Your data matches 77 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000655
(load all 9 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => 1
[1,2] => [1,0,1,0]
 => 1
[2,1] => [1,1,0,0]
 => 2
[1,2,3] => [1,0,1,0,1,0]
 => 1
[1,3,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [1,1,0,0,1,0]
 => 1
[2,3,1] => [1,1,0,1,0,0]
 => 1
[3,1,2] => [1,1,1,0,0,0]
 => 3
[3,2,1] => [1,1,1,0,0,0]
 => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => 1
Description
The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].
Matching statistic: St000657
(load all 10 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => 1
[1,2] => [1,0,1,0]
 => [1,1] => 1
[2,1] => [1,1,0,0]
 => [2] => 2
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,2] => 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1] => 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,1] => 1
[3,1,2] => [1,1,1,0,0,0]
 => [3] => 3
[3,2,1] => [1,1,1,0,0,0]
 => [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,2,1] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,1,1] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1,1] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,2] => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,2] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,1] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,1] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,1] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,1] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [3,1] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [3,1] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000297
Mapping
Mp00108: Permutations —cycle type⟶ 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
[1,2] => [1,1]
 => [2]
 => 100 => 1
[2,1] => [2]
 => [1,1]
 => 110 => 2
[1,2,3] => [1,1,1]
 => [3]
 => 1000 => 1
[1,3,2] => [2,1]
 => [2,1]
 => 1010 => 1
[2,1,3] => [2,1]
 => [2,1]
 => 1010 => 1
[2,3,1] => [3]
 => [1,1,1]
 => 1110 => 3
[3,1,2] => [3]
 => [1,1,1]
 => 1110 => 3
[3,2,1] => [2,1]
 => [2,1]
 => 1010 => 1
[1,2,3,4] => [1,1,1,1]
 => [4]
 => 10000 => 1
[1,2,4,3] => [2,1,1]
 => [3,1]
 => 10010 => 1
[1,3,2,4] => [2,1,1]
 => [3,1]
 => 10010 => 1
[1,3,4,2] => [3,1]
 => [2,1,1]
 => 10110 => 1
[1,4,2,3] => [3,1]
 => [2,1,1]
 => 10110 => 1
[1,4,3,2] => [2,1,1]
 => [3,1]
 => 10010 => 1
[2,1,3,4] => [2,1,1]
 => [3,1]
 => 10010 => 1
[2,1,4,3] => [2,2]
 => [2,2]
 => 1100 => 2
[2,3,1,4] => [3,1]
 => [2,1,1]
 => 10110 => 1
[2,3,4,1] => [4]
 => [1,1,1,1]
 => 11110 => 4
[2,4,1,3] => [4]
 => [1,1,1,1]
 => 11110 => 4
[2,4,3,1] => [3,1]
 => [2,1,1]
 => 10110 => 1
[3,1,2,4] => [3,1]
 => [2,1,1]
 => 10110 => 1
[3,1,4,2] => [4]
 => [1,1,1,1]
 => 11110 => 4
[3,2,1,4] => [2,1,1]
 => [3,1]
 => 10010 => 1
[3,2,4,1] => [3,1]
 => [2,1,1]
 => 10110 => 1
[3,4,1,2] => [2,2]
 => [2,2]
 => 1100 => 2
[3,4,2,1] => [4]
 => [1,1,1,1]
 => 11110 => 4
[4,1,2,3] => [4]
 => [1,1,1,1]
 => 11110 => 4
[4,1,3,2] => [3,1]
 => [2,1,1]
 => 10110 => 1
[4,2,1,3] => [3,1]
 => [2,1,1]
 => 10110 => 1
[4,2,3,1] => [2,1,1]
 => [3,1]
 => 10010 => 1
[4,3,1,2] => [4]
 => [1,1,1,1]
 => 11110 => 4
[4,3,2,1] => [2,2]
 => [2,2]
 => 1100 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => 100000 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,2,4,5,3] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,2,5,3,4] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 10100 => 1
[1,3,4,2,5] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,3,4,5,2] => [4,1]
 => [2,1,1,1]
 => 101110 => 1
[1,3,5,2,4] => [4,1]
 => [2,1,1,1]
 => 101110 => 1
[1,3,5,4,2] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,4,2,3,5] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,4,2,5,3] => [4,1]
 => [2,1,1,1]
 => 101110 => 1
[1,4,3,2,5] => [2,1,1,1]
 => [4,1]
 => 100010 => 1
[1,4,3,5,2] => [3,1,1]
 => [3,1,1]
 => 100110 => 1
[1,4,5,2,3] => [2,2,1]
 => [3,2]
 => 10100 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ 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
[1,2] => [1,1]
 => 110 => 110 => 1
[2,1] => [2]
 => 100 => 010 => 2
[1,2,3] => [1,1,1]
 => 1110 => 1110 => 1
[1,3,2] => [2,1]
 => 1010 => 1100 => 1
[2,1,3] => [2,1]
 => 1010 => 1100 => 1
[2,3,1] => [3]
 => 1000 => 0010 => 3
[3,1,2] => [3]
 => 1000 => 0010 => 3
[3,2,1] => [2,1]
 => 1010 => 1100 => 1
[1,2,3,4] => [1,1,1,1]
 => 11110 => 11110 => 1
[1,2,4,3] => [2,1,1]
 => 10110 => 11010 => 1
