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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St001062
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(load all 10 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001062: 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}}
=> 2
[2,1,3] => {{1,2},{3}}
=> 2
[2,3,1] => {{1,2,3}}
=> 3
[3,1,2] => {{1,3},{2}}
=> 2
[3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,2,4,3] => {{1},{2},{3,4}}
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> 3
[1,4,2,3] => {{1},{2,4},{3}}
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> 3
[2,3,4,1] => {{1,2,3,4}}
=> 4
[2,4,1,3] => {{1,2,4},{3}}
=> 3
[2,4,3,1] => {{1,2,4},{3}}
=> 3
[3,1,2,4] => {{1,3},{2},{4}}
=> 2
[3,1,4,2] => {{1,3,4},{2}}
=> 3
[3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> 3
[3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> 2
[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}}
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 4
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> 2
Description
The maximal size of a block of a set partition.
Matching statistic: St000381
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00128: Set partitions —to composition⟶ Integer compositions
St000381: 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] => 2
[2,1,3] => {{1,2},{3}}
=> [2,1] => 2
[2,3,1] => {{1,2,3}}
=> [3] => 3
[3,1,2] => {{1,3},{2}}
=> [2,1] => 2
[3,2,1] => {{1,3},{2}}
=> [2,1] => 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => 2
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => 3
[1,4,2,3] => {{1},{2,4},{3}}
=> [1,2,1] => 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => 3
[2,3,4,1] => {{1,2,3,4}}
=> [4] => 4
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1] => 3
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => 3
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1] => 2
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1] => 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => 3
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2] => 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1] => 2
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1] => 2
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1] => 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => 2
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2] => 2
[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] => 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => 4
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [1,3,1] => 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [1,3,1] => 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [1,2,2] => 2
Description
The largest part of an integer composition.
Matching statistic: St000392
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0 => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => 1 => 1 = 2 - 1
[1,2,3] => [1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => 01 => 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,1,3] => 10 => 1 = 2 - 1
[2,3,1] => [3,1,2] => [3,2,1] => 11 => 2 = 3 - 1
[3,1,2] => [2,3,1] => [3,1,2] => 10 => 1 = 2 - 1
[3,2,1] => [3,2,1] => [2,3,1] => 01 => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 001 => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 011 => 2 = 3 - 1
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 010 => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 001 => 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 100 => 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 101 => 1 = 2 - 1
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 110 => 2 = 3 - 1
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 111 => 3 = 4 - 1
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 110 => 2 = 3 - 1
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 011 => 2 = 3 - 1
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 100 => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 110 => 2 = 3 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 010 => 1 = 2 - 1
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 011 => 2 = 3 - 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 101 => 1 = 2 - 1
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => 101 => 1 = 2 - 1
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 100 => 1 = 2 - 1
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => 010 => 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => 010 => 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 001 => 1 = 2 - 1
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 101 => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 101 => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 0011 => 2 = 3 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0001 => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1 = 2 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0110 => 2 = 3 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 0111 => 3 = 4 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 0110 => 2 = 3 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 0011 => 2 = 3 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0100 => 1 = 2 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => 0110 => 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0010 => 1 = 2 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => 0011 => 2 = 3 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 0101 => 1 = 2 - 1
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,3,4,2] => 0101 => 1 = 2 - 1
[3,2,5,1,6,7,8,4] => [4,2,1,8,3,5,6,7] => [2,8,7,6,5,3,1,4] => ? => ? = 6 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000147
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
St000147: Integer partitions ⟶ 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}}
=> [2,1]
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> 2
[2,3,1] => {{1,2,3}}
=> [3]
=> 3
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> 2
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> 3
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> 3
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> 4
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> 3
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> 3
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> 2
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> 3
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> 2
[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}}
=> [2,1,1,1]
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> 4
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> 2
[4,2,3,5,6,7,8,1] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ? = 6
[2,3,4,7,6,5,8,1] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ? = 6
[2,5,4,3,6,7,8,1] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ? = 6
[2,3,4,5,7,8,1,6] => {{1,2,3,4,5,7},{6,8}}
=> ?
=> ? = 6
[4,1,2,5,6,7,8,3] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ? = 6
[2,5,4,1,6,7,8,3] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ? = 6
[2,3,4,7,6,1,8,5] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ? = 6
[3,6,4,5,7,2,8,1] => {{1,3,4,5,7,8},{2,6}}
=> ?
