Processing math: 100%

Your data matches 148 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000993
(load all 6 compositions to match this statistic)
Mapping
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 1
[1,1]
 => 2
[3]
 => 1
[2,1]
 => 1
[1,1,1]
 => 3
[4]
 => 1
[3,1]
 => 1
[2,2]
 => 2
[2,1,1]
 => 1
[1,1,1,1]
 => 4
[5]
 => 1
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 1
[2,2,1]
 => 2
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 5
[6]
 => 1
[5,1]
 => 1
[4,2]
 => 1
[4,1,1]
 => 1
[3,3]
 => 2
[3,2,1]
 => 1
[3,1,1,1]
 => 1
[2,2,2]
 => 3
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 6
[7]
 => 1
[6,1]
 => 1
[5,2]
 => 1
[5,1,1]
 => 1
[4,3]
 => 1
[4,2,1]
 => 1
[4,1,1,1]
 => 1
[3,3,1]
 => 2
[3,2,2]
 => 1
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 3
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 7
[8]
 => 1
[7,1]
 => 1
[6,2]
 => 1
[6,1,1]
 => 1
[5,3]
 => 1
[5,2,1]
 => 1
[5,1,1,1]
 => 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000297
(load all 13 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => => ? = 2
[1,1]
 => [1]
 => [1]
 => 10 => 1
[3]
 => []
 => []
 => => ? = 3
[2,1]
 => [1]
 => [1]
 => 10 => 1
[1,1,1]
 => [1,1]
 => [2]
 => 100 => 1
[4]
 => []
 => []
 => => ? = 4
[3,1]
 => [1]
 => [1]
 => 10 => 1
[2,2]
 => [2]
 => [1,1]
 => 110 => 2
[2,1,1]
 => [1,1]
 => [2]
 => 100 => 1
[1,1,1,1]
 => [1,1,1]
 => [3]
 => 1000 => 1
[5]
 => []
 => []
 => => ? = 5
[4,1]
 => [1]
 => [1]
 => 10 => 1
[3,2]
 => [2]
 => [1,1]
 => 110 => 2
[3,1,1]
 => [1,1]
 => [2]
 => 100 => 1
[2,2,1]
 => [2,1]
 => [2,1]
 => 1010 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => 1000 => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 10000 => 1
[6]
 => []
 => []
 => => ? = 6
[5,1]
 => [1]
 => [1]
 => 10 => 1
[4,2]
 => [2]
 => [1,1]
 => 110 => 2
[4,1,1]
 => [1,1]
 => [2]
 => 100 => 1
[3,3]
 => [3]
 => [1,1,1]
 => 1110 => 3
[3,2,1]
 => [2,1]
 => [2,1]
 => 1010 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => 1000 => 1
[2,2,2]
 => [2,2]
 => [2,2]
 => 1100 => 2
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 10010 => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 10000 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 100000 => 1
[7]
 => []
 => []
 => => ? = 7
[6,1]
 => [1]
 => [1]
 => 10 => 1
[5,2]
 => [2]
 => [1,1]
 => 110 => 2
[5,1,1]
 => [1,1]
 => [2]
 => 100 => 1
[4,3]
 => [3]
 => [1,1,1]
 => 1110 => 3
[4,2,1]
 => [2,1]
 => [2,1]
 => 1010 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => 1000 => 1
[3,3,1]
 => [3,1]
 => [2,1,1]
 => 10110 => 1
[3,2,2]
 => [2,2]
 => [2,2]
 => 1100 => 2
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => 10010 => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 10000 => 1
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => 10100 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 100000 => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => 1000000 => 1
[8]
 => []
 => []
 => => ? = 8
[7,1]
 => [1]
 => [1]
 => 10 => 1
[6,2]
 => [2]
 => [1,1]
 => 110 => 2
[6,1,1]
 => [1,1]
 => [2]
 => 100 => 1
[5,3]
 => [3]
 => [1,1,1]
 => 1110 => 3
[5,2,1]
 => [2,1]
 => [2,1]
 => 1010 => 1
[5,1,1,1]
 => [1,1,1]
 => [3]
 => 1000 => 1
[4,4]
 => [4]
 => [1,1,1,1]
 => 11110 => 4
[4,3,1]
 => [3,1]
 => [2,1,1]
 => 10110 => 1
[4,2,2]
 => [2,2]
 => [2,2]
 => 1100 => 2
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => 10010 => 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 10000 => 1
[3,3,2]
 => [3,2]
 => [2,2,1]
 => 11010 => 2
[3,3,1,1]
 => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[9]
 => []
 => []
 => => ? = 9
[10]
 => []
 => []
 => => ? = 10
[11]
 => []
 => []
 => => ? = 11
[12]
 => []
 => []
 => => ? ∊ {1,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [11]
 => 100000000000 => ? ∊ {1,12}
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 17 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => => => ? = 2