[1,3,2,4] => [2,1,1]
 => 10110 => 11010 => 1
[1,3,4,2] => [3,1]
 => 10010 => 10100 => 1
[1,4,2,3] => [3,1]
 => 10010 => 10100 => 1
[1,4,3,2] => [2,1,1]
 => 10110 => 11010 => 1
[2,1,3,4] => [2,1,1]
 => 10110 => 11010 => 1
[2,1,4,3] => [2,2]
 => 1100 => 0110 => 2
[2,3,1,4] => [3,1]
 => 10010 => 10100 => 1
[2,3,4,1] => [4]
 => 10000 => 00010 => 4
[2,4,1,3] => [4]
 => 10000 => 00010 => 4
[2,4,3,1] => [3,1]
 => 10010 => 10100 => 1
[3,1,2,4] => [3,1]
 => 10010 => 10100 => 1
[3,1,4,2] => [4]
 => 10000 => 00010 => 4
[3,2,1,4] => [2,1,1]
 => 10110 => 11010 => 1
[3,2,4,1] => [3,1]
 => 10010 => 10100 => 1
[3,4,1,2] => [2,2]
 => 1100 => 0110 => 2
[3,4,2,1] => [4]
 => 10000 => 00010 => 4
[4,1,2,3] => [4]
 => 10000 => 00010 => 4
[4,1,3,2] => [3,1]
 => 10010 => 10100 => 1
[4,2,1,3] => [3,1]
 => 10010 => 10100 => 1
[4,2,3,1] => [2,1,1]
 => 10110 => 11010 => 1
[4,3,1,2] => [4]
 => 10000 => 00010 => 4
[4,3,2,1] => [2,2]
 => 1100 => 0110 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => 111110 => 111110 => 1
[1,2,3,5,4] => [2,1,1,1]
 => 101110 => 110110 => 1
[1,2,4,3,5] => [2,1,1,1]
 => 101110 => 110110 => 1
[1,2,4,5,3] => [3,1,1]
 => 100110 => 101010 => 1
[1,2,5,3,4] => [3,1,1]
 => 100110 => 101010 => 1
[1,2,5,4,3] => [2,1,1,1]
 => 101110 => 110110 => 1
[1,3,2,4,5] => [2,1,1,1]
 => 101110 => 110110 => 1
[1,3,2,5,4] => [2,2,1]
 => 11010 => 11100 => 1
[1,3,4,2,5] => [3,1,1]
 => 100110 => 101010 => 1
[1,3,4,5,2] => [4,1]
 => 100010 => 100100 => 1
[1,3,5,2,4] => [4,1]
 => 100010 => 100100 => 1
[1,3,5,4,2] => [3,1,1]
 => 100110 => 101010 => 1
[1,4,2,3,5] => [3,1,1]
 => 100110 => 101010 => 1
[1,4,2,5,3] => [4,1]
 => 100010 => 100100 => 1
[1,4,3,2,5] => [2,1,1,1]
 => 101110 => 110110 => 1
[1,4,3,5,2] => [3,1,1]
 => 100110 => 101010 => 1
[1,4,5,2,3] => [2,2,1]
 => 11010 => 11100 => 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
Mapping
Mp00108: Permutations —cycle type⟶ 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
[1,2] => [1,1]
 => [[1],[2]]
 => [1,1] => 1
[2,1] => [2]
 => [[1,2]]
 => [2] => 2
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[2,3,1] => [3]
 => [[1,2,3]]
 => [3] => 3
[3,1,2] => [3]
 => [[1,2,3]]
 => [3] => 3
[3,2,1] => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[1,3,4,2] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[1,4,2,3] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[1,4,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[2,1,3,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[2,3,1,4] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[2,3,4,1] => [4]
 => [[1,2,3,4]]
 => [4] => 4
[2,4,1,3] => [4]
 => [[1,2,3,4]]
 => [4] => 4
[2,4,3,1] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[3,1,2,4] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[3,1,4,2] => [4]
 => [[1,2,3,4]]
 => [4] => 4
[3,2,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[3,2,4,1] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[3,4,2,1] => [4]
 => [[1,2,3,4]]
 => [4] => 4
[4,1,2,3] => [4]
 => [[1,2,3,4]]
 => [4] => 4
[4,1,3,2] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[4,2,1,3] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[4,2,3,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 1
[4,3,1,2] => [4]
 => [[1,2,3,4]]
 => [4] => 4
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,2,4,5,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,2,5,3,4] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,2,5,4,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,3,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
[1,3,4,2,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,3,4,5,2] => [4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[1,3,5,2,4] => [4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[1,3,5,4,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,4,2,3,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,4,2,5,3] => [4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[1,4,3,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 1
[1,4,3,5,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 1
[1,4,5,2,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ 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
[1,2] => [1,1]
 => 110 => [2,1] => 1
[2,1] => [2]
 => 100 => [1,2] => 2
[1,2,3] => [1,1,1]
 => 1110 => [3,1] => 1
[1,3,2] => [2,1]
 => 1010 => [1,1,1,1] => 1
[2,1,3] => [2,1]
 => 1010 => [1,1,1,1] => 1
