Your data matches 92 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000657
(load all 13 compositions to match this statistic)
Mapping
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 1
[2] => 2
[1,1,1] => 1
[1,2] => 1
[2,1] => 1
[3] => 3
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 4
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 1
[2,1,2] => 1
[2,2,1] => 1
[2,3] => 2
[3,1,1] => 1
[3,2] => 2
[4,1] => 1
[5] => 5
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 1
[1,2,1,2] => 1
[1,2,2,1] => 1
[1,2,3] => 1
[1,3,1,1] => 1
[1,3,2] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000297
Mapping
Mp00040: Integer compositions —to partition⟶ 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%
Values
[1] => [1]
 => [1]
 => 10 => 1
[1,1] => [1,1]
 => [2]
 => 100 => 1
[2] => [2]
 => [1,1]
 => 110 => 2
[1,1,1] => [1,1,1]
 => [3]
 => 1000 => 1
[1,2] => [2,1]
 => [2,1]
 => 1010 => 1
[2,1] => [2,1]
 => [2,1]
 => 1010 => 1
[3] => [3]
 => [1,1,1]
 => 1110 => 3
[1,1,1,1] => [1,1,1,1]
 => [4]
 => 10000 => 1
[1,1,2] => [2,1,1]
 => [3,1]
 => 10010 => 1
[1,2,1] => [2,1,1]
 => [3,1]
 => 10010 => 1
[1,3] => [3,1]
 => [2,1,1]
 => 10110 => 1
[2,1,1] => [2,1,1]
 => [3,1]
 => 10010 => 1
[2,2] => [2,2]
 => [2,2]
 => 1100 => 2
[3,1] => [3,1]
 => [2,1,1]
 => 10110 => 1
[4] => [4]
 => [1,1,1,1]
 => 11110 => 4
[1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 100000 => 1
[1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,1,3] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,2,2] => [2,2,1]
 => [3,2]
 => 10100 => 1
[1,3,1] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,4] => [4,1]
 => [2,1,1,1]
 => 101110 => 1
[2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[2,1,2] => [2,2,1]
 => [3,2]
 => 10100 => 1
[2,2,1] => [2,2,1]
 => [3,2]
 => 10100 => 1
[2,3] => [3,2]
 => [2,2,1]
 => 11010 => 2
[3,1,1] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[3,2] => [3,2]
 => [2,2,1]
 => 11010 => 2
[4,1] => [4,1]
 => [2,1,1,1]
 => 101110 => 1
[5] => [5]
 => [1,1,1,1,1]
 => 111110 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [6]
 => 1000000 => 1
[1,1,1,1,2] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1
[1,1,1,3] => [3,1,1,1]
 => [4,1,1]
 => 1000110 => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1
[1,1,2,2] => [2,2,1,1]
 => [4,2]
 => 100100 => 1
[1,1,3,1] => [3,1,1,1]
 => [4,1,1]
 => 1000110 => 1
[1,1,4] => [4,1,1]
 => [3,1,1,1]
 => 1001110 => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1
[1,2,1,2] => [2,2,1,1]
 => [4,2]
 => 100100 => 1
[1,2,2,1] => [2,2,1,1]
 => [4,2]
 => 100100 => 1
[1,2,3] => [3,2,1]
 => [3,2,1]
 => 101010 => 1
[1,3,1,1] => [3,1,1,1]
 => [4,1,1]
 => 1000110 => 1
[1,3,2] => [3,2,1]
 => [3,2,1]
 => 101010 => 1
[1,4,1] => [4,1,1]
 => [3,1,1,1]
 => 1001110 => 1
[1,5] => [5,1]
 => [2,1,1,1,1]
 => 1011110 => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1
[2,1,1,2] => [2,2,1,1]
 => [4,2]
 => 100100 => 1
[2,1,2,1] => [2,2,1,1]
 => [4,2]
 => 100100 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ 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%
Values
[1] => [1]
 => 10 => 10 => 1
[1,1] => [1,1]
 => 110 => 110 => 1
[2] => [2]
 => 100 => 010 => 2
[1,1,1] => [1,1,1]
 => 1110 => 1110 => 1
[1,2] => [2,1]
 => 1010 => 1100 => 1
[2,1] => [2,1]
 => 1010 => 1100 => 1
[3] => [3]
 => 1000 => 0010 => 3
[1,1,1,1] => [1,1,1,1]
 => 11110 => 11110 => 1
[1,1,2] => [2,1,1]
 => 10110 => 11010 => 1
[1,2,1] => [2,1,1]
 => 10110 => 11010 => 1
[1,3] => [3,1]
 => 10010 => 10100 => 1
[2,1,1] => [2,1,1]
 => 10110 => 11010 => 1
[2,2] => [2,2]
 => 1100 => 0110 => 2
[3,1] => [3,1]
 => 10010 => 10100 => 1
[4] => [4]
 => 10000 => 00010 => 4
[1,1,1,1,1] => [1,1,1,1,1]
 => 111110 => 111110 => 1
[1,1,1,2] => [2,1,1,1]
 => 101110 => 110110 => 1
[1,1,2,1] => [2,1,1,1]
 => 101110 => 110110 => 1
[1,1,3] => [3,1,1]
 => 100110 => 101010 => 1
[1,2,1,1] => [2,1,1,1]
 => 101110 => 110110 => 1
[1,2,2] => [2,2,1]
 => 11010 => 11100 => 1
[1,3,1] => [3,1,1]
 => 100110 => 101010 => 1
[1,4] => [4,1]
 => 100010 => 100100 => 1
[2,1,1,1] => [2,1,1,1]
 => 101110 => 110110 => 1
[2,1,2] => [2,2,1]
 => 11010 => 11100 => 1
[2,2,1] => [2,2,1]
 => 11010 => 11100 => 1
[2,3] => [3,2]
 => 10100 => 01100 => 2
[3,1,1] => [3,1,1]
 => 100110 => 101010 => 1
[3,2] => [3,2]
 => 10100 => 01100 => 2
[4,1] => [4,1]
 => 100010 => 100100 => 1
[5] => [5]
 => 100000 => 000010 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1111110 => 1111110 => 1
[1,1,1,1,2] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1
[1,1,1,3] => [3,1,1,1]
 => 1001110 => 1010110 => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1
[1,1,2,2] => [2,2,1,1]
 => 110110 => 111010 => 1
[1,1,3,1] => [3,1,1,1]
 => 1001110 => 1010110 => 1
[1,1,4] => [4,1,1]
 => 1000110 => 1001010 => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1
[1,2,1,2] => [2,2,1,1]
 => 110110 => 111010 => 1
[1,2,2,1] => [2,2,1,1]