[1,1]
 => [1]
 => 10 => 10 => 1
[3]
 => []
 => => => ? = 3
[2,1]
 => [1]
 => 10 => 10 => 1
[1,1,1]
 => [1,1]
 => 110 => 110 => 1
[4]
 => []
 => => => ? = 4
[3,1]
 => [1]
 => 10 => 10 => 1
[2,2]
 => [2]
 => 100 => 010 => 2
[2,1,1]
 => [1,1]
 => 110 => 110 => 1
[1,1,1,1]
 => [1,1,1]
 => 1110 => 1110 => 1
[5]
 => []
 => => => ? = 5
[4,1]
 => [1]
 => 10 => 10 => 1
[3,2]
 => [2]
 => 100 => 010 => 2
[3,1,1]
 => [1,1]
 => 110 => 110 => 1
[2,2,1]
 => [2,1]
 => 1010 => 1100 => 1
[2,1,1,1]
 => [1,1,1]
 => 1110 => 1110 => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 11110 => 1
[6]
 => []
 => => => ? = 6
[5,1]
 => [1]
 => 10 => 10 => 1
[4,2]
 => [2]
 => 100 => 010 => 2
[4,1,1]
 => [1,1]
 => 110 => 110 => 1
[3,3]
 => [3]
 => 1000 => 0010 => 3
[3,2,1]
 => [2,1]
 => 1010 => 1100 => 1
[3,1,1,1]
 => [1,1,1]
 => 1110 => 1110 => 1
[2,2,2]
 => [2,2]
 => 1100 => 0110 => 2
[2,2,1,1]
 => [2,1,1]
 => 10110 => 11010 => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 11110 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 111110 => 1
[7]
 => []
 => => => ? = 7
[6,1]
 => [1]
 => 10 => 10 => 1
[5,2]
 => [2]
 => 100 => 010 => 2
[5,1,1]
 => [1,1]
 => 110 => 110 => 1
[4,3]
 => [3]
 => 1000 => 0010 => 3
[4,2,1]
 => [2,1]
 => 1010 => 1100 => 1
[4,1,1,1]
 => [1,1,1]
 => 1110 => 1110 => 1
[3,3,1]
 => [3,1]
 => 10010 => 10100 => 1
[3,2,2]
 => [2,2]
 => 1100 => 0110 => 2
[3,2,1,1]
 => [2,1,1]
 => 10110 => 11010 => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 11110 => 1
[2,2,2,1]
 => [2,2,1]
 => 11010 => 11100 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 110110 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 111110 => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 1111110 => 1
[8]
 => []
 => => => ? = 8
[7,1]
 => [1]
 => 10 => 10 => 1
[6,2]
 => [2]
 => 100 => 010 => 2
[6,1,1]
 => [1,1]
 => 110 => 110 => 1
[5,3]
 => [3]
 => 1000 => 0010 => 3
[5,2,1]
 => [2,1]
 => 1010 => 1100 => 1
[5,1,1,1]
 => [1,1,1]
 => 1110 => 1110 => 1
[4,4]
 => [4]
 => 10000 => 00010 => 4
[4,3,1]
 => [3,1]
 => 10010 => 10100 => 1
[4,2,2]
 => [2,2]
 => 1100 => 0110 => 2
[4,2,1,1]
 => [2,1,1]
 => 10110 => 11010 => 1
[4,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 11110 => 1
[3,3,2]
 => [3,2]
 => 10100 => 01100 => 2
[3,3,1,1]
 => [3,1,1]
 => 100110 => 101010 => 1
[9]
 => []
 => => => ? = 9
[10]
 => []
 => => => ? = 10
[11]
 => []
 => => => ? = 11
[12]
 => []
 => => => ? ∊ {1,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => 111111111110 => ? => ? ∊ {1,12}
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of {1,…,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000382
(load all 16 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => [0] => ? = 2
[1,1]
 => [1]
 => [[1]]
 => [1] => 1
[3]
 => []
 => []
 => [0] => ? = 3
[2,1]
 => [1]
 => [[1]]
 => [1] => 1
[1,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[4]
 => []
 => []
 => [0] => ? = 4
[3,1]
 => [1]
 => [[1]]
 => [1] => 1
[2,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[2,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[5]
 => []
 => []
 => [0] => ? = 5
[4,1]
 => [1]
 => [[1]]
 => [1] => 1
[3,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[3,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[2,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[6]
 => []
 => []
 => [0] => ? = 6
[5,1]
 => [1]
 => [[1]]
 => [1] => 1
[4,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[4,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[3,3]
 => [3]
 => [[1,2,3]]
 => [3] => 3
[3,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[3,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[2,2,1,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[7]
 => []
 => []
 => [0] => ? = 7
[6,1]
 => [1]
 => [[1]]
 => [1] => 1
[5,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[5,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[4,3]
 => [3]
 => [[1,2,3]]
 => [3] => 3