[2,3,1] => [3]
 => 1000 => [1,3] => 3
[3,1,2] => [3]
 => 1000 => [1,3] => 3
[3,2,1] => [2,1]
 => 1010 => [1,1,1,1] => 1
[1,2,3,4] => [1,1,1,1]
 => 11110 => [4,1] => 1
[1,2,4,3] => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[1,3,2,4] => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[1,3,4,2] => [3,1]
 => 10010 => [1,2,1,1] => 1
[1,4,2,3] => [3,1]
 => 10010 => [1,2,1,1] => 1
[1,4,3,2] => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[2,1,3,4] => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[2,1,4,3] => [2,2]
 => 1100 => [2,2] => 2
[2,3,1,4] => [3,1]
 => 10010 => [1,2,1,1] => 1
[2,3,4,1] => [4]
 => 10000 => [1,4] => 4
[2,4,1,3] => [4]
 => 10000 => [1,4] => 4
[2,4,3,1] => [3,1]
 => 10010 => [1,2,1,1] => 1
[3,1,2,4] => [3,1]
 => 10010 => [1,2,1,1] => 1
[3,1,4,2] => [4]
 => 10000 => [1,4] => 4
[3,2,1,4] => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[3,2,4,1] => [3,1]
 => 10010 => [1,2,1,1] => 1
[3,4,1,2] => [2,2]
 => 1100 => [2,2] => 2
[3,4,2,1] => [4]
 => 10000 => [1,4] => 4
[4,1,2,3] => [4]
 => 10000 => [1,4] => 4
[4,1,3,2] => [3,1]
 => 10010 => [1,2,1,1] => 1
[4,2,1,3] => [3,1]
 => 10010 => [1,2,1,1] => 1
[4,2,3,1] => [2,1,1]
 => 10110 => [1,1,2,1] => 1
[4,3,1,2] => [4]
 => 10000 => [1,4] => 4
[4,3,2,1] => [2,2]
 => 1100 => [2,2] => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => 111110 => [5,1] => 1
[1,2,3,5,4] => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[1,2,4,3,5] => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[1,2,4,5,3] => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[1,2,5,3,4] => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[1,2,5,4,3] => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[1,3,2,4,5] => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[1,3,2,5,4] => [2,2,1]
 => 11010 => [2,1,1,1] => 1
[1,3,4,2,5] => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[1,3,4,5,2] => [4,1]
 => 100010 => [1,3,1,1] => 1
[1,3,5,2,4] => [4,1]
 => 100010 => [1,3,1,1] => 1
[1,3,5,4,2] => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[1,4,2,3,5] => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[1,4,2,5,3] => [4,1]
 => 100010 => [1,3,1,1] => 1
[1,4,3,2,5] => [2,1,1,1]
 => 101110 => [1,1,3,1] => 1
[1,4,3,5,2] => [3,1,1]
 => 100110 => [1,2,2,1] => 1
[1,4,5,2,3] => [2,2,1]
 => 11010 => [2,1,1,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ 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
[1,2] => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[2,1] => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
[1,3,2] => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[2,1,3] => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[2,3,1] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[3,1,2] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[3,2,1] => [2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
[1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[1,3,4,2] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[1,4,2,3] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[1,4,3,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[2,3,1,4] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[2,3,4,1] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[2,4,1,3] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[2,4,3,1] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[3,1,2,4] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[3,1,4,2] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[3,2,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[3,2,4,1] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,4,2,1] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[4,1,2,3] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[4,1,3,2] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[4,2,1,3] => [3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 1
[4,2,3,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 1
[4,3,1,2] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[1,3,4,5,2] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 1
[1,3,5,2,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 1
[1,3,5,4,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[1,4,2,5,3] => [4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 1
[1,4,3,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mapping
Mp00108: Permutations —cycle type⟶ 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