 => 110110 => 111010 => 1
[1,2,3] => [3,2,1]
 => 101010 => 111000 => 1
[1,3,1,1] => [3,1,1,1]
 => 1001110 => 1010110 => 1
[1,3,2] => [3,2,1]
 => 101010 => 111000 => 1
[1,4,1] => [4,1,1]
 => 1000110 => 1001010 => 1
[1,5] => [5,1]
 => 1000010 => 1000100 => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => 1011110 => 1101110 => 1
[2,1,1,2] => [2,2,1,1]
 => 110110 => 111010 => 1
[2,1,2,1] => [2,2,1,1]
 => 110110 => 111010 => 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
Mapping
Mp00040: Integer compositions —to partition⟶ 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%
Values
[1] => [1]
 => [[1]]
 => [1] => 1
[1,1] => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[2] => [2]
 => [[1,2]]
 => [2] => 2
[1,1,1] => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[1,2] => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[2,1] => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[3] => [3]
 => [[1,2,3]]
 => [3] => 3
[1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[1,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,2,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,3] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[2,1,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[2,2] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[3,1] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[4] => [4]
 => [[1,2,3,4]]
 => [4] => 4
[1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[1,1,1,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,1,2,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,1,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,2,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,2,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[1,3,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,4] => [4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[2,1,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[2,1,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[2,3] => [3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 2
[3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 2
[4,1] => [4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[5] => [5]
 => [[1,2,3,4,5]]
 => [5] => 5
[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
[1,1,1,1,2] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,1,1,3] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,1,2,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 1
[1,1,3,1] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 1
[1,1,4] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[1,2,1,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 1
[1,2,2,1] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 1
[1,2,3] => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 1
[1,3,1,1] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 1
[1,3,2] => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 1
[1,4,1] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 1
[1,5] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 1
[2,1,1,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 1
[2,1,2,1] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 10 => [1,1] => 1
[1,1] => [1,1]
 => 110 => [2,1] => 1
[2] => [2]
 => 100 => [1,2] => 2
[1,1,1] => [1,1,1]
 => 1110 => [3,1] => 1
[1,2] => [2,1]
 => 1010 => [1,1,1,1] => 1
[2,1] => [2,1]
 => 1010 => [1,1,1,1] => 1
[3] => [3]
 => 1000 => [1,3] => 3
[1,1,1,1] => [1,1,1,1]
 => 11110 => [4,1] => 1
[1,1,2] => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[1,2,1] => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[1,3] => [3,1]
 => 10010 => [1,2,1,1] => 1
[2,1,1] => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[2,2] => [2,2]
 => 1100 => [2,2] => 2
[3,1] => [3,1]
 => 10010 => [1,2,1,1] => 1
[4] => [4]
 => 10000 => [1,4] => 4
[1,1,1,1,1] => [1,1,1,1,1]
 => 111110 => [5,1] => 1
[1,1,1,2] => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[1,1,2,1] => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[1,1,3] => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[1,2,1,1] => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[1,2,2] => [2,2,1]
 => 11010 => [2,1,1,1] => 1
[1,3,1] => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[1,4] => [4,1]
 => 100010 => [1,3,1,1] => 1
[2,1,1,1] => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[2,1,2] => [2,2,1]
 => 11010 => [2,1,1,1] => 1
[2,2,1] => [2,2,1]
 => 11010 => [2,1,1,1] => 1
[2,3] => [3,2]
 => 10100 => [1,1,1,2] => 2
[3,1,1] => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[3,2] => [3,2]
 => 10100 => [1,1,1,2] => 2
[4,1] => [4,1]
 => 100010 => [1,3,1,1] => 1
[5] => [5]
 => 100000 => [1,5] => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1111110 => [6,1] => 1
[1,1,1,1,2] => [2,1,1,1,1]
 => 1011110 => [1,1,4,1] => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => 1011110 => [1,1,4,1] => 1
[1,1,1,3] => [3,1,1,1]