[4,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[3,3,1]
 => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[3,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[3,2,1,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[2,2,2,1]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 1
[8]
 => []
 => []
 => [0] => ? = 8
[7,1]
 => [1]
 => [[1]]
 => [1] => 1
[6,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[6,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[5,3]
 => [3]
 => [[1,2,3]]
 => [3] => 3
[5,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[5,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[4,4]
 => [4]
 => [[1,2,3,4]]
 => [4] => 4
[4,3,1]
 => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[4,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[4,2,1,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[3,3,2]
 => [3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 2
[3,3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[9]
 => []
 => []
 => [0] => ? = 9
[10]
 => []
 => []
 => [0] => ? = 10
[11]
 => []
 => []
 => [0] => ? = 11
[12]
 => []
 => []
 => [0] => ? ∊ {1,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? => ? ∊ {1,12}
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 23 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => => [] => ? = 2
[1,1]
 => [1]
 => 10 => [1,1] => 1
[3]
 => []
 => => [] => ? = 3
[2,1]
 => [1]
 => 10 => [1,1] => 1
[1,1,1]
 => [1,1]
 => 110 => [2,1] => 1
[4]
 => []
 => => [] => ? = 4
[3,1]
 => [1]
 => 10 => [1,1] => 1
[2,2]
 => [2]
 => 100 => [1,2] => 2
[2,1,1]
 => [1,1]
 => 110 => [2,1] => 1
[1,1,1,1]
 => [1,1,1]
 => 1110 => [3,1] => 1
[5]
 => []
 => => [] => ? = 5
[4,1]
 => [1]
 => 10 => [1,1] => 1
[3,2]
 => [2]
 => 100 => [1,2] => 2
[3,1,1]
 => [1,1]
 => 110 => [2,1] => 1
[2,2,1]
 => [2,1]
 => 1010 => [1,1,1,1] => 1
[2,1,1,1]
 => [1,1,1]
 => 1110 => [3,1] => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => [4,1] => 1
[6]
 => []
 => => [] => ? = 6
[5,1]
 => [1]
 => 10 => [1,1] => 1
[4,2]
 => [2]
 => 100 => [1,2] => 2
[4,1,1]
 => [1,1]
 => 110 => [2,1] => 1
[3,3]
 => [3]
 => 1000 => [1,3] => 3
[3,2,1]
 => [2,1]
 => 1010 => [1,1,1,1] => 1
[3,1,1,1]
 => [1,1,1]
 => 1110 => [3,1] => 1
[2,2,2]
 => [2,2]
 => 1100 => [2,2] => 2
[2,2,1,1]
 => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => [4,1] => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => [5,1] => 1
[7]
 => []
 => => [] => ? = 7
[6,1]
 => [1]
 => 10 => [1,1] => 1
[5,2]
 => [2]
 => 100 => [1,2] => 2
[5,1,1]
 => [1,1]
 => 110 => [2,1] => 1
[4,3]
 => [3]
 => 1000 => [1,3] => 3
[4,2,1]
 => [2,1]
 => 1010 => [1,1,1,1] => 1
[4,1,1,1]
 => [1,1,1]
 => 1110 => [3,1] => 1
[3,3,1]
 => [3,1]
 => 10010 => [1,2,1,1] => 1
[3,2,2]
 => [2,2]
 => 1100 => [2,2] => 2
[3,2,1,1]
 => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => [4,1] => 1
[2,2,2,1]
 => [2,2,1]
 => 11010 => [2,1,1,1] => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => [5,1] => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => [6,1] => 1
[8]
 => []
 => => [] => ? = 8
[7,1]
 => [1]
 => 10 => [1,1] => 1
[6,2]
 => [2]
 => 100 => [1,2] => 2
[6,1,1]
 => [1,1]
 => 110 => [2,1] => 1
[5,3]
 => [3]
 => 1000 => [1,3] => 3
[5,2,1]
 => [2,1]
 => 1010 => [1,1,1,1] => 1
[5,1,1,1]
 => [1,1,1]
 => 1110 => [3,1] => 1
[4,4]
 => [4]
 => 10000 => [1,4] => 4
[4,3,1]
 => [3,1]
 => 10010 => [1,2,1,1] => 1
[4,2,2]
 => [2,2]
 => 1100 => [2,2] => 2
[4,2,1,1]
 => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[4,1,1,1,1]
 => [1,1,1,1]
 => 11110 => [4,1] => 1
[3,3,2]
 => [3,2]
 => 10100 => [1,1,1,2] => 2
[3,3,1,1]
 => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[9]
 => []
 => => [] => ? = 9
[10]
 => []
 => => [] => ? = 10
[11]
 => []
 => => [] => ? = 11
[12]
 => []
 => => [] => ? ∊ {1,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => 111111111110 => ? => ? ∊ {1,12}
Description
The last part of an integer composition.