[1,2] => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 1
[2,1] => [2]
 => [[1,2]]
 => [[1],[2]]
 => 2
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[2,3,1] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[3,1,2] => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[3,2,1] => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[1,3,4,2] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[1,4,2,3] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[1,4,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[2,1,3,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[2,3,1,4] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[2,3,4,1] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[2,4,1,3] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[2,4,3,1] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[3,1,2,4] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[3,1,4,2] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[3,2,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[3,2,4,1] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[3,4,1,2] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,4,2,1] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[4,1,2,3] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[4,1,3,2] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[4,2,1,3] => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 1
[4,2,3,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[4,3,1,2] => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 4
[4,3,2,1] => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[1,2,3,4,5] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[1,2,5,3,4] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[1,2,5,4,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[1,3,4,5,2] => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 1
[1,3,5,2,4] => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 1
[1,3,5,4,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[1,4,2,3,5] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[1,4,2,5,3] => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 1
[1,4,3,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
[1,4,5,2,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001803
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [1]
 => [[1]]
 => 0 = 1 - 1
[1,2] => [1,1]
 => [2]
 => [[1,2]]
 => 0 = 1 - 1
[2,1] => [2]
 => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[1,2,3] => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
[1,3,2] => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 0 = 1 - 1
[2,1,3] => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 0 = 1 - 1
[2,3,1] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[3,1,2] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[3,2,1] => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 0 = 1 - 1
[1,2,3,4] => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
[1,2,4,3] => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 0 = 1 - 1
[1,3,2,4] => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 0 = 1 - 1
[1,3,4,2] => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0 = 1 - 1
[1,4,2,3] => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0 = 1 - 1
[1,4,3,2] => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 0 = 1 - 1
[2,1,3,4] => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 0 = 1 - 1
[2,1,4,3] => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[2,3,1,4] => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0 = 1 - 1
[2,3,4,1] => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
[2,4,1,3] => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
[2,4,3,1] => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0 = 1 - 1
[3,1,2,4] => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0 = 1 - 1
[3,1,4,2] => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
[3,2,1,4] => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 0 = 1 - 1
[3,2,4,1] => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0 = 1 - 1
[3,4,1,2] => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[3,4,2,1] => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
[4,1,2,3] => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
[4,1,3,2] => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0 = 1 - 1
[4,2,1,3] => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 0 = 1 - 1