=> ? = 6
[3,2,5,1,6,7,8,4] => {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ? = 6
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> [1,1]
=> [2]
=> 1
[2,1] => {{1,2}}
=> [2]
=> [1,1]
=> 2
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> [2,1]
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> [2,1]
=> 2
[2,3,1] => {{1,2,3}}
=> [3]
=> [1,1,1]
=> 3
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> [2,1]
=> 2
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> [2,1]
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 3
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 3
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 4
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 3
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 3
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 2
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 3
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [5]
=> 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> [2,1,1,1]
=> 4
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [3,1,1]
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [3,2]
=> 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [3,2]
=> 2
[4,2,3,5,6,7,8,1] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 6
[2,3,4,7,6,5,8,1] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 6
[2,5,4,3,6,7,8,1] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 6
[2,3,4,5,7,8,1,6] => {{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ? = 6
[4,1,2,5,6,7,8,3] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 6
[2,5,4,1,6,7,8,3] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 6
[2,3,4,7,6,1,8,5] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 6
[3,6,4,5,7,2,8,1] => {{1,3,4,5,7,8},{2,6}}
=> ?
=> ?
=> ? = 6
[3,2,5,1,6,7,8,4] => {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 6
Description
The length of the partition.
Matching statistic: St000676
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1] => {{1,2}}
=> [2]
=> [1,0,1,0]
=> 2
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => {{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,2,3,4,5] => {{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] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,3,5,6,7,8,1] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 6
[2,3,4,7,6,5,8,1] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 6
[2,5,4,3,6,7,8,1] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 6
[2,3,4,5,7,8,1,6] => {{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ? = 6
[4,1,2,5,6,7,8,3] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 6
[2,5,4,1,6,7,8,3] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 6
[2,3,4,7,6,1,8,5] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 6
[3,6,4,5,7,2,8,1] => {{1,3,4,5,7,8},{2,6}}
=> ?
=> ?
=> ? = 6
[3,2,5,1,6,7,8,4] => {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 6
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000734
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ 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
[2,1] => {{1,2}}
=> [2]
=> [[1,2]]
=> 2
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,3,1] => {{1,2,3}}
=> [3]
=> [[1,2,3]]
=> 3
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> 2
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> 4
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[4,2,3,5,6,7,8,1] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 6
[2,3,4,7,6,5,8,1] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 6
[2,5,4,3,6,7,8,1] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 6
[2,3,4,5,7,8,1,6] => {{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ? = 6
[4,1,2,5,6,7,8,3] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 6
[2,5,4,1,6,7,8,3] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 6
[2,3,4,7,6,1,8,5] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 6
[3,6,4,5,7,2,8,1] => {{1,3,4,5,7,8},{2,6}}
=> ?
=> ?
=> ? = 6
[3,2,5,1,6,7,8,4] => {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 6
Description
The last entry in the first row of a standard tableau.
Matching statistic: St001039
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1] => {{1,2}}
=> [2]
=> [1,0,1,0]
=> 2
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => {{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,2,3,4,5] => {{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] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,3,5,6,7,8,1] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 6
[2,3,4,7,6,5,8,1] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 6
[2,5,4,3,6,7,8,1] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 6
[2,3,4,5,7,8,1,6] => {{1,2,3,4,5,7},{6,8}}
=> ?
=> ?
=> ? = 6
[4,1,2,5,6,7,8,3] => {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 6
[2,5,4,1,6,7,8,3] => {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 6
[2,3,4,7,6,1,8,5] => {{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ? = 6
[3,6,4,5,7,2,8,1] => {{1,3,4,5,7,8},{2,6}}
=> ?
=> ?
=> ? = 6
[3,2,5,1,6,7,8,4] => {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 6
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001291
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 97%●distinct values known / distinct values provided: 62%
Values
[1,2] => {{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1] => {{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,2,3] => {{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2] => {{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,3] => {{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,1] => {{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,1,2] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1] => {{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,4,2] => {{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,4,2,3] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[2,3,4,1] => {{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[2,4,1,3] => {{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,1,2,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,4,2] => {{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => {{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[4,1,2,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,2,1,3] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[4,3,1,2] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[2,3,4,5,6,1] => {{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}}
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,4,5,7,6] => {{1},{2},{3},{4},{5},{6,7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,4,6,5,7] => {{1},{2},{3},{4},{5,6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,4,7,5,6] => {{1},{2},{3},{4},{5,7},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,4,7,6,5] => {{1},{2},{3},{4},{5,7},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,5,4,6,7] => {{1},{2},{3},{4,5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,6,4,5,7] => {{1},{2},{3},{4,6},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,6,5,4,7] => {{1},{2},{3},{4,6},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,7,4,5,6] => {{1},{2},{3},{4,7},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,7,4,6,5] => {{1},{2},{3},{4,7},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,7,5,4,6] => {{1},{2},{3},{4,7},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,3,7,5,6,4] => {{1},{2},{3},{4,7},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,4,3,5,6,7] => {{1},{2},{3,4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,5,3,4,6,7] => {{1},{2},{3,5},{4},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,5,4,3,6,7] => {{1},{2},{3,5},{4},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,6,3,4,5,7] => {{1},{2},{3,6},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,6,3,5,4,7] => {{1},{2},{3,6},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,6,4,3,5,7] => {{1},{2},{3,6},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,6,4,5,3,7] => {{1},{2},{3,6},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,7,3,4,5,6] => {{1},{2},{3,7},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,7,3,4,6,5] => {{1},{2},{3,7},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,7,3,5,4,6] => {{1},{2},{3,7},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,7,3,5,6,4] => {{1},{2},{3,7},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,7,4,3,5,6] => {{1},{2},{3,7},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,7,4,3,6,5] => {{1},{2},{3,7},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,7,4,5,3,6] => {{1},{2},{3,7},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,2,7,4,5,6,3] => {{1},{2},{3,7},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,3,2,4,5,6,7] => {{1},{2,3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,3,4,5,6,7,2] => {{1},{2,3,4,5,6,7}}
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 + 1
[1,4,2,3,5,6,7] => {{1},{2,4},{3},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,4,3,2,5,6,7] => {{1},{2,4},{3},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,5,2,3,4,6,7] => {{1},{2,5},{3},{4},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,5,2,4,3,6,7] => {{1},{2,5},{3},{4},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,5,3,2,4,6,7] => {{1},{2,5},{3},{4},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,5,3,4,2,6,7] => {{1},{2,5},{3},{4},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,2,3,4,5,7] => {{1},{2,6},{3},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,2,3,5,4,7] => {{1},{2,6},{3},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,2,4,3,5,7] => {{1},{2,6},{3},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,2,4,5,3,7] => {{1},{2,6},{3},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,3,2,4,5,7] => {{1},{2,6},{3},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,3,2,5,4,7] => {{1},{2,6},{3},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,3,4,2,5,7] => {{1},{2,6},{3},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,6,3,4,5,2,7] => {{1},{2,6},{3},{4},{5},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,7,2,3,4,5,6] => {{1},{2,7},{3},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,7,2,3,4,6,5] => {{1},{2,7},{3},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,7,2,3,5,4,6] => {{1},{2,7},{3},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,7,2,3,5,6,4] => {{1},{2,7},{3},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
[1,7,2,4,3,5,6] => {{1},{2,7},{3},{4},{5},{6}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000485
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 75%
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 75%
Values
[1,2] => {{1},{2}}
=> {{1},{2}}
=> [1,2] => 1
[2,1] => {{1,2}}
=> {{1,2}}
=> [2,1] => 2
[1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => 1
[1,3,2] => {{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => 2
[2,1,3] => {{1,2},{3}}
=> {{1,3},{2}}
=> [3,2,1] => 2
[2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 3
[3,1,2] => {{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => 2
[3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 1
[1,2,4,3] => {{1},{2},{3,4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => 2
[1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 2
[1,3,4,2] => {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 3
[1,4,2,3] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 2
[1,4,3,2] => {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 2
[2,1,3,4] => {{1,2},{3},{4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 2
[2,1,4,3] => {{1,2},{3,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 3
[2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 4
[2,4,1,3] => {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 3
[2,4,3,1] => {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 3
[3,1,2,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 2
[3,1,4,2] => {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 3
[3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 2
[3,2,4,1] => {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 3
[3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 2
[3,4,2,1] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 2
[4,1,2,3] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 2
[4,1,3,2] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 2
[4,2,1,3] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 2
[4,2,3,1] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 2
[4,3,1,2] => {{1,4},{2,3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2
[4,3,2,1] => {{1,4},{2,3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 2
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 2
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 3
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 2
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 2
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 4
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 3
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 3
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 2
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 3
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 2
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 2
[1,4,5,3,2] => {{1},{2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 2
[1,2,6,3,4,5,7] => {{1},{2},{3,6},{4},{5},{7}}
=> {{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ? = 2
[1,2,6,3,4,7,5] => {{1},{2},{3,6,7},{4},{5}}
=> {{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ? = 3
[1,2,6,3,5,4,7] => {{1},{2},{3,6},{4},{5},{7}}
=> {{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ? = 2
[1,2,6,3,5,7,4] => {{1},{2},{3,6,7},{4},{5}}
=> {{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ? = 3
[1,2,6,3,7,4,5] => {{1},{2},{3,6},{4},{5,7}}
=> {{1},{2,5},{3},{4,6},{7}}
=> [1,5,3,6,2,4,7] => ? = 2
[1,2,6,3,7,5,4] => {{1},{2},{3,6},{4},{5,7}}
=> {{1},{2,5},{3},{4,6},{7}}
=> [1,5,3,6,2,4,7] => ? = 2
[1,2,6,4,3,5,7] => {{1},{2},{3,6},{4},{5},{7}}
=> {{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ? = 2
[1,2,6,4,3,7,5] => {{1},{2},{3,6,7},{4},{5}}
=> {{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ? = 3
[1,2,6,4,5,3,7] => {{1},{2},{3,6},{4},{5},{7}}
=> {{1},{2,5},{3},{4},{6},{7}}
=> [1,5,3,4,2,6,7] => ? = 2
[1,2,6,4,5,7,3] => {{1},{2},{3,6,7},{4},{5}}
=> {{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => ? = 3
[1,2,6,4,7,3,5] => {{1},{2},{3,6},{4},{5,7}}
=> {{1},{2,5},{3},{4,6},{7}}
=> [1,5,3,6,2,4,7] => ? = 2
[1,2,6,4,7,5,3] => {{1},{2},{3,6},{4},{5,7}}
=> {{1},{2,5},{3},{4,6},{7}}
=> [1,5,3,6,2,4,7] => ? = 2
[1,2,6,5,3,4,7] => {{1},{2},{3,6},{4,5},{7}}
=> {{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => ? = 2
[1,2,6,5,3,7,4] => {{1},{2},{3,6,7},{4,5}}
=> {{1},{2,5,6},{3,4},{7}}
=> [1,5,4,3,6,2,7] => ? = 3
[1,2,6,5,4,3,7] => {{1},{2},{3,6},{4,5},{7}}
=> {{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => ? = 2
[1,2,6,5,4,7,3] => {{1},{2},{3,6,7},{4,5}}
=> {{1},{2,5,6},{3,4},{7}}
=> [1,5,4,3,6,2,7] => ? = 3
[1,2,6,5,7,3,4] => {{1},{2},{3,6},{4,5,7}}
=> {{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => ? = 3
[1,2,6,5,7,4,3] => {{1},{2},{3,6},{4,5,7}}
=> {{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => ? = 3
[1,2,6,7,3,4,5] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,3,5,4] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,4,3,5] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,4,5,3] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,5,3,4] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,6,7,5,4,3] => {{1},{2},{3,6},{4,7},{5}}
=> {{1},{2,5},{3,6},{4},{7}}
=> [1,5,6,4,2,3,7] => ? = 2
[1,2,7,3,4,5,6] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 2
[1,2,7,3,4,6,5] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 2
[1,2,7,3,5,4,6] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 2
[1,2,7,3,5,6,4] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 2
[1,2,7,3,6,4,5] => {{1},{2},{3,7},{4},{5,6}}
=> {{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ? = 2
[1,2,7,3,6,5,4] => {{1},{2},{3,7},{4},{5,6}}
=> {{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ? = 2
[1,2,7,4,3,5,6] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 2
[1,2,7,4,3,6,5] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 2
[1,2,7,4,5,3,6] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 2
[1,2,7,4,5,6,3] => {{1},{2},{3,7},{4},{5},{6}}
=> {{1},{2,6},{3},{4},{5},{7}}
=> [1,6,3,4,5,2,7] => ? = 2
[1,2,7,4,6,3,5] => {{1},{2},{3,7},{4},{5,6}}
=> {{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ? = 2
[1,2,7,4,6,5,3] => {{1},{2},{3,7},{4},{5,6}}
=> {{1},{2,6},{3},{4,5},{7}}
=> [1,6,3,5,4,2,7] => ? = 2
[1,2,7,5,3,4,6] => {{1},{2},{3,7},{4,5},{6}}
=> {{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ? = 2
[1,2,7,5,3,6,4] => {{1},{2},{3,7},{4,5},{6}}
=> {{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ? = 2
[1,2,7,5,4,3,6] => {{1},{2},{3,7},{4,5},{6}}
=> {{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ? = 2
[1,2,7,5,4,6,3] => {{1},{2},{3,7},{4,5},{6}}
=> {{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => ? = 2
[1,2,7,5,6,3,4] => {{1},{2},{3,7},{4,5,6}}
=> {{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => ? = 3
[1,2,7,5,6,4,3] => {{1},{2},{3,7},{4,5,6}}
=> {{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => ? = 3
[1,2,7,6,3,4,5] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,3,5,4] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,4,3,5] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,4,5,3] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,5,3,4] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,2,7,6,5,4,3] => {{1},{2},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,5},{4},{7}}
=> [1,6,5,4,3,2,7] => ? = 2
[1,3,2,4,5,6,7] => {{1},{2,3},{4},{5},{6},{7}}
=> {{1,2},{3},{4},{5},{6},{7}}
=> [2,1,3,4,5,6,7] => ? = 2
[1,3,2,4,5,7,6] => {{1},{2,3},{4},{5},{6,7}}
=> {{1,2},{3},{4},{5,6},{7}}
=> [2,1,3,4,6,5,7] => ? = 2
Description
The length of the longest cycle of a permutation.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001651The Frankl number of a lattice. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001624The breadth of a lattice.
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