Matching statistic: St000733
(load all 6 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => []
 => ? = 2
[1,1]
 => [1]
 => [1]
 => [[1]]
 => 1
[3]
 => []
 => []
 => []
 => ? = 3
[2,1]
 => [1]
 => [1]
 => [[1]]
 => 1
[1,1,1]
 => [1,1]
 => [2]
 => [[1,2]]
 => 1
[4]
 => []
 => []
 => []
 => ? = 4
[3,1]
 => [1]
 => [1]
 => [[1]]
 => 1
[2,2]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 2
[2,1,1]
 => [1,1]
 => [2]
 => [[1,2]]
 => 1
[1,1,1,1]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 1
[5]
 => []
 => []
 => []
 => ? = 5
[4,1]
 => [1]
 => [1]
 => [[1]]
 => 1
[3,2]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 2
[3,1,1]
 => [1,1]
 => [2]
 => [[1,2]]
 => 1
[2,2,1]
 => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 1
[6]
 => []
 => []
 => []
 => ? = 6
[5,1]
 => [1]
 => [1]
 => [[1]]
 => 1
[4,2]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 2
[4,1,1]
 => [1,1]
 => [2]
 => [[1,2]]
 => 1
[3,3]
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[3,2,1]
 => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 1
[2,2,2]
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 1
[7]
 => []
 => []
 => []
 => ? = 7
[6,1]
 => [1]
 => [1]
 => [[1]]
 => 1
[5,2]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 2
[5,1,1]
 => [1,1]
 => [2]
 => [[1,2]]
 => 1
[4,3]
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[4,2,1]
 => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 1
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 1
[3,2,2]
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 1
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => 1
[8]
 => []
 => []
 => []
 => ? = 8
[7,1]
 => [1]
 => [1]
 => [[1]]
 => 1
[6,2]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 2
[6,1,1]
 => [1,1]
 => [2]
 => [[1,2]]
 => 1
[5,3]
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[5,2,1]
 => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 1
[5,1,1,1]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 1
[4,4]
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
[4,3,1]
 => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 1
[4,2,2]
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 1
[3,3,2]
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
[3,3,1,1]
 => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 1
[9]
 => []
 => []
 => []
 => ? = 9
[10]
 => []
 => []
 => []
 => ? = 10
[11]
 => []
 => []
 => []
 => ? = 11
[12]
 => []
 => []
 => []
 => ? ∊ {1,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [11]
 => [[1,2,3,4,5,6,7,8,9,10,11]]
 => ? ∊ {1,12}
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000657
(load all 13 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => [0] => ? = 2
[1,1]
 => [1]
 => [[1]]
 => [1] => 1
[3]
 => []
 => []
 => [0] => ? = 3
[2,1]
 => [1]
 => [[1]]
 => [1] => 1
[1,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[4]
 => []
 => []
 => [0] => ? = 4
[3,1]
 => [1]
 => [[1]]
 => [1] => 1
[2,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[2,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[5]
 => []
 => []
 => [0] => ? = 5
[4,1]
 => [1]
 => [[1]]
 => [1] => 1
[3,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[3,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[6]
 => []
 => []
 => [0] => ? = 6
[5,1]
 => [1]
 => [[1]]
 => [1] => 1
[4,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[4,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[3,3]
 => [3]
 => [[1,2,3]]
 => [3] => 3
[3,2,1]
 => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[3,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[7]
 => []
 => []
 => [0] => ? = 7
[6,1]
 => [1]
 => [[1]]
 => [1] => 1
[5,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[5,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[4,3]
 => [3]
 => [[1,2,3]]
 => [3] => 3
[4,2,1]
 => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[3,3,1]
 => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1
[3,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[3,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[2,2,2,1]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 1
[8]
 => []
 => []
 => [0] => ? = 8
[7,1]
 => [1]
 => [[1]]
 => [1] => 1
[6,2]
 => [2]
 => [[1,2]]
 => [2] => 2
[6,1,1]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[5,3]
 => [3]
 => [[1,2,3]]
 => [3] => 3
[5,2,1]
 => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[5,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[4,4]
 => [4]
 => [[1,2,3,4]]
 => [4] => 4
[4,3,1]
 => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1
[4,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[4,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[3,3,2]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => 2
[3,3,1,1]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 1
[9]
 => []
 => []
 => [0] => ? = 9
[10]
 => []
 => []
 => [0] => ? = 10
[11]
 => []
 => []
 => [0] => ? ∊ {1,11}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => [1,1,1,1,1,1,1,1,1,1] => ? ∊ {1,11}
[12]
 => []
 => []
 => [0] => ? ∊ {1,1,1,1,1,1,2,12}
[2,2,2,2,2,2]
 => [2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10]]
 => [2,2,2,2,2] => ? ∊ {1,1,1,1,1,1,2,12}
[2,2,2,2,2,1,1]
 => [2,2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7,8],[9],[10]]
 => [2,2,2,2,1,1] => ? ∊ {1,1,1,1,1,1,2,12}
[2,2,2,2,1,1,1,1]
 => [2,2,2,1,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
