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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1]
=> 10 => 1
{{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
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> 10 => 10 => 1
{{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
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 —shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [[1]]
=> [1] => 1
{{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
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [[1]]
=> [1] => 1
{{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]
=> [[1,2],[3]]
=> [2,1] => 1
{{1,3},{2}}
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
{{1},{2,3}}
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 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]
=> [[1,2,3],[4]]
=> [3,1] => 1
{{1,2,4},{3}}
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
{{1,3,4},{2}}
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1},{2,3,4}}
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 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]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
Description
The last part of an integer composition.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1]
=> [[1]]
=> 1
{{1,2}}
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
{{1},{2}}
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000993
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1]
=> ? = 1
{{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.
Matching statistic: St001038
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 98%●distinct values known / distinct values provided: 71%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 98%●distinct values known / distinct values provided: 71%
Values
{{1}}
=> [1]
=> []
=> []
=> ? = 1
{{1,2}}
=> [2]
=> []
=> []
=> ? = 2
{{1},{2}}
=> [1,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3}}
=> [3]
=> []
=> []
=> ? = 3
{{1,2},{3}}
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3},{2}}
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3}}
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> []
=> []
=> ? = 4
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> []
=> ? = 5
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2,3,4,5,6}}
=> [6]
=> []
=> []
=> ? = 6
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,7}}
=> [7]
=> []
=> []
=> ? = 7
{{1,2,3,4,5,6},{7}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,7},{6}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,6,7},{5}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,5,6,7},{4}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4,5,6,7},{3}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4,5,6,7},{2}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4,5,6,7}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4,5,6,7,8}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4,5,6,7,8},{2}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4,5,6,7,8},{3}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,5,6,7,8},{4}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,6,7,8},{5}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,7,8},{6}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,7},{8}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,8},{7}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1,2,3,4,5,6,7,8},{9}}
=> [8,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4,5,6,7,8,9}}
=> [8,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4,5,6,7,8,9,10}}
=> [9,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4,5,6,7,8,9},{2}}
=> [8,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,7,9},{8}}
=> [8,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,7,8,10},{9}}
=> [9,1]
=> [1]
=> [1,0]
=> ? = 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000745
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [[1]]
=> [[1]]
=> 1
{{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
{{1,3,7},{2,4,8},{5,10},{6,12},{9},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,4,8},{5,11},{6,12},{9},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,4,9},{5,10},{6,12},{8},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,4,9},{5,11},{6,12},{8},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,4,10},{5,11},{6,12},{8},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,8},{2,4,9},{5,10},{6,12},{7},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,8},{2,4,9},{5,11},{6,12},{7},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,8},{2,4,10},{5,11},{6,12},{7},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,9},{2,4,10},{5,11},{6,12},{7},{8}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,5,10},{4,8},{6,12},{9},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,5,11},{4,8},{6,12},{9},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,5,10},{4,9},{6,12},{8},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,5,11},{4,9},{6,12},{8},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,5,11},{4,10},{6,12},{8},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,8},{2,5,10},{4,9},{6,12},{7},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,8},{2,5,11},{4,9},{6,12},{7},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,8},{2,5,11},{4,10},{6,12},{7},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,9},{2,5,11},{4,10},{6,12},{7},{8}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,6,12},{4,8},{5,10},{9},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,6,12},{4,8},{5,11},{9},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,6,12},{4,9},{5,10},{8},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,6,12},{4,9},{5,11},{8},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,7},{2,6,12},{4,10},{5,11},{8},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,8},{2,6,12},{4,9},{5,10},{7},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,8},{2,6,12},{4,9},{5,11},{7},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,8},{2,6,12},{4,10},{5,11},{7},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,3,9},{2,6,12},{4,10},{5,11},{7},{8}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,8},{2,5,10},{3,7},{6,12},{9},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,8},{2,5,11},{3,7},{6,12},{9},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,9},{2,5,10},{3,7},{6,12},{8},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,9},{2,5,11},{3,7},{6,12},{8},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,10},{2,5,11},{3,7},{6,12},{8},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,9},{2,5,10},{3,8},{6,12},{7},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,9},{2,5,11},{3,8},{6,12},{7},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,10},{2,5,11},{3,8},{6,12},{7},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,10},{2,5,11},{3,9},{6,12},{7},{8}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,8},{2,6,12},{3,7},{5,10},{9},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,8},{2,6,12},{3,7},{5,11},{9},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,9},{2,6,12},{3,7},{5,10},{8},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,9},{2,6,12},{3,7},{5,11},{8},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,10},{2,6,12},{3,7},{5,11},{8},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,9},{2,6,12},{3,8},{5,10},{7},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,9},{2,6,12},{3,8},{5,11},{7},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,10},{2,6,12},{3,8},{5,11},{7},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,4,10},{2,6,12},{3,9},{5,11},{7},{8}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,5,10},{2,6,12},{3,7},{4,8},{9},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,5,11},{2,6,12},{3,7},{4,8},{9},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,5,10},{2,6,12},{3,7},{4,9},{8},{11}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,5,11},{2,6,12},{3,7},{4,9},{8},{10}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
{{1,5,11},{2,6,12},{3,7},{4,10},{8},{9}}
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000657
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
St000657: Integer compositions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 1
{{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},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => ? = 1
{{1},{2,3,4,5,6,7,8,9,10}}
=> [1,9] => ? = 1
{{1,2,3,4,5,6,7,9},{8},{10}}
=> [8,1,1] => ? = 1
{{1},{2,3,4,5,6,7,8,10},{9}}
=> [1,8,1] => ? = 1
{{1,2,3,4,5},{6,7,8,9,10}}
=> [5,5] => ? = 5
{{1,2,3,4,5,6,7,8},{9},{10}}
=> [8,1,1] => ? = 1
{{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => ? = 1
{{1},{2,4,5,6,7,8,9,10},{3}}
=> [1,8,1] => ? = 1
{{1,3,4,5,6,7,8,9},{2},{10}}
=> [8,1,1] => ? = 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
{{1,2,3,4,5,6,7,8,10},{9}}
=> [9,1] => ? = 1
{{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}}
=> [2,2,2,2,2,2] => ? = 2
{{1,2,3,4,5,6,7,9,10},{8}}
=> [9,1] => ? = 1
{{1,10},{2,3},{4},{5},{6},{7},{8},{9}}
=> [2,2,1,1,1,1,1,1] => ? = 1
{{1,10},{2,9},{3},{4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1,1] => ? = 1
{{1,3,7},{2,4,8},{5,10},{6,12},{9},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,4,8},{5,11},{6,12},{9},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,4,9},{5,10},{6,12},{8},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,4,9},{5,11},{6,12},{8},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,4,10},{5,11},{6,12},{8},{9}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,8},{2,4,9},{5,10},{6,12},{7},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,8},{2,4,9},{5,11},{6,12},{7},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,8},{2,4,10},{5,11},{6,12},{7},{9}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,9},{2,4,10},{5,11},{6,12},{7},{8}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,5,10},{4,8},{6,12},{9},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,5,11},{4,8},{6,12},{9},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,5,10},{4,9},{6,12},{8},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,5,11},{4,9},{6,12},{8},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,5,11},{4,10},{6,12},{8},{9}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,8},{2,5,10},{4,9},{6,12},{7},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,8},{2,5,11},{4,9},{6,12},{7},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,8},{2,5,11},{4,10},{6,12},{7},{9}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,9},{2,5,11},{4,10},{6,12},{7},{8}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,6,12},{4,8},{5,10},{9},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,6,12},{4,8},{5,11},{9},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,6,12},{4,9},{5,10},{8},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,6,12},{4,9},{5,11},{8},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,7},{2,6,12},{4,10},{5,11},{8},{9}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,8},{2,6,12},{4,9},{5,10},{7},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,8},{2,6,12},{4,9},{5,11},{7},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,8},{2,6,12},{4,10},{5,11},{7},{9}}
=> [3,3,2,2,1,1] => ? = 1
{{1,3,9},{2,6,12},{4,10},{5,11},{7},{8}}
=> [3,3,2,2,1,1] => ? = 1
{{1,4,8},{2,5,10},{3,7},{6,12},{9},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,4,8},{2,5,11},{3,7},{6,12},{9},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,4,9},{2,5,10},{3,7},{6,12},{8},{11}}
=> [3,3,2,2,1,1] => ? = 1
{{1,4,9},{2,5,11},{3,7},{6,12},{8},{10}}
=> [3,3,2,2,1,1] => ? = 1
{{1,4,10},{2,5,11},{3,7},{6,12},{8},{9}}
=> [3,3,2,2,1,1] => ? = 1
{{1,4,9},{2,5,10},{3,8},{6,12},{7},{11}}
=> [3,3,2,2,1,1] => ? = 1
Description
The smallest part of an integer composition.
Matching statistic: St001803
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
{{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},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 1 - 1
{{1,2,3,4,5,6,7,8},{9}}
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 1
{{1},{2,3,4,5,6,7,8,9}}
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 1
{{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
{{1,2,3,4,5,6,8},{7},{9}}
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 1 - 1
{{1},{2,3,4,5,6,7,9},{8}}
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 1 - 1
{{1},{2,3,4,5,6,7,8,9,10}}
=> [9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
{{1,2,3,4,5,6,7,9},{8},{10}}
=> [8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 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
{{1,2,3,4,5,7,8},{6},{9}}
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[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},{2,3,4,5,6,8,9},{7}}
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 1 - 1
{{1},{2,3,4,5,6,7,8,10},{9}}
=> [8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 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}}
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 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
{{1,2,3,4,5,6,7,8},{9},{10}}
=> [8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 1
{{1,3,4,5,6,7,8,9},{2}}
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 1
{{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
{{1},{2,4,5,6,7,8,9},{3}}
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 1 - 1
{{1,3,4,5,6,7,8},{2},{9}}
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 1 - 1
{{1},{2,4,5,6,7,8,9,10},{3}}
=> [8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 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
{{1,3,4,5,6,7,8,9},{2},{10}}
=> [8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 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
{{1,9},{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},{2,3,4,5,7,8},{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
{{1,2,3,4,5,6,7,9},{8}}
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 1
{{1,2,3,4,5,6,7,8,10},{9}}
=> [9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 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
{{1,3,4,5,6,7,9},{2},{8}}
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[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}}
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 1 - 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},{2,3,4,5,6,8},{7,9}}
=> [6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 1 - 1
{{1,2,3,4,5,6,8,9},{7}}
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 1
{{1,2,4,5,6,7,8,9},{3}}
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1 - 1
{{1,2,3,4,5,6,7,9,10},{8}}
=> [9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1 - 1
{{1,10},{2,3},{4},{5},{6},{7},{8},{9}}
=> [2,2,1,1,1,1,1,1]
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> ? = 1 - 1
{{1,10},{2,9},{3},{4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1,1]
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> ? = 1 - 1
{{1,3,7},{2,4,8},{5,10},{6,12},{9},{11}}
=> [3,3,2,2,1,1]
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> ? = 1 - 1
{{1,3,7},{2,4,8},{5,11},{6,12},{9},{10}}
=> [3,3,2,2,1,1]
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> ? = 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000655The length of the minimal rise of a Dyck path. 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. St000700The protection number of an ordered tree. 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. St000210Minimum over maximum difference of elements in cycles. St000487The length of the shortest cycle of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000314The number of left-to-right-maxima of a permutation.
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