[4,2,3,1] => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 0 = 1 - 1
[4,3,1,2] => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
[4,3,2,1] => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 0 = 1 - 1
[1,2,5,3,4] => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 0 = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 0 = 1 - 1
[1,3,4,2,5] => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 0 = 1 - 1
[1,3,4,5,2] => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 0 = 1 - 1
[1,3,5,2,4] => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 0 = 1 - 1
[1,3,5,4,2] => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 0 = 1 - 1
[1,4,2,3,5] => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 0 = 1 - 1
[1,4,2,5,3] => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 0 = 1 - 1
[1,4,3,2,5] => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 0 = 1 - 1
[1,4,5,2,3] => [2,2,1]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 0 = 1 - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001075
(load all 10 compositions to match this statistic)
Mapping
Mp00151: Permutations —to cycle type⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
 => ? = 1
[1,2] => {{1},{2}}
 => 1
[2,1] => {{1,2}}
 => 2
[1,2,3] => {{1},{2},{3}}
 => 1
[1,3,2] => {{1},{2,3}}
 => 1
[2,1,3] => {{1,2},{3}}
 => 1
[2,3,1] => {{1,2,3}}
 => 3
[3,1,2] => {{1,2,3}}
 => 3
[3,2,1] => {{1,3},{2}}
 => 1
[1,2,3,4] => {{1},{2},{3},{4}}
 => 1
[1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,3,2,4] => {{1},{2,3},{4}}
 => 1
[1,3,4,2] => {{1},{2,3,4}}
 => 1
[1,4,2,3] => {{1},{2,3,4}}
 => 1
[1,4,3,2] => {{1},{2,4},{3}}
 => 1
[2,1,3,4] => {{1,2},{3},{4}}
 => 1
[2,1,4,3] => {{1,2},{3,4}}
 => 2
[2,3,1,4] => {{1,2,3},{4}}
 => 1
[2,3,4,1] => {{1,2,3,4}}
 => 4
[2,4,1,3] => {{1,2,3,4}}
 => 4
[2,4,3,1] => {{1,2,4},{3}}
 => 1
[3,1,2,4] => {{1,2,3},{4}}
 => 1
[3,1,4,2] => {{1,2,3,4}}
 => 4
[3,2,1,4] => {{1,3},{2},{4}}
 => 1
[3,2,4,1] => {{1,3,4},{2}}
 => 1
[3,4,1,2] => {{1,3},{2,4}}
 => 2
[3,4,2,1] => {{1,2,3,4}}
 => 4
[4,1,2,3] => {{1,2,3,4}}
 => 4
[4,1,3,2] => {{1,2,4},{3}}
 => 1
[4,2,1,3] => {{1,3,4},{2}}
 => 1
[4,2,3,1] => {{1,4},{2},{3}}
 => 1
[4,3,1,2] => {{1,2,3,4}}
 => 4
[4,3,2,1] => {{1,4},{2,3}}
 => 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
 => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
 => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
 => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
 => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
 => 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
 => 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
 => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
 => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
 => 1
[1,4,5,3,2] => {{1},{2,3,4,5}}
 => 1
Description
The minimal size of a block of a set partition.
The following 67 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000667The greatest common divisor of the parts of the partition. St000990The first ascent of a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St001890The maximum magnitude of the Möbius function of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001128The exponens consonantiae of a partition. 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. St000210Minimum over maximum difference of elements in cycles. St000487The length of the shortest cycle of a permutation. St000668The least common multiple of the parts of the partition. 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. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer 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. St000654The first descent of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000456The monochromatic index of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000090The variation of a composition. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000314The number of left-to-right-maxima of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000618The number of self-evacuating tableaux of given shape. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000706The product of the factorials of the multiplicities of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001096The size of the overlap set of a permutation. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001884The number of borders of a binary word.