 => [2,2,2,1,1,1,1] => ? ∊ {1,1,1,1,1,1,2,12}
[2,2,2,1,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
 => [2,2,1,1,1,1,1,1] => ? ∊ {1,1,1,1,1,1,2,12}
[2,2,1,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => [2,1,1,1,1,1,1,1,1] => ? ∊ {1,1,1,1,1,1,2,12}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => [1,1,1,1,1,1,1,1,1,1] => ? ∊ {1,1,1,1,1,1,2,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? => ? ∊ {1,1,1,1,1,1,2,12}
Description
The smallest part of an integer composition.
Matching statistic: St001038
(load all 10 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Values
[2]
 => []
 => []
 => ? ∊ {1,2}
[1,1]
 => [1]
 => [1,0]
 => ? ∊ {1,2}
[3]
 => []
 => []
 => ? ∊ {1,3}
[2,1]
 => [1]
 => [1,0]
 => ? ∊ {1,3}
[1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[4]
 => []
 => []
 => ? ∊ {1,4}
[3,1]
 => [1]
 => [1,0]
 => ? ∊ {1,4}
[2,2]
 => [2]
 => [1,0,1,0]
 => 2
[2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[5]
 => []
 => []
 => ? ∊ {1,5}
[4,1]
 => [1]
 => [1,0]
 => ? ∊ {1,5}
[3,2]
 => [2]
 => [1,0,1,0]
 => 2
[3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[6]
 => []
 => []
 => ? ∊ {1,6}
[5,1]
 => [1]
 => [1,0]
 => ? ∊ {1,6}
[4,2]
 => [2]
 => [1,0,1,0]
 => 2
[4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[7]
 => []
 => []
 => ? ∊ {1,7}
[6,1]
 => [1]
 => [1,0]
 => ? ∊ {1,7}
[5,2]
 => [2]
 => [1,0,1,0]
 => 2
[5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[4,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[8]
 => []
 => []
 => ? ∊ {1,8}
[7,1]
 => [1]
 => [1,0]
 => ? ∊ {1,8}
[6,2]
 => [2]
 => [1,0,1,0]
 => 2
[6,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[5,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[5,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[5,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[4,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 2
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 1
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[9]
 => []
 => []
 => ? ∊ {1,1,9}
[8,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,9}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,9}
[10]
 => []
 => []
 => ? ∊ {1,1,1,10}
[9,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,1,10}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,10}
[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]
 => ? ∊ {1,1,1,10}
[11]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,11}
[10,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,1,1,1,1,11}
[3,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,11}
[2,2,2,1,1,1,1,1]
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,11}
[2,2,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,11}
[2,1,1,1,1,1,1,1,1,1]
 => [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]
 => ? ∊ {1,1,1,1,1,1,11}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,11}
[12]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[11,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[4,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[3,3,1,1,1,1,1,1]
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[3,2,2,1,1,1,1,1]
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[3,2,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[3,1,1,1,1,1,1,1,1,1]
 => [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]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[2,2,2,2,1,1,1,1]
 => [2,2,2,1,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[2,2,2,1,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[2,2,1,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,12}
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000745
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Values
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => [[1,3],[2]]
 => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => [[1,2,3]]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
 => 4
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [[1,6],[2],[3],[4],[5]]
 => 5
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [[1,2,5],[3],[4]]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [[1,2,3],[4],[5]]
 => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => [[1,7],[2],[3],[4],[5],[6]]
 => 6
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,3,2,1,6] => [[1,2,6],[3],[4],[5]]
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [[1,3,5],[2],[4]]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [[1,2,5],[3],[4]]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [[1,4],[2,5],[3]]
 => 3
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [[1,2,4],[3],[5]]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [[1,3],[2,4],[5]]
 => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [[1,2,3],[4],[5]]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,4,3,2] => [[1,2,3],[4],[5],[6]]
 => 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7]]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => [[1,8],[2],[3],[4],[5],[6],[7]]
 => 7
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,6,4,3,2,1,7] => [[1,2,7],[3],[4],[5],[6]]
 => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,3,5,2,1,6] => [[1,3,6],[2],[4],[5]]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,5,4,2,1,6] => [[1,2,6],[3],[4],[5]]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [[1,4,5],[2],[3]]
 => 3
