> ?
=> ? => ? = 1
[4,3,5,7,8,6,2,1] => ?
=> ?
=> ? => ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ? => ? = 8
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ? => ? = 8
[2,3,5,7,6,8,4,1] => ?
=> ?
=> ? => ? = 2
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ? => ? = 8
[4,3,5,2,8,7,6,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,4,7,8,6,1] => ?
=> ?
=> ? => ? = 1
[2,3,5,4,7,8,6,1] => ?
=> ?
=> ? => ? = 1
[3,4,6,7,5,2,8,1] => ?
=> ?
=> ? => ? = 1
[2,5,6,7,4,3,8,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ? => ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ? => ? = 2
[3,4,2,7,6,5,8,1] => ?
=> ?
=> ? => ? = 2
[3,5,4,2,7,6,8,1] => ?
=> ?
=> ? => ? = 1
[3,4,2,5,7,6,8,1] => ?
=> ?
=> ? => ? = 1
[3,5,4,6,2,7,8,1] => ?
=> ?
=> ? => ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ? => ? = 1
[2,5,4,3,6,7,8,1] => ?
=> ?
=> ? => ? = 2
[1,4,3,6,8,7,5,2] => ?
=> ?
=> ? => ? = 1
[1,4,3,7,6,8,5,2] => ?
=> ?
=> ? => ? = 1
[1,4,6,5,3,8,7,2] => ?
=> ?
=> ? => ? = 1
[1,3,4,5,7,6,8,2] => ?
=> ?
=> ? => ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ? => ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ? => ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ? => ? = 1
[2,6,1,7,3,8,4,5] => ?
=> ?
=> ? => ? = 2
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> 110 => [2,1] => 1
[2,1] => [2]
=> 100 => [1,2] => 2
[1,2,3] => [1,1,1]
=> 1110 => [3,1] => 1
[1,3,2] => [2,1]
=> 1010 => [1,1,1,1] => 1
[2,1,3] => [2,1]
=> 1010 => [1,1,1,1] => 1
[2,3,1] => [3]
=> 1000 => [1,3] => 3
[3,1,2] => [3]
=> 1000 => [1,3] => 3
[3,2,1] => [2,1]
=> 1010 => [1,1,1,1] => 1
[1,2,3,4] => [1,1,1,1]
=> 11110 => [4,1] => 1
[1,2,4,3] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,3,2,4] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[1,3,4,2] => [3,1]
=> 10010 => [1,2,1,1] => 1
[1,4,2,3] => [3,1]
=> 10010 => [1,2,1,1] => 1
[1,4,3,2] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[2,1,3,4] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[2,1,4,3] => [2,2]
=> 1100 => [2,2] => 2
[2,3,1,4] => [3,1]
=> 10010 => [1,2,1,1] => 1
[2,3,4,1] => [4]
=> 10000 => [1,4] => 4
[2,4,1,3] => [4]
=> 10000 => [1,4] => 4
[2,4,3,1] => [3,1]
=> 10010 => [1,2,1,1] => 1
[3,1,2,4] => [3,1]
=> 10010 => [1,2,1,1] => 1
[3,1,4,2] => [4]
=> 10000 => [1,4] => 4
[3,2,1,4] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[3,2,4,1] => [3,1]
=> 10010 => [1,2,1,1] => 1
[3,4,1,2] => [2,2]
=> 1100 => [2,2] => 2
[3,4,2,1] => [4]
=> 10000 => [1,4] => 4
[4,1,2,3] => [4]
=> 10000 => [1,4] => 4
[4,1,3,2] => [3,1]
=> 10010 => [1,2,1,1] => 1
[4,2,1,3] => [3,1]
=> 10010 => [1,2,1,1] => 1
[4,2,3,1] => [2,1,1]
=> 10110 => [1,1,2,1] => 1
[4,3,1,2] => [4]
=> 10000 => [1,4] => 4
[4,3,2,1] => [2,2]
=> 1100 => [2,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
=> 111110 => [5,1] => 1
[1,2,3,5,4] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,2,4,3,5] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,2,4,5,3] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,2,5,3,4] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,2,5,4,3] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,3,2,4,5] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,3,2,5,4] => [2,2,1]
=> 11010 => [2,1,1,1] => 1
[1,3,4,2,5] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,3,4,5,2] => [4,1]
=> 100010 => [1,3,1,1] => 1
[1,3,5,2,4] => [4,1]
=> 100010 => [1,3,1,1] => 1
[1,3,5,4,2] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,4,2,3,5] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,4,2,5,3] => [4,1]
=> 100010 => [1,3,1,1] => 1
[1,4,3,2,5] => [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
[1,4,3,5,2] => [3,1,1]
=> 100110 => [1,2,2,1] => 1
[1,4,5,2,3] => [2,2,1]
=> 11010 => [2,1,1,1] => 1
[1,4,5,3,2] => [4,1]
=> 100010 => [1,3,1,1] => 1
[6,7,5,8,4,2,3,1] => ?