 => 1001110 => [1,2,3,1] => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => 1011110 => [1,1,4,1] => 1
[1,1,2,2] => [2,2,1,1]
 => 110110 => [2,1,2,1] => 1
[1,1,3,1] => [3,1,1,1]
 => 1001110 => [1,2,3,1] => 1
[1,1,4] => [4,1,1]
 => 1000110 => [1,3,2,1] => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => 1011110 => [1,1,4,1] => 1
[1,2,1,2] => [2,2,1,1]
 => 110110 => [2,1,2,1] => 1
[1,2,2,1] => [2,2,1,1]
 => 110110 => [2,1,2,1] => 1
[1,2,3] => [3,2,1]
 => 101010 => [1,1,1,1,1,1] => 1
[1,3,1,1] => [3,1,1,1]
 => 1001110 => [1,2,3,1] => 1
[1,3,2] => [3,2,1]
 => 101010 => [1,1,1,1,1,1] => 1
[1,4,1] => [4,1,1]
 => 1000110 => [1,3,2,1] => 1
[1,5] => [5,1]
 => 1000010 => [1,4,1,1] => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => 1011110 => [1,1,4,1] => 1
[2,1,1,2] => [2,2,1,1]
 => 110110 => [2,1,2,1] => 1
[2,1,2,1] => [2,2,1,1]
 => 110110 => [2,1,2,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ 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
[1,1] => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[2] => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[1,1,1] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
[1,2] => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[2,1] => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[3] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
[1,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[1,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[1,3] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[2,1,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[2,2] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,1] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[4] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 1
[1,1,1,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[1,1,2,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[1,1,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[1,2,1,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[1,2,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 1
[1,3,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[1,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 1
[2,1,1,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[2,1,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 1
[2,2,1] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 1
[2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 2
[3,1,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[3,2] => [3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 2
[4,1] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 1
[5] => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 1
[1,1,1,1,2] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,3,4,5,6],[2]]
 => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,3,4,5,6],[2]]
 => 1
[1,1,1,3] => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [[1,4,5,6],[2],[3]]
 => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,3,4,5,6],[2]]
 => 1
[1,1,2,2] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[1,3,5,6],[2,4]]
 => 1
[1,1,3,1] => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [[1,4,5,6],[2],[3]]
 => 1
[1,1,4] => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [[1,5,6],[2],[3],[4]]
 => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,3,4,5,6],[2]]
 => 1
[1,2,1,2] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[1,3,5,6],[2,4]]
 => 1
[1,2,2,1] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[1,3,5,6],[2,4]]
 => 1
[1,2,3] => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [[1,4,6],[2,5],[3]]
 => 1
[1,3,1,1] => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [[1,4,5,6],[2],[3]]
 => 1
[1,3,2] => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [[1,4,6],[2,5],[3]]
 => 1
[1,4,1] => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [[1,5,6],[2],[3],[4]]
 => 1
[1,5] => [5,1]
 => [[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,3,4,5,6],[2]]
 => 1
[2,1,1,2] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[1,3,5,6],[2,4]]
 => 1
[2,1,2,1] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[1,3,5,6],[2,4]]
 => 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [[1]]
 => 1
[1,1] => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[2] => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[1,1,1] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
[1,2] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[2,1] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[3] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
[1,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[1,2,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[1,3] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[2,1,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[2,2] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,1] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[4] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 1