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [[1,2,5],[3],[4]]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,4,6,3,2] => [[1,2,4],[3],[5],[6]]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,5,3,2] => [[1,2,3],[4],[5],[6]]
 => 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,5,4,3,2] => [[1,2,3],[4],[5],[6],[7]]
 => 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8]]
 => 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,7,6,5,4,3,2,1,9] => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
 => 8
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [6,7,5,4,3,2,1,8] => [[1,2,8],[3],[4],[5],[6],[7]]
 => 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [5,4,6,3,2,1,7] => [[1,3,7],[2],[4],[5],[6]]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [4,6,5,3,2,1,7] => [[1,2,7],[3],[4],[5],[6]]
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => [[1,4,6],[2],[3],[5]]
 => 3
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [3,4,5,2,1,6] => [[1,2,3,6],[4],[5]]
 => 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,4,3,1,6] => [[1,2,6],[3],[4],[5]]
 => 1
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? ∊ {1,1,2,10}
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
 => ? ∊ {1,1,2,10}
[3,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,8,7,9,6,5,4,3,2] => [[1,2,4],[3],[5],[6],[7],[8],[9]]
 => ? ∊ {1,1,2,10}
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? ∊ {1,1,2,10}
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,10,9,8,7,6,5,4,3,2,1,12] => [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,10,8,7,6,5,4,3,2,1,11] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => [8,7,9,6,5,4,3,2,1,10] => [[1,3,10],[2],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[9,1,1]
 => [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => [7,9,8,6,5,4,3,2,1,10] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => [7,6,5,8,4,3,2,1,9] => [[1,4,9],[2],[3],[5],[6],[7],[8]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [6,7,8,5,4,3,2,1,9] => [[1,2,3,9],[4],[5],[6],[7],[8]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[8,1,1,1]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
 => [5,8,7,6,4,3,2,1,9] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[7,1,1,1,1]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
 => [3,7,6,5,4,2,1,8] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,8,7,6,9,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,5,7,8,6,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [1,9,8,10,7,6,5,4,3,2] => [[1,2,4],[3],[5],[6],[7],[8],[9],[10]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,6,9,8,7,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,8,10,9,7,6,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1,10,11,9,8,7,6,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,3,11}
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[11,1]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,11,9,8,7,6,5,4,3,2,1,12] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [9,8,10,7,6,5,4,3,2,1,11] => [[1,3,11],[2],[4],[5],[6],[7],[8],[9],[10]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [8,10,9,7,6,5,4,3,2,1,11] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[9,3]
 => [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
 => [8,7,6,9,5,4,3,2,1,10] => [[1,4,10],[2],[3],[5],[6],[7],[8],[9]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [7,8,9,6,5,4,3,2,1,10] => [[1,2,3,10],[4],[5],[6],[7],[8],[9]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[9,1,1,1]
 => [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [6,9,8,7,5,4,3,2,1,10] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [7,6,5,4,8,3,2,1,9] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [6,7,5,8,4,3,2,1,9] => [[1,2,4,9],[3],[5],[6],[7],[8]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[8,2,2]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
 => [6,5,8,7,4,3,2,1,9] => [[1,3,9],[2,4],[5],[6],[7],[8]]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [5,7,8,6,4,3,2,1,9] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [3,6,7,5,4,2,1,8] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,6,7,5,4,8,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,8,7,6,5,9,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[4,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,7,8,6,9,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[4,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [1,9,8,7,10,6,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[3,3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,5,6,8,7,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[3,3,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [1,7,6,9,8,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [1,6,8,9,7,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [1,10,9,11,8,7,6,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [1,5,9,8,7,6,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,7,10,9,8,6,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [1,9,11,10,8,7,6,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [1,11,12,10,9,8,7,6,5,4,3,2] => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,4,12}
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000667
(load all 4 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 83%●distinct values known / distinct values provided: 33%
Values
[2]
 => []
 => ?
