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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000487
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
St000487: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 4
[2,4,1,3] => 4
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 4
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 4
[4,1,2,3] => 4
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 4
[4,3,2,1] => 2
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
[1,4,5,3,2] => 1
Description
The length of the shortest cycle of a permutation.
Matching statistic: St001075
(load all 12 compositions to match this statistic)
(load all 12 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: 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●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
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: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●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
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: 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: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●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
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
Description
The last part of an integer composition.
Matching statistic: St000655
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,4,2,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2,1,3] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3,1,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,4,5,2,3] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,4,5,3,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000733The row containing the largest entry of a standard tableau. 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. 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. St000314The number of left-to-right-maxima of a permutation. St000654The first descent of a permutation. St001236The dominant dimension of the corresponding Comp-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. St000260The radius of a connected graph. St000090The variation of a composition. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition.
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