[1,1,1,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[1,1,2,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[1,1,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[1,2,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[1,2,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
[1,3,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[1,4] => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 1
[2,1,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[2,1,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
[2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
[2,3] => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
[3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
[4,1] => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 1
[5] => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 1
[1,1,1,1,2] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 1
[1,1,1,3] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 1
[1,1,2,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 1
[1,1,3,1] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => 1
[1,1,4] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [[1,2,3],[4],[5],[6]]
 => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 1
[1,2,1,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 1
[1,2,2,1] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 1
[1,2,3] => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 1
[1,3,1,1] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => 1
[1,3,2] => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 1
[1,4,1] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [[1,2,3],[4],[5],[6]]
 => 1
[1,5] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 1
[2,1,1,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 1
[2,1,2,1] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000993
(load all 2 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ 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%
Values
[1] => [1]
 => [1]
 => ? = 1
[1,1] => [1,1]
 => [2]
 => 1
[2] => [2]
 => [1,1]
 => 2
[1,1,1] => [1,1,1]
 => [3]
 => 1
[1,2] => [2,1]
 => [2,1]
 => 1
[2,1] => [2,1]
 => [2,1]
 => 1
[3] => [3]
 => [1,1,1]
 => 3
[1,1,1,1] => [1,1,1,1]
 => [4]
 => 1
[1,1,2] => [2,1,1]
 => [3,1]
 => 1
[1,2,1] => [2,1,1]
 => [3,1]
 => 1
[1,3] => [3,1]
 => [2,1,1]
 => 1
[2,1,1] => [2,1,1]
 => [3,1]
 => 1
[2,2] => [2,2]
 => [2,2]
 => 2
[3,1] => [3,1]
 => [2,1,1]
 => 1
[4] => [4]
 => [1,1,1,1]
 => 4
[1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1
[1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 1
[1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 1
[1,1,3] => [3,1,1]
 => [3,1,1]
 => 1
[1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 1
[1,2,2] => [2,2,1]
 => [3,2]
 => 1
[1,3,1] => [3,1,1]
 => [3,1,1]
 => 1
[1,4] => [4,1]
 => [2,1,1,1]
 => 1
[2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 1
[2,1,2] => [2,2,1]
 => [3,2]
 => 1
[2,2,1] => [2,2,1]
 => [3,2]
 => 1
[2,3] => [3,2]
 => [2,2,1]
 => 2
[3,1,1] => [3,1,1]
 => [3,1,1]
 => 1
[3,2] => [3,2]
 => [2,2,1]
 => 2
[4,1] => [4,1]
 => [2,1,1,1]
 => 1
[5] => [5]
 => [1,1,1,1,1]
 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [6]
 => 1
[1,1,1,1,2] => [2,1,1,1,1]
 => [5,1]
 => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [5,1]
 => 1
[1,1,1,3] => [3,1,1,1]
 => [4,1,1]
 => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [5,1]
 => 1
[1,1,2,2] => [2,2,1,1]
 => [4,2]
 => 1
[1,1,3,1] => [3,1,1,1]
 => [4,1,1]
 => 1
[1,1,4] => [4,1,1]
 => [3,1,1,1]
 => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [5,1]
 => 1
[1,2,1,2] => [2,2,1,1]
 => [4,2]
 => 1
[1,2,2,1] => [2,2,1,1]
 => [4,2]
 => 1
[1,2,3] => [3,2,1]
 => [3,2,1]
 => 1
[1,3,1,1] => [3,1,1,1]
 => [4,1,1]
 => 1
[1,3,2] => [3,2,1]
 => [3,2,1]
 => 1
[1,4,1] => [4,1,1]
 => [3,1,1,1]
 => 1
[1,5] => [5,1]
 => [2,1,1,1,1]
 => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [5,1]
 => 1
[2,1,1,2] => [2,2,1,1]
 => [4,2]
 => 1
[2,1,2,1] => [2,2,1,1]
 => [4,2]
 => 1
[2,1,3] => [3,2,1]
 => [3,2,1]
 => 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001038
(load all 6 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 95%●distinct values known / distinct values provided: 44%
Values
[1] => [1]
 => []
 => []
 => ? = 1
[1,1] => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {1,2}
[2] => [2]
 => []
 => []
 => ? ∊ {1,2}
[1,1,1] => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,2] => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,3}
[2,1] => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,3}
[3] => [3]
 => []
 => []
 => ? ∊ {1,1,3}
[1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,2] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,2,1] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,3] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,4}