 => ? ∊ {1,2}
[1,1]
 => [1]
 => []
 => ? ∊ {1,2}
[3]
 => []
 => ?
 => ? ∊ {1,3}
[2,1]
 => [1]
 => []
 => ? ∊ {1,3}
[1,1,1]
 => [1,1]
 => [1]
 => 1
[4]
 => []
 => ?
 => ? ∊ {1,2,4}
[3,1]
 => [1]
 => []
 => ? ∊ {1,2,4}
[2,2]
 => [2]
 => []
 => ? ∊ {1,2,4}
[2,1,1]
 => [1,1]
 => [1]
 => 1
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[5]
 => []
 => ?
 => ? ∊ {1,2,5}
[4,1]
 => [1]
 => []
 => ? ∊ {1,2,5}
[3,2]
 => [2]
 => []
 => ? ∊ {1,2,5}
[3,1,1]
 => [1,1]
 => [1]
 => 1
[2,2,1]
 => [2,1]
 => [1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[6]
 => []
 => ?
 => ? ∊ {1,2,3,6}
[5,1]
 => [1]
 => []
 => ? ∊ {1,2,3,6}
[4,2]
 => [2]
 => []
 => ? ∊ {1,2,3,6}
[4,1,1]
 => [1,1]
 => [1]
 => 1
[3,3]
 => [3]
 => []
 => ? ∊ {1,2,3,6}
[3,2,1]
 => [2,1]
 => [1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[2,2,2]
 => [2,2]
 => [2]
 => 2
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[7]
 => []
 => ?
 => ? ∊ {1,2,3,7}
[6,1]
 => [1]
 => []
 => ? ∊ {1,2,3,7}
[5,2]
 => [2]
 => []
 => ? ∊ {1,2,3,7}
[5,1,1]
 => [1,1]
 => [1]
 => 1
[4,3]
 => [3]
 => []
 => ? ∊ {1,2,3,7}
[4,2,1]
 => [2,1]
 => [1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[3,3,1]
 => [3,1]
 => [1]
 => 1
[3,2,2]
 => [2,2]
 => [2]
 => 2
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[8]
 => []
 => ?
 => ? ∊ {1,2,3,4,8}
[7,1]
 => [1]
 => []
 => ? ∊ {1,2,3,4,8}
[6,2]
 => [2]
 => []
 => ? ∊ {1,2,3,4,8}
[6,1,1]
 => [1,1]
 => [1]
 => 1
[5,3]
 => [3]
 => []
 => ? ∊ {1,2,3,4,8}
[5,2,1]
 => [2,1]
 => [1]
 => 1
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[4,4]
 => [4]
 => []
 => ? ∊ {1,2,3,4,8}
[4,3,1]
 => [3,1]
 => [1]
 => 1
[4,2,2]
 => [2,2]
 => [2]
 => 2
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,3,2]
 => [3,2]
 => [2]
 => 2
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 1
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => 2
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => 1
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
[9]
 => []
 => ?
 => ? ∊ {1,2,3,4,9}
[8,1]
 => [1]
 => []
 => ? ∊ {1,2,3,4,9}
[7,2]
 => [2]
 => []
 => ? ∊ {1,2,3,4,9}
[7,1,1]
 => [1,1]
 => [1]
 => 1
[6,3]
 => [3]
 => []
 => ? ∊ {1,2,3,4,9}
[6,2,1]
 => [2,1]
 => [1]
 => 1
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[5,4]
 => [4]
 => []
 => ? ∊ {1,2,3,4,9}
[5,3,1]
 => [3,1]
 => [1]
 => 1
[5,2,2]
 => [2,2]
 => [2]
 => 2
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[4,4,1]
 => [4,1]
 => [1]
 => 1
[10]
 => []
 => ?