=> ? => ? => ? = 8
[6,7,4,5,8,2,3,1] => ?
=> ? => ? => ? = 8
[5,6,4,2,3,7,8,1] => ?
=> ? => ? => ? = 8
[5,4,6,2,3,7,8,1] => ?
=> ? => ? => ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ? => ? => ? = 8
[7,5,6,8,4,2,1,3] => ?
=> ? => ? => ? = 2
[8,6,4,5,7,1,2,3] => ?
=> ? => ? => ? = 8
[6,7,8,5,2,3,1,4] => ?
=> ? => ? => ? = 8
[8,7,5,6,2,3,1,4] => ?
=> ? => ? => ? = 8
[8,6,7,5,3,1,2,4] => ?
=> ? => ? => ? = 8
[7,8,5,6,3,1,2,4] => ?
=> ? => ? => ? = 2
[7,5,6,8,3,1,2,4] => ?
=> ? => ? => ? = 2
[6,7,8,4,2,3,1,5] => ?
=> ? => ? => ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ? => ? => ? = 8
[7,8,6,2,3,1,4,5] => ?
=> ? => ? => ? = 8
[7,8,6,2,1,3,4,5] => ?
=> ? => ? => ? = 2
[8,7,4,5,2,3,1,6] => ?
=> ? => ? => ? = 8
[8,7,4,2,1,3,5,6] => ?
=> ? => ? => ? = 8
[8,5,4,3,6,1,2,7] => ?
=> ? => ? => ? = 2
[6,7,5,1,2,3,4,8] => ?
=> ? => ? => ? = 1
[3,5,7,6,8,4,2,1] => ?
=> ? => ? => ? = 2
[3,5,4,7,8,6,2,1] => ?
=> ? => ? => ? = 1
[4,3,5,7,8,6,2,1] => ?
=> ? => ? => ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ? => ? => ? = 8
[3,5,4,7,6,8,2,1] => ?
=> ? => ? => ? = 8
[2,3,5,7,6,8,4,1] => ?
=> ? => ? => ? = 2
[4,3,2,6,8,7,5,1] => ?
=> ? => ? => ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ? => ? => ? = 8
[4,3,5,2,8,7,6,1] => ?
=> ? => ? => ? = 2
[3,2,5,4,7,8,6,1] => ?
=> ? => ? => ? = 1
[2,3,5,4,7,8,6,1] => ?
=> ? => ? => ? = 1
[3,4,6,7,5,2,8,1] => ?
=> ? => ? => ? = 1
[2,5,6,7,4,3,8,1] => ?
=> ? => ? => ? = 2
[3,2,5,7,6,4,8,1] => ?
=> ? => ? => ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ? => ? => ? = 2
[3,4,2,7,6,5,8,1] => ?
=> ? => ? => ? = 2
[3,5,4,2,7,6,8,1] => ?
=> ? => ? => ? = 1
[3,4,2,5,7,6,8,1] => ?
=> ? => ? => ? = 1
[3,5,4,6,2,7,8,1] => ?
=> ? => ? => ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ? => ? => ? = 1
[2,5,4,3,6,7,8,1] => ?
=> ? => ? => ? = 2
[1,4,3,6,8,7,5,2] => ?
=> ? => ? => ? = 1
[1,4,3,7,6,8,5,2] => ?
=> ? => ? => ? = 1
[1,4,6,5,3,8,7,2] => ?
=> ? => ? => ? = 1
[1,3,4,5,7,6,8,2] => ?
=> ? => ? => ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ? => ? => ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ? => ? => ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ? => ? => ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ? => ? => ? = 1
[2,6,1,7,3,8,4,5] => ?
=> ? => ? => ? = 2
Description
The last part of an integer composition.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[2,1] => [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[3,1,2] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[1,4,5,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
[6,7,5,8,4,2,3,1] => ?
=> ?
=> ?
=> ? = 8
[6,7,4,5,8,2,3,1] => ?
=> ?
=> ?
=> ? = 8
[5,6,4,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 8
[5,4,6,2,3,7,8,1] => ?
=> ?
=> ?
=> ? = 2
[6,7,4,5,8,3,1,2] => ?
=> ?
=> ?
=> ? = 8
[7,5,6,8,4,2,1,3] => ?
=> ?
=> ?
=> ? = 2
[8,6,4,5,7,1,2,3] => ?
=> ?
=> ?
=> ? = 8
[6,7,8,5,2,3,1,4] => ?
=> ?
=> ?
=> ? = 8
[8,7,5,6,2,3,1,4] => ?
=> ?
=> ?
=> ? = 8
[8,6,7,5,3,1,2,4] => ?
=> ?
=> ?
=> ? = 8
[7,8,5,6,3,1,2,4] => ?
=> ?
=> ?
=> ? = 2
[7,5,6,8,3,1,2,4] => ?
=> ?
=> ?
=> ? = 2
[6,7,8,4,2,3,1,5] => ?
=> ?
=> ?
=> ? = 1
[7,8,6,3,4,1,2,5] => ?
=> ?
=> ?
=> ? = 8
[7,8,6,2,3,1,4,5] => ?
=> ?
=> ?
=> ? = 8
[7,8,6,2,1,3,4,5] => ?
=> ?
=> ?
=> ? = 2
[8,7,4,5,2,3,1,6] => ?
=> ?
=> ?
=> ? = 8
[8,7,4,2,1,3,5,6] => ?
=> ?
=> ?
=> ? = 8
[8,5,4,3,6,1,2,7] => ?
=> ?
=> ?
=> ? = 2
[6,7,5,1,2,3,4,8] => ?
=> ?
=> ?
=> ? = 1
[3,5,7,6,8,4,2,1] => ?
=> ?
=> ?
=> ? = 2
[3,5,4,7,8,6,2,1] => ?
=> ?
=> ?
=> ? = 1
[4,3,5,7,8,6,2,1] => ?
=> ?
=> ?
=> ? = 1
[4,3,5,6,8,7,2,1] => ?
=> ?
=> ?
=> ? = 8
[3,5,4,7,6,8,2,1] => ?
=> ?
=> ?
=> ? = 8
[2,3,5,7,6,8,4,1] => ?
=> ?
=> ?
=> ? = 2
[4,3,2,6,8,7,5,1] => ?
=> ?
=> ?
=> ? = 2
[3,5,4,2,7,8,6,1] => ?
=> ?
=> ?
=> ? = 8
[4,3,5,2,8,7,6,1] => ?
=> ?
=> ?
=> ? = 2
[3,2,5,4,7,8,6,1] => ?
=> ?
=> ?
=> ? = 1
[2,3,5,4,7,8,6,1] => ?
=> ?
=> ?
=> ? = 1
[3,4,6,7,5,2,8,1] => ?
=> ?
=> ?
=> ? = 1
[2,5,6,7,4,3,8,1] => ?
=> ?
=> ?
=> ? = 2
[3,2,5,7,6,4,8,1] => ?
=> ?
=> ?
=> ? = 1
[4,3,2,6,7,5,8,1] => ?
=> ?
=> ?
=> ? = 2
[3,4,2,7,6,5,8,1] => ?
=> ?
=> ?
=> ? = 2
[3,5,4,2,7,6,8,1] => ?
=> ?
=> ?
=> ? = 1
[3,4,2,5,7,6,8,1] => ?
=> ?
=> ?
=> ? = 1
[3,5,4,6,2,7,8,1] => ?
=> ?
=> ?
=> ? = 2
[3,2,5,6,4,7,8,1] => ?
=> ?
=> ?
=> ? = 1
[2,5,4,3,6,7,8,1] => ?
=> ?
=> ?
=> ? = 2
[1,4,3,6,8,7,5,2] => ?
=> ?
=> ?
=> ? = 1
[1,4,3,7,6,8,5,2] => ?
=> ?
=> ?
=> ? = 1
[1,4,6,5,3,8,7,2] => ?
=> ?
=> ?
=> ? = 1
[1,3,4,5,7,6,8,2] => ?
=> ?
=> ?
=> ? = 1
[2,5,7,6,4,3,1,8] => ?
=> ?
=> ?
=> ? = 1
[3,5,4,2,6,7,1,8] => ?
=> ?
=> ?
=> ? = 1
[2,5,1,6,7,3,4,8] => ?
=> ?
=> ?
=> ? = 1
[2,5,1,3,7,4,6,8] => ?
=> ?
=> ?
=> ? = 1
[2,6,1,7,3,8,4,5] => ?
=> ?
=> ?
=> ? = 2
Description
The row containing the largest entry of a standard tableau.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000667The greatest common divisor of the parts of the partition. St000990The first ascent of a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000260The radius of a connected graph. St000654The first descent of a permutation. St000487The length of the shortest cycle of a permutation. St000090The variation of a composition. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000314The number of left-to-right-maxima of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra.
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