[2,1,1] => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,2] => [2,2]
 => [2]
 => [1,0,1,0]
 => 2
[3,1] => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,4}
[4] => [4]
 => []
 => []
 => ? ∊ {1,1,4}
[1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,3] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,2] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,3,1] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,4] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,5}
[2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[2,1,2] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1] => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[3,1,1] => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[3,2] => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
[4,1] => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,5}
[5] => [5]
 => []
 => []
 => ? ∊ {1,1,5}
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,1,1,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,2,2] => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,3,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,1,4] => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,2,1,2] => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,2,2,1] => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,2,3] => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,3,1,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,3,2] => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,4,1] => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,5] => [5,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,6}
[2,1,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1,2] => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,2,1] => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,3] => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1,1] => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,2,2] => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
[2,3,1] => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,4] => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
[3,1,1,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[3,1,2] => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,2,1] => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,3] => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
[4,1,1] => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[4,2] => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
[5,1] => [5,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,6}
[6] => [6]
 => []
 => []
 => ? ∊ {1,1,6}
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,6] => [6,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,7}
[6,1] => [6,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,7}
[7] => [7]
 => []
 => []
 => ? ∊ {1,1,7}
[1,7] => [7,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,8}
[7,1] => [7,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,8}
[8] => [8]
 => []
 => []
 => ? ∊ {1,1,8}
[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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1,9}
[1,8] => [8,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,1,9}
[8,1] => [8,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,1,9}
[9] => [9]
 => []
 => []
 => ? ∊ {1,1,1,9}
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000667
(load all 2 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
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: 91%●distinct values known / distinct values provided: 33%
Values
[1] => [1]
 => []
 => ?
 => ? = 1
[1,1] => [1,1]
 => [1]
 => []
 => ? ∊ {1,2}
[2] => [2]
 => []
 => ?
 => ? ∊ {1,2}
[1,1,1] => [1,1,1]
 => [1,1]
 => [1]
 => 1
[1,2] => [2,1]
 => [1]
 => []
 => ? ∊ {1,1,3}
[2,1] => [2,1]
 => [1]
 => []
 => ? ∊ {1,1,3}
[3] => [3]
 => []
 => ?
 => ? ∊ {1,1,3}
[1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,1] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[1,3] => [3,1]
 => [1]
 => []
 => ? ∊ {1,1,2,4}
[2,1,1] => [2,1,1]
 => [1,1]
 => [1]
 => 1
[2,2] => [2,2]
 => [2]
 => []
 => ? ∊ {1,1,2,4}
[3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {1,1,2,4}
[4] => [4]
 => []
 => ?
 => ? ∊ {1,1,2,4}
[1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,3] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,2] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[1,3,1] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[1,4] => [4,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,5}
[2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,2] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,2,1] => [2,2,1]
 => [2,1]
 => [1]
 => 1
[2,3] => [3,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,5}
[3,1,1] => [3,1,1]
 => [1,1]
 => [1]
 => 1
[3,2] => [3,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,5}
[4,1] => [4,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,5}
[5] => [5]
 => []
 => ?
 => ? ∊ {1,1,2,2,5}
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,1,1,1,2] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,2,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,2,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,2,2] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,1,3,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,4] => [4,1,1]
 => [1,1]
 => [1]
 => 1
[1,2,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,1,2] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,2,2,1] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,2,3] => [3,2,1]
 => [2,1]
 => [1]
 => 1
[1,3,1,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => [3,2,1]
 => [2,1]
 => [1]
 => 1
[1,4,1] => [4,1,1]
 => [1,1]
 => [1]
 => 1
[1,5] => [5,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,3,6}
[2,1,1,1,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,1,1,2] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[2,1,2,1] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[2,1,3] => [3,2,1]
 => [2,1]
 => [1]
 => 1
[2,2,1,1] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[2,2,2] => [2,2,2]
 => [2,2]
 => [2]
 => 2
[2,3,1] => [3,2,1]
 => [2,1]
 => [1]
 => 1
[2,4] => [4,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,3,6}
[3,1,1,1] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[3,1,2] => [3,2,1]
 => [2,1]
 => [1]
 => 1
[3,2,1] => [3,2,1]
 => [2,1]
 => [1]
 => 1
[3,3] => [3,3]
 => [3]
 => []
 => ? ∊ {1,1,2,2,3,6}
[4,1,1] => [4,1,1]
 => [1,1]
 => [1]
 => 1
[4,2] => [4,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,3,6}
[5,1] => [5,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,3,6}
[6] => [6]
 => []
 => ?
 => ? ∊ {1,1,2,2,3,6}
[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
[1,1,1,1,1,2] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,1,1,1,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,1,1,2,2] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,3,1] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,4] => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,6] => [6,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,3,3,7}
[2,5] => [5,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,3,3,7}
[3,4] => [4,3]
 => [3]
 => []
 => ? ∊ {1,1,2,2,3,3,7}
[4,3] => [4,3]
 => [3]
 => []
 => ? ∊ {1,1,2,2,3,3,7}
[5,2] => [5,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,3,3,7}
[6,1] => [6,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,3,3,7}
[7] => [7]
 => []
 => ?
 => ? ∊ {1,1,2,2,3,3,7}
[1,7] => [7,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,3,3,4,8}
[2,6] => [6,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,3,3,4,8}
[3,5] => [5,3]
 => [3]
 => []
 => ? ∊ {1,1,2,2,3,3,4,8}
[4,4] => [4,4]
 => [4]
 => []
 => ? ∊ {1,1,2,2,3,3,4,8}
[5,3] => [5,3]
 => [3]
 => []
 => ? ∊ {1,1,2,2,3,3,4,8}
[6,2] => [6,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,3,3,4,8}
[7,1] => [7,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,3,3,4,8}
[8] => [8]
 => []
 => ?
 => ? ∊ {1,1,2,2,3,3,4,8}
[1,8] => [8,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,3,3,4,4,9}
[2,7] => [7,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,3,3,4,4,9}
[3,6] => [6,3]
 => [3]
 => []
 => ? ∊ {1,1,2,2,3,3,4,4,9}
[4,5] => [5,4]
 => [4]
 => []
 => ? ∊ {1,1,2,2,3,3,4,4,9}
[5,4] => [5,4]
 => [4]
 => []
 => ? ∊ {1,1,2,2,3,3,4,4,9}
[6,3] => [6,3]
 => [3]
 => []
 => ? ∊ {1,1,2,2,3,3,4,4,9}
[7,2] => [7,2]
 => [2]
 => []
 => ? ∊ {1,1,2,2,3,3,4,4,9}
[8,1] => [8,1]
 => [1]
 => []
 => ? ∊ {1,1,2,2,3,3,4,4,9}
[9] => [9]
 => []
 => ?
 => ? ∊ {1,1,2,2,3,3,4,4,9}
Description
The greatest common divisor of the parts of the partition.
The following 82 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001571The Cartan determinant of the integer partition. St000990The first ascent of a permutation. St000655The length of the minimal rise of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000781The number of proper colouring schemes of a Ferrers diagram. St000700The protection number of an ordered tree. St000617The number of global maxima of a Dyck path. St000627The exponent of a binary word. St000847The number of standard Young tableaux whose descent set is the binary word. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001256Number of simple reflexive modules that are 2-stable reflexive. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000654The first descent of a permutation. St001075The minimal size of a block of a set partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001829The common independence number of a graph. St000914The sum of the values of the Möbius function of a poset. St001119The length of a shortest maximal path in a graph. St001316The domatic number of a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000908The length of the shortest maximal antichain in a poset. St001322The size of a minimal independent dominating set in a graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001901The largest multiplicity of an irreducible representation 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001890The maximum magnitude of the Möbius function of a poset. 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. St000210Minimum over maximum difference of elements in cycles. St000487The length of the shortest cycle of a permutation. St000906The length of the shortest maximal chain in a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St000090The variation of a composition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000314The number of left-to-right-maxima of a permutation. St000310The minimal degree of a vertex of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001363The Euler characteristic of a graph according to Knill. St001339The irredundance number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.