 => ? ∊ {1,2,3,4,5,10}
[9,1]
 => [1]
 => []
 => ? ∊ {1,2,3,4,5,10}
[8,2]
 => [2]
 => []
 => ? ∊ {1,2,3,4,5,10}
[7,3]
 => [3]
 => []
 => ? ∊ {1,2,3,4,5,10}
[6,4]
 => [4]
 => []
 => ? ∊ {1,2,3,4,5,10}
[5,5]
 => [5]
 => []
 => ? ∊ {1,2,3,4,5,10}
[11]
 => []
 => ?
 => ? ∊ {2,2,3,4,5,11}
[10,1]
 => [1]
 => []
 => ? ∊ {2,2,3,4,5,11}
[9,2]
 => [2]
 => []
 => ? ∊ {2,2,3,4,5,11}
[8,3]
 => [3]
 => []
 => ? ∊ {2,2,3,4,5,11}
[7,4]
 => [4]
 => []
 => ? ∊ {2,2,3,4,5,11}
[6,5]
 => [5]
 => []
 => ? ∊ {2,2,3,4,5,11}
[12]
 => []
 => ?
 => ? ∊ {2,2,3,4,5,6,12}
[11,1]
 => [1]
 => []
 => ? ∊ {2,2,3,4,5,6,12}
[10,2]
 => [2]
 => []
 => ? ∊ {2,2,3,4,5,6,12}
[9,3]
 => [3]
 => []
 => ? ∊ {2,2,3,4,5,6,12}
[8,4]
 => [4]
 => []
 => ? ∊ {2,2,3,4,5,6,12}
[7,5]
 => [5]
 => []
 => ? ∊ {2,2,3,4,5,6,12}
[6,6]
 => [6]
 => []
 => ? ∊ {2,2,3,4,5,6,12}
Description
The greatest common divisor of the parts of the partition.
The following 138 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000990The first ascent of a permutation. 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. St000974The length of the trunk of an ordered tree. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001571The Cartan determinant of the integer partition. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001568The smallest positive integer that does not appear twice in the partition. St000011The number of touch points (or returns) of a Dyck path. St001075The minimal size of a block of a set partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000823The number of unsplittable factors of the set partition. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001733The number of weak left to right maxima of a Dyck path. St001829The common independence number of a graph. St000918The 2-limited packing number of a graph. St000439The position of the first down step of a Dyck path. St000234The number of global ascents of a permutation. St000654The first descent of a permutation. St001432The order dimension of the partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000314The number of left-to-right-maxima of a permutation. St000546The number of global descents of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000054The first entry of the permutation. St000007The number of saliances of the permutation. St000883The number of longest increasing subsequences of a permutation. St000759The smallest missing part in an integer partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000989The number of final rises of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000260The radius of a connected graph. St000700The protection number of an ordered tree. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000287The number of connected components of a graph. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000221The number of strong fixed points of a permutation. St000315The number of isolated vertices of a graph. St000756The sum of the positions of the left to right maxima of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000025The number of initial rises of a Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001884The number of borders of a binary word. St000068The number of minimal elements in a poset. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000765The number of weak records in an integer composition. St000911The number of maximal antichains of maximal size in a poset. St000908The length of the shortest maximal antichain in a poset. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000061The number of nodes on the left branch of a binary tree. St000026The position of the first return of a Dyck path. St000084The number of subtrees. St000991The number of right-to-left minima of a permutation. St000090The variation of a composition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000069The number of maximal elements of a poset. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000505The biggest entry in the block containing the 1. St000031The number of cycles in the cycle decomposition of a permutation. St000504The cardinality of the first block of a set partition. St000729The minimal arc length of a set partition. St000909The number of maximal chains of maximal size in a poset. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000363The number of minimal vertex covers of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) [c0,c1,...,cn−1] by adding c0 to cn−1. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000873The aix statistic of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000338The number of pixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000906The length of the shortest maximal chain in a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000258The burning number of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000487The length of the shortest cycle of a permutation. St000740The last entry of a permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St000210Minimum over maximum difference of elements in cycles. St000015The number of peaks of a Dyck path. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001530The depth of a Dyck path. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000456The monochromatic index of a connected graph. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA.