Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001176
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => 0
[1,-2] => [1]
 => 0
[-1,2] => [1]
 => 0
[-1,-2] => [1,1]
 => 1
[2,-1] => [2]
 => 0
[-2,1] => [2]
 => 0
[1,2,-3] => [1]
 => 0
[1,-2,3] => [1]
 => 0
[1,-2,-3] => [1,1]
 => 1
[-1,2,3] => [1]
 => 0
[-1,2,-3] => [1,1]
 => 1
[-1,-2,3] => [1,1]
 => 1
[-1,-2,-3] => [1,1,1]
 => 2
[1,3,-2] => [2]
 => 0
[1,-3,2] => [2]
 => 0
[-1,3,2] => [1]
 => 0
[-1,3,-2] => [2,1]
 => 1
[-1,-3,2] => [2,1]
 => 1
[-1,-3,-2] => [1]
 => 0
[2,1,-3] => [1]
 => 0
[2,-1,3] => [2]
 => 0
[2,-1,-3] => [2,1]
 => 1
[-2,1,3] => [2]
 => 0
[-2,1,-3] => [2,1]
 => 1
[-2,-1,-3] => [1]
 => 0
[2,3,-1] => [3]
 => 0
[2,-3,1] => [3]
 => 0
[-2,3,1] => [3]
 => 0
[-2,-3,-1] => [3]
 => 0
[3,1,-2] => [3]
 => 0
[3,-1,2] => [3]
 => 0
[-3,1,2] => [3]
 => 0
[-3,-1,-2] => [3]
 => 0
[3,2,-1] => [2]
 => 0
[3,-2,1] => [1]
 => 0
[3,-2,-1] => [2,1]
 => 1
[-3,2,1] => [2]
 => 0
[-3,-2,1] => [2,1]
 => 1
[-3,-2,-1] => [1]
 => 0
[1,2,3,-4] => [1]
 => 0
[1,2,-3,4] => [1]
 => 0
[1,2,-3,-4] => [1,1]
 => 1
[1,-2,3,4] => [1]
 => 0
[1,-2,3,-4] => [1,1]
 => 1
[1,-2,-3,4] => [1,1]
 => 1
[1,-2,-3,-4] => [1,1,1]
 => 2
[-1,2,3,4] => [1]
 => 0
[-1,2,3,-4] => [1,1]
 => 1
[-1,2,-3,4] => [1,1]
 => 1
[-1,2,-3,-4] => [1,1,1]
 => 2
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
(load all 6 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => []
 => 0
[1,-2] => [1]
 => []
 => 0
[-1,2] => [1]
 => []
 => 0
[-1,-2] => [1,1]
 => [1]
 => 1
[2,-1] => [2]
 => []
 => 0
[-2,1] => [2]
 => []
 => 0
[1,2,-3] => [1]
 => []
 => 0
[1,-2,3] => [1]
 => []
 => 0
[1,-2,-3] => [1,1]
 => [1]
 => 1
[-1,2,3] => [1]
 => []
 => 0
[-1,2,-3] => [1,1]
 => [1]
 => 1
[-1,-2,3] => [1,1]
 => [1]
 => 1
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => 2
[1,3,-2] => [2]
 => []
 => 0
[1,-3,2] => [2]
 => []
 => 0
[-1,3,2] => [1]
 => []
 => 0
[-1,3,-2] => [2,1]
 => [1]
 => 1
[-1,-3,2] => [2,1]
 => [1]
 => 1
[-1,-3,-2] => [1]
 => []
 => 0
[2,1,-3] => [1]
 => []
 => 0
[2,-1,3] => [2]
 => []
 => 0
[2,-1,-3] => [2,1]
 => [1]
 => 1
[-2,1,3] => [2]
 => []
 => 0
[-2,1,-3] => [2,1]
 => [1]
 => 1
[-2,-1,-3] => [1]
 => []
 => 0
[2,3,-1] => [3]
 => []
 => 0
[2,-3,1] => [3]
 => []
 => 0
[-2,3,1] => [3]
 => []
 => 0
[-2,-3,-1] => [3]
 => []
 => 0
[3,1,-2] => [3]
 => []
 => 0
[3,-1,2] => [3]
 => []
 => 0
[-3,1,2] => [3]
 => []
 => 0
[-3,-1,-2] => [3]
 => []
 => 0
[3,2,-1] => [2]
 => []
 => 0
[3,-2,1] => [1]
 => []
 => 0
[3,-2,-1] => [2,1]
 => [1]
 => 1
[-3,2,1] => [2]
 => []
 => 0
[-3,-2,1] => [2,1]
 => [1]
 => 1
[-3,-2,-1] => [1]
 => []
 => 0
[1,2,3,-4] => [1]
 => []
 => 0
[1,2,-3,4] => [1]
 => []
 => 0
[1,2,-3,-4] => [1,1]
 => [1]
 => 1
[1,-2,3,4] => [1]
 => []
 => 0
[1,-2,3,-4] => [1,1]
 => [1]
 => 1
[1,-2,-3,4] => [1,1]
 => [1]
 => 1
[1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => 2
[-1,2,3,4] => [1]
 => []
 => 0
[-1,2,3,-4] => [1,1]
 => [1]
 => 1
[-1,2,-3,4] => [1,1]
 => [1]
 => 1
[-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => 2
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000738
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: 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,-2] => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,2] => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,-2] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[2,-1] => [2]
