Your data matches 27 different statistics following compositions of up to 3 maps.
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Mp00079: Set partitions shapeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,1]
=> 2
{{1},{2}}
=> [1,1]
=> [2]
=> 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> 3
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Mp00079: Set partitions shapeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 2
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 3
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,2},{3},{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
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000297
Mp00079: Set partitions shapeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000297: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,1]
=> 110 => 2
{{1},{2}}
=> [1,1]
=> [2]
=> 100 => 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> 1110 => 3
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> 1010 => 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> 1010 => 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> 1010 => 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1000 => 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 11110 => 4
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 10110 => 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 10110 => 1
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 1100 => 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 10110 => 1
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> 1100 => 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 1100 => 2
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 10110 => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 10000 => 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> 111110 => 5
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
Description
The number of leading ones in a binary word.
Mp00079: Set partitions shapeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00096: Binary words Foata bijectionBinary words
St000326: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> 100 => 010 => 2
{{1},{2}}
=> [1,1]
=> 110 => 110 => 1
{{1,2,3}}
=> [3]
=> 1000 => 0010 => 3
{{1,2},{3}}
=> [2,1]
=> 1010 => 1100 => 1
{{1,3},{2}}
=> [2,1]
=> 1010 => 1100 => 1
{{1},{2,3}}
=> [2,1]
=> 1010 => 1100 => 1
{{1},{2},{3}}
=> [1,1,1]
=> 1110 => 1110 => 1
{{1,2,3,4}}
=> [4]
=> 10000 => 00010 => 4
{{1,2,3},{4}}
=> [3,1]
=> 10010 => 10100 => 1
{{1,2,4},{3}}
=> [3,1]
=> 10010 => 10100 => 1
{{1,2},{3,4}}
=> [2,2]
=> 1100 => 0110 => 2
{{1,2},{3},{4}}
=> [2,1,1]
=> 10110 => 11010 => 1
{{1,3,4},{2}}
=> [3,1]
=> 10010 => 10100 => 1
{{1,3},{2,4}}
=> [2,2]
=> 1100 => 0110 => 2
{{1,3},{2},{4}}
=> [2,1,1]
=> 10110 => 11010 => 1
{{1,4},{2,3}}
=> [2,2]
=> 1100 => 0110 => 2
{{1},{2,3,4}}
=> [3,1]
=> 10010 => 10100 => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> 10110 => 11010 => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 10110 => 11010 => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 10110 => 11010 => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 10110 => 11010 => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 11110 => 11110 => 1
{{1,2,3,4,5}}
=> [5]
=> 100000 => 000010 => 5
{{1,2,3,4},{5}}
=> [4,1]
=> 100010 => 100100 => 1
{{1,2,3,5},{4}}
=> [4,1]
=> 100010 => 100100 => 1
{{1,2,3},{4,5}}
=> [3,2]
=> 10100 => 01100 => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 100110 => 101010 => 1
{{1,2,4,5},{3}}
=> [4,1]
=> 100010 => 100100 => 1
{{1,2,4},{3,5}}
=> [3,2]
=> 10100 => 01100 => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 100110 => 101010 => 1
{{1,2,5},{3,4}}
=> [3,2]
=> 10100 => 01100 => 2
{{1,2},{3,4,5}}
=> [3,2]
=> 10100 => 01100 => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 11010 => 11100 => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 100110 => 101010 => 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 11010 => 11100 => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 11010 => 11100 => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 101110 => 110110 => 1
{{1,3,4,5},{2}}
=> [4,1]
=> 100010 => 100100 => 1
{{1,3,4},{2,5}}
=> [3,2]
=> 10100 => 01100 => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 100110 => 101010 => 1
{{1,3,5},{2,4}}
=> [3,2]
=> 10100 => 01100 => 2
{{1,3},{2,4,5}}
=> [3,2]
=> 10100 => 01100 => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 11010 => 11100 => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 100110 => 101010 => 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 11010 => 11100 => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 11010 => 11100 => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 101110 => 110110 => 1
{{1,4,5},{2,3}}
=> [3,2]
=> 10100 => 01100 => 2
{{1,4},{2,3,5}}
=> [3,2]
=> 10100 => 01100 => 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 11010 => 11100 => 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
Mp00079: Set partitions shapeInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [[1,2]]
=> [2] => 2
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> [3] => 3
{{1,2},{3}}
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
{{1,3},{2}}
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
{{1},{2,3}}
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> [4] => 4
{{1,2,3},{4}}
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
{{1,2,4},{3}}
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1,2},{3},{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
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1},{2,3,4}}
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 5
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,2,7},{3,8},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,2,7},{3,8},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,2,7},{3,8},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,2,7},{3,9},{4,10},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,2,8},{3,9},{4,10},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,3,8},{2,7},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,3,8},{2,7},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,3,8},{2,7},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,3,9},{2,7},{4,10},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,3,9},{2,8},{4,10},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,4,9},{2,7},{3,8},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,4,9},{2,7},{3,8},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,4,10},{2,7},{3,8},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,4,10},{2,7},{3,9},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,4,10},{2,8},{3,9},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,5,10},{2,7},{3,8},{4,9},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,5,11},{2,7},{3,8},{4,9},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,5,11},{2,7},{3,8},{4,10},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,5,11},{2,7},{3,9},{4,10},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,5,11},{2,8},{3,9},{4,10},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,6,12},{2,7},{3,8},{4,9},{5,10},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,6,12},{2,7},{3,8},{4,9},{5,11},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,6,12},{2,7},{3,8},{4,10},{5,11},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,6,12},{2,7},{3,9},{4,10},{5,11},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,6,12},{2,8},{3,9},{4,10},{5,11},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,3,8},{2,9},{4,10},{5,11},{6},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,4,8},{2,9},{3,10},{5},{6,11},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,5,8},{2,9},{3},{4,10},{6,11},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,6,11},{2,7},{3,8},{4,9},{5},{10,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,6,10},{2,7},{3,8},{4},{5,11},{9,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,6,9},{2,7},{3},{4,10},{5,11},{8,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,6,8},{2},{3,9},{4,10},{5,11},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
Description
The first part of an integer composition.
Mp00079: Set partitions shapeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
{{1,2},{3}}
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
{{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
{{1},{2,3}}
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
{{1,2},{3},{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
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
{{1,2,7},{3,8},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,2,7},{3,8},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,2,7},{3,8},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,2,7},{3,9},{4,10},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,2,8},{3,9},{4,10},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,3,8},{2,7},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,3,8},{2,7},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,3,8},{2,7},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,3,9},{2,7},{4,10},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,3,9},{2,8},{4,10},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,4,9},{2,7},{3,8},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,4,9},{2,7},{3,8},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,4,10},{2,7},{3,8},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,4,10},{2,7},{3,9},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,4,10},{2,8},{3,9},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,5,10},{2,7},{3,8},{4,9},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,5,11},{2,7},{3,8},{4,9},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,5,11},{2,7},{3,8},{4,10},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,5,11},{2,7},{3,9},{4,10},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,5,11},{2,8},{3,9},{4,10},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,6,12},{2,7},{3,8},{4,9},{5,10},{11}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,6,12},{2,7},{3,8},{4,9},{5,11},{10}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,6,12},{2,7},{3,8},{4,10},{5,11},{9}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,6,12},{2,7},{3,9},{4,10},{5,11},{8}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,6,12},{2,8},{3,9},{4,10},{5,11},{7}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,3,8},{2,9},{4,10},{5,11},{6},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,4,8},{2,9},{3,10},{5},{6,11},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,5,8},{2,9},{3},{4,10},{6,11},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,6,11},{2,7},{3,8},{4,9},{5},{10,12}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,6,10},{2,7},{3,8},{4},{5,11},{9,12}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,6,9},{2,7},{3},{4,10},{5,11},{8,12}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
{{1,6,8},{2},{3,9},{4,10},{5,11},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,4,6,8,10,12],[2,5,7,9,11],[3]]
=> ? = 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mp00079: Set partitions shapeInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00084: Standard tableaux conjugateStandard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
{{1,2},{3}}
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
{{1,3},{2}}
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
{{1},{2,3}}
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,2,7},{3,8},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,2,7},{3,8},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,2,7},{3,8},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,2,7},{3,9},{4,10},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,2,8},{3,9},{4,10},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,3,8},{2,7},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,3,8},{2,7},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,3,8},{2,7},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,3,9},{2,7},{4,10},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,3,9},{2,8},{4,10},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,4,9},{2,7},{3,8},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,4,9},{2,7},{3,8},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,4,10},{2,7},{3,8},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,4,10},{2,7},{3,9},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,4,10},{2,8},{3,9},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,5,10},{2,7},{3,8},{4,9},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,5,11},{2,7},{3,8},{4,9},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,5,11},{2,7},{3,8},{4,10},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,5,11},{2,7},{3,9},{4,10},{6,12},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,5,11},{2,8},{3,9},{4,10},{6,12},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,6,12},{2,7},{3,8},{4,9},{5,10},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,6,12},{2,7},{3,8},{4,9},{5,11},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,6,12},{2,7},{3,8},{4,10},{5,11},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,6,12},{2,7},{3,9},{4,10},{5,11},{8}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,6,12},{2,8},{3,9},{4,10},{5,11},{7}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,3,8},{2,9},{4,10},{5,11},{6},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,4,8},{2,9},{3,10},{5},{6,11},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,5,8},{2,9},{3},{4,10},{6,11},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,6,11},{2,7},{3,8},{4,9},{5},{10,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,6,10},{2,7},{3,8},{4},{5,11},{9,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,6,9},{2,7},{3},{4,10},{5,11},{8,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,6,8},{2},{3,9},{4,10},{5,11},{7,12}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Mp00079: Set partitions shapeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00097: Binary words delta morphismInteger compositions
St000383: Integer compositions ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> 100 => [1,2] => 2
{{1},{2}}
=> [1,1]
=> 110 => [2,1] => 1
{{1,2,3}}
=> [3]
=> 1000 => [1,3] => 3
{{1,2},{3}}
=> [2,1]
=> 1010 => [1,1,1,1] => 1
{{1,3},{2}}
=> [2,1]
=> 1010 => [1,1,1,1] => 1
{{1},{2,3}}
=> [2,1]
=> 1010 => [1,1,1,1] => 1
{{1},{2},{3}}
=> [1,1,1]
=> 1110 => [3,1] => 1
{{1,2,3,4}}
=> [4]
=> 10000 => [1,4] => 4
{{1,2,3},{4}}
=> [3,1]
=> 10010 => [1,2,1,1] => 1
{{1,2,4},{3}}
=> [3,1]
=> 10010 => [1,2,1,1] => 1
{{1,2},{3,4}}
=> [2,2]
=> 1100 => [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1,3,4},{2}}
=> [3,1]
=> 10010 => [1,2,1,1] => 1
{{1,3},{2,4}}
=> [2,2]
=> 1100 => [2,2] => 2
{{1,3},{2},{4}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1,4},{2,3}}
=> [2,2]
=> 1100 => [2,2] => 2
{{1},{2,3,4}}
=> [3,1]
=> 10010 => [1,2,1,1] => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 11110 => [4,1] => 1
{{1,2,3,4,5}}
=> [5]
=> 100000 => [1,5] => 5
{{1,2,3,4},{5}}
=> [4,1]
=> 100010 => [1,3,1,1] => 1
{{1,2,3,5},{4}}
=> [4,1]
=> 100010 => [1,3,1,1] => 1
{{1,2,3},{4,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,2,4,5},{3}}
=> [4,1]
=> 100010 => [1,3,1,1] => 1
{{1,2,4},{3,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,2,5},{3,4}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,2},{3,4,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
{{1,3,4,5},{2}}
=> [4,1]
=> 100010 => [1,3,1,1] => 1
{{1,3,4},{2,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,3,5},{2,4}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,3},{2,4,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
{{1,4,5},{2,3}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,4},{2,3,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}}
=> [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
{{1,2,7},{3,8},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,2,7},{3,8},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,2,7},{3,8},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,2,7},{3,9},{4,10},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,2,8},{3,9},{4,10},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,3,8},{2,7},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,3,8},{2,7},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,3,8},{2,7},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,3,9},{2,7},{4,10},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,3,9},{2,8},{4,10},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,4,9},{2,7},{3,8},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,4,9},{2,7},{3,8},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,4,10},{2,7},{3,8},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,4,10},{2,7},{3,9},{5,11},{6,12},{8}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,4,10},{2,8},{3,9},{5,11},{6,12},{7}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,5,10},{2,7},{3,8},{4,9},{6,12},{11}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,5,11},{2,7},{3,8},{4,9},{6,12},{10}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,5,11},{2,7},{3,8},{4,10},{6,12},{9}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,5,11},{2,7},{3,9},{4,10},{6,12},{8}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,5,11},{2,8},{3,9},{4,10},{6,12},{7}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,6,12},{2,7},{3,8},{4,9},{5,10},{11}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,6,12},{2,7},{3,8},{4,9},{5,11},{10}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,6,12},{2,7},{3,8},{4,10},{5,11},{9}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,6,12},{2,7},{3,9},{4,10},{5,11},{8}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,6,12},{2,8},{3,9},{4,10},{5,11},{7}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,3,8},{2,9},{4,10},{5,11},{6},{7,12}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,4,8},{2,9},{3,10},{5},{6,11},{7,12}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,5,8},{2,9},{3},{4,10},{6,11},{7,12}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,6,11},{2,7},{3,8},{4,9},{5},{10,12}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,6,10},{2,7},{3,8},{4},{5,11},{9,12}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,6,9},{2,7},{3},{4,10},{5,11},{8,12}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,6,8},{2},{3,9},{4,10},{5,11},{7,12}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
Description
The last part of an integer composition.
Mp00128: Set partitions to compositionInteger compositions
St000657: Integer compositions ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => 2
{{1},{2}}
=> [1,1] => 1
{{1,2,3}}
=> [3] => 3
{{1,2},{3}}
=> [2,1] => 1
{{1,3},{2}}
=> [2,1] => 1
{{1},{2,3}}
=> [1,2] => 1
{{1},{2},{3}}
=> [1,1,1] => 1
{{1,2,3,4}}
=> [4] => 4
{{1,2,3},{4}}
=> [3,1] => 1
{{1,2,4},{3}}
=> [3,1] => 1
{{1,2},{3,4}}
=> [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,1] => 1
{{1,3,4},{2}}
=> [3,1] => 1
{{1,3},{2,4}}
=> [2,2] => 2
{{1,3},{2},{4}}
=> [2,1,1] => 1
{{1,4},{2,3}}
=> [2,2] => 2
{{1},{2,3,4}}
=> [1,3] => 1
{{1},{2,3},{4}}
=> [1,2,1] => 1
{{1,4},{2},{3}}
=> [2,1,1] => 1
{{1},{2,4},{3}}
=> [1,2,1] => 1
{{1},{2},{3,4}}
=> [1,1,2] => 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1
{{1,2,3,4,5}}
=> [5] => 5
{{1,2,3,4},{5}}
=> [4,1] => 1
{{1,2,3,5},{4}}
=> [4,1] => 1
{{1,2,3},{4,5}}
=> [3,2] => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => 1
{{1,2,4,5},{3}}
=> [4,1] => 1
{{1,2,4},{3,5}}
=> [3,2] => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => 1
{{1,2,5},{3,4}}
=> [3,2] => 2
{{1,2},{3,4,5}}
=> [2,3] => 2
{{1,2},{3,4},{5}}
=> [2,2,1] => 1
{{1,2,5},{3},{4}}
=> [3,1,1] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => 1
{{1,2},{3},{4,5}}
=> [2,1,2] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 1
{{1,3,4,5},{2}}
=> [4,1] => 1
{{1,3,4},{2,5}}
=> [3,2] => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => 1
{{1,3,5},{2,4}}
=> [3,2] => 2
{{1,3},{2,4,5}}
=> [2,3] => 2
{{1,3},{2,4},{5}}
=> [2,2,1] => 1
{{1,3,5},{2},{4}}
=> [3,1,1] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => 1
{{1,3},{2},{4,5}}
=> [2,1,2] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 1
{{1,4,5},{2,3}}
=> [3,2] => 2
{{1,4},{2,3,5}}
=> [2,3] => 2
{{1,4},{2,3},{5}}
=> [2,2,1] => 1
{{1,6},{2,7},{3,8},{4,9},{5,10}}
=> [2,2,2,2,2] => ? = 2
{{1,2,3,4,5},{6,7,8,9,10}}
=> [5,5] => ? = 5
{{1,2,3,4,5,6,7},{8,9,10}}
=> [7,3] => ? = 3
{{1,2,3,4,5,6},{7,8,9,10}}
=> [6,4] => ? = 4
{{1,2,3,4},{5,6,7,8,9,10}}
=> [4,6] => ? = 4
{{1,2,4,8},{3,6},{5,10},{7},{9}}
=> [4,2,2,1,1] => ? = 1
{{1,9},{2,8},{3,7},{4,6},{5},{10}}
=> [2,2,2,2,1,1] => ? = 1
{{1},{2,10},{3,9},{4,8},{5,7},{6}}
=> [1,2,2,2,2,1] => ? = 1
{{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}}
=> [2,2,2,2,2,2] => ? = 2
{{1,2,3},{4,5,6,7,8,9,10}}
=> [3,7] => ? = 3
{{1,2,4,9},{3,6},{5,10},{7},{8}}
=> [4,2,2,1,1] => ? = 1
{{1,2,4,8},{3,7},{5,10},{6},{9}}
=> [4,2,2,1,1] => ? = 1
{{1,2,4,9},{3,7},{5,10},{6},{8}}
=> [4,2,2,1,1] => ? = 1
{{1,2,4,9},{3,8},{5,10},{6},{7}}
=> [4,2,2,1,1] => ? = 1
{{1,2,5,10},{3,6},{4,8},{7},{9}}
=> [4,2,2,1,1] => ? = 1
{{1,2,5,10},{3,6},{4,9},{7},{8}}
=> [4,2,2,1,1] => ? = 1
{{1,2,5,10},{3,7},{4,8},{6},{9}}
=> [4,2,2,1,1] => ? = 1
{{1,2,5,10},{3,7},{4,9},{6},{8}}
=> [4,2,2,1,1] => ? = 1
{{1,2,5,10},{3,8},{4,9},{6},{7}}
=> [4,2,2,1,1] => ? = 1
{{1,2,6},{3,7},{4,8},{5,10},{9}}
=> [3,2,2,2,1] => ? = 1
{{1,2,6},{3,7},{4,9},{5,10},{8}}
=> [3,2,2,2,1] => ? = 1
{{1,2,6},{3,8},{4,9},{5,10},{7}}
=> [3,2,2,2,1] => ? = 1
{{1,2,7},{3,8},{4,9},{5,10},{6}}
=> [3,2,2,2,1] => ? = 1
{{1,3,6},{2,4,8},{5,10},{7},{9}}
=> [3,3,2,1,1] => ? = 1
{{1,3,6},{2,4,9},{5,10},{7},{8}}
=> [3,3,2,1,1] => ? = 1
{{1,3,7},{2,4,8},{5,10},{6},{9}}
=> [3,3,2,1,1] => ? = 1
{{1,3,7},{2,4,9},{5,10},{6},{8}}
=> [3,3,2,1,1] => ? = 1
{{1,3,8},{2,4,9},{5,10},{6},{7}}
=> [3,3,2,1,1] => ? = 1
{{1,3,6},{2,5,10},{4,8},{7},{9}}
=> [3,3,2,1,1] => ? = 1
{{1,3,6},{2,5,10},{4,9},{7},{8}}
=> [3,3,2,1,1] => ? = 1
{{1,3,7},{2,5,10},{4,8},{6},{9}}
=> [3,3,2,1,1] => ? = 1
{{1,3,7},{2,5,10},{4,9},{6},{8}}
=> [3,3,2,1,1] => ? = 1
{{1,3,8},{2,5,10},{4,9},{6},{7}}
=> [3,3,2,1,1] => ? = 1
{{1,3,7},{2,6},{4,8},{5,10},{9}}
=> [3,2,2,2,1] => ? = 1
{{1,3,7},{2,6},{4,9},{5,10},{8}}
=> [3,2,2,2,1] => ? = 1
{{1,3,8},{2,6},{4,9},{5,10},{7}}
=> [3,2,2,2,1] => ? = 1
{{1,3,8},{2,7},{4,9},{5,10},{6}}
=> [3,2,2,2,1] => ? = 1
{{1,4,8},{2,5,10},{3,6},{7},{9}}
=> [3,3,2,1,1] => ? = 1
{{1,4,9},{2,5,10},{3,6},{7},{8}}
=> [3,3,2,1,1] => ? = 1
{{1,4,8},{2,5,10},{3,7},{6},{9}}
=> [3,3,2,1,1] => ? = 1
{{1,4,9},{2,5,10},{3,7},{6},{8}}
=> [3,3,2,1,1] => ? = 1
{{1,4,9},{2,5,10},{3,8},{6},{7}}
=> [3,3,2,1,1] => ? = 1
{{1,4,8},{2,6},{3,7},{5,10},{9}}
=> [3,2,2,2,1] => ? = 1
{{1,4,9},{2,6},{3,7},{5,10},{8}}
=> [3,2,2,2,1] => ? = 1
{{1,4,9},{2,6},{3,8},{5,10},{7}}
=> [3,2,2,2,1] => ? = 1
{{1,4,9},{2,7},{3,8},{5,10},{6}}
=> [3,2,2,2,1] => ? = 1
{{1,5,10},{2,6},{3,7},{4,8},{9}}
=> [3,2,2,2,1] => ? = 1
{{1,5,10},{2,6},{3,7},{4,9},{8}}
=> [3,2,2,2,1] => ? = 1
{{1,5,10},{2,6},{3,8},{4,9},{7}}
=> [3,2,2,2,1] => ? = 1
{{1,5,10},{2,7},{3,8},{4,9},{6}}
=> [3,2,2,2,1] => ? = 1
Description
The smallest part of an integer composition.
Matching statistic: St001803
Mp00079: Set partitions shapeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 94% values known / values provided: 94%distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
{{1},{2}}
=> [1,1]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0 = 1 - 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0 = 1 - 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0 = 1 - 1
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0 = 1 - 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 2 - 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0 = 1 - 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 2 - 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 2 - 1
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 2 - 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 2 - 1
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 2 - 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0 = 1 - 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 2 - 1
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1 = 2 - 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0 = 1 - 1
{{1,6},{2,7},{3,8},{4,9},{5,10}}
=> [2,2,2,2,2]
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> ? = 2 - 1
{{1,2,3,4,5,8},{6,7},{9}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1},{2,3,4,5,6,9},{7,8}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,2,3,4,5},{6,7,8,9,10}}
=> [5,5]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 5 - 1
{{1,2,3,4,5,6,7},{8,9}}
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 2 - 1
{{1,2,3,4,5,6},{7,8,9}}
=> [6,3]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 3 - 1
{{1,2,3,4,5,6},{7,8},{9}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,2,3,4,5},{6,7,8,9}}
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 4 - 1
{{1,2,3,4,5},{6,7,8},{9}}
=> [5,3,1]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 1 - 1
{{1,2,3,4,5,6,7},{8,9,10}}
=> [7,3]
=> [2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4],[5],[7],[9]]
=> ? = 3 - 1
{{1,2,3,4,5,6},{7,8,9,10}}
=> [6,4]
=> [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 4 - 1
{{1},{2,5,6,7,8,9},{3,4}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,4,5,6,7,8},{2,3},{9}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,2,3,4},{5,6,7,8,9,10}}
=> [6,4]
=> [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 4 - 1
{{1,2,4,8},{3,6},{5,10},{7},{9}}
=> [4,2,2,1,1]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> ? = 1 - 1
{{1,2,3,9},{4,5},{6},{7},{8}}
=> [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
{{1,2,3,4,9},{5,6},{7},{8}}
=> [5,2,1,1]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 1 - 1
{{1,2,8,9},{3,7},{4},{5},{6}}
=> [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
{{1,2,9},{3,8},{4,5},{6},{7}}
=> [3,2,2,1,1]
=> [5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> ? = 1 - 1
{{1,2,5,9},{3,4},{6},{7},{8}}
=> [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
{{1,9},{2,3,4,5,6,7,8}}
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 2 - 1
{{1,8},{2,3,4,5,6,7},{9}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,7},{2,3,4,5,6},{8},{9}}
=> [5,2,1,1]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 1 - 1
{{1,6},{2,3,4,5},{7},{8},{9}}
=> [4,2,1,1,1]
=> [5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> ? = 1 - 1
{{1,5},{2,3,4},{6},{7},{8},{9}}
=> [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
{{1,9},{2,3,4,5,6,8},{7}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,9},{2,3,4,5,8},{6,7}}
=> [5,2,2]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 2 - 1
{{1,9},{2,3,4,5,7,8},{6}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,9},{2,8},{3,7},{4,6},{5},{10}}
=> [2,2,2,2,1,1]
=> [6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> ? = 1 - 1
{{1},{2,10},{3,9},{4,8},{5,7},{6}}
=> [2,2,2,2,1,1]
=> [6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> ? = 1 - 1
{{1,2},{3,4,5,6,7,8},{9}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}}
=> [2,2,2,2,2,2]
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 2 - 1
{{1,3,4,5,6,7},{2,9},{8}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,2,3,4,5,7},{6,8},{9}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,3,4,5,6,8},{2,9},{7}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,3,4,5,6,9},{2,8},{7}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,2},{3,4,5,6,7,8,9}}
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 2 - 1
{{1,2,3},{4,5,6,7,8,9}}
=> [6,3]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 3 - 1
{{1,2,3,4},{5,6,7,8,9}}
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 4 - 1
{{1,2},{3,5,6,7,8,9},{4}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,2,4,5,6,7},{3},{8,9}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,2,3},{4,5,6,7,8,9,10}}
=> [7,3]
=> [2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4],[5],[7],[9]]
=> ? = 3 - 1
{{1,2,3},{4,5,7,8,9},{6}}
=> [5,3,1]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 1 - 1
{{1,2,3,5,6},{4},{7,8,9}}
=> [5,3,1]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 1 - 1
{{1,2,3,4},{5,6,7,9},{8}}
=> [4,4,1]
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> ? = 1 - 1
{{1},{2,3,4,5,6,8},{7,9}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,2,7},{3},{4},{5},{6},{8,9}}
=> [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
{{1,2},{3,4,9},{5},{6},{7},{8}}
=> [3,2,1,1,1,1]
=> [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1 - 1
{{1,2,4,9},{3,6},{5,10},{7},{8}}
=> [4,2,2,1,1]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> ? = 1 - 1
{{1,2,4,8},{3,7},{5,10},{6},{9}}
=> [4,2,2,1,1]
=> [5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> ? = 1 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000667The greatest common divisor of the parts of the partition. 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. St000990The first ascent of a permutation. St000655The length of the minimal rise of a Dyck path. St000700The protection number of an ordered tree. 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. St000654The first descent of a permutation. St001075The minimal size of a block of a set partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000090The variation of a composition. St000260The radius of a connected graph. St000487The length of the shortest cycle of a permutation. St000210Minimum over maximum difference of elements in cycles. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000314The number of left-to-right-maxima of a permutation.