 => [[1,2]]
 => 1 = 0 + 1
[-2,1] => [2]
 => [[1,2]]
 => 1 = 0 + 1
[1,2,-3] => [1]
 => [[1]]
 => 1 = 0 + 1
[1,-2,3] => [1]
 => [[1]]
 => 1 = 0 + 1
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[-1,2,3] => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[1,3,-2] => [2]
 => [[1,2]]
 => 1 = 0 + 1
[1,-3,2] => [2]
 => [[1,2]]
 => 1 = 0 + 1
[-1,3,2] => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,3,-2] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-1,-3,2] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-1,-3,-2] => [1]
 => [[1]]
 => 1 = 0 + 1
[2,1,-3] => [1]
 => [[1]]
 => 1 = 0 + 1
[2,-1,3] => [2]
 => [[1,2]]
 => 1 = 0 + 1
[2,-1,-3] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-2,1,3] => [2]
 => [[1,2]]
 => 1 = 0 + 1
[-2,1,-3] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-2,-1,-3] => [1]
 => [[1]]
 => 1 = 0 + 1
[2,3,-1] => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[2,-3,1] => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[-2,3,1] => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[3,1,-2] => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[3,-1,2] => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[-3,1,2] => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[3,2,-1] => [2]
 => [[1,2]]
 => 1 = 0 + 1
[3,-2,1] => [1]
 => [[1]]
 => 1 = 0 + 1
[3,-2,-1] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-3,2,1] => [2]
 => [[1,2]]
 => 1 = 0 + 1
[-3,-2,1] => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-3,-2,-1] => [1]
 => [[1]]
 => 1 = 0 + 1
[1,2,3,-4] => [1]
 => [[1]]
 => 1 = 0 + 1
[1,2,-3,4] => [1]
 => [[1]]
 => 1 = 0 + 1
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[1,-2,3,4] => [1]
 => [[1]]
 => 1 = 0 + 1
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[-1,2,3,4] => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St000141
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [[1]]
 => [1] => 0
[1,-2] => [1]
 => [[1]]
 => [1] => 0
[-1,2] => [1]
 => [[1]]
 => [1] => 0
[-1,-2] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[2,-1] => [2]
 => [[1,2]]
 => [1,2] => 0
[-2,1] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,2,-3] => [1]
 => [[1]]
 => [1] => 0
[1,-2,3] => [1]
 => [[1]]
 => [1] => 0
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,3] => [1]
 => [[1]]
 => [1] => 0
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[1,3,-2] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,-3,2] => [2]
 => [[1,2]]
 => [1,2] => 0
[-1,3,2] => [1]
 => [[1]]
 => [1] => 0
[-1,3,-2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[-1,-3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[-1,-3,-2] => [1]
 => [[1]]
 => [1] => 0
[2,1,-3] => [1]
 => [[1]]
 => [1] => 0
[2,-1,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[2,-1,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[-2,1,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[-2,1,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[-2,-1,-3] => [1]
 => [[1]]
 => [1] => 0
[2,3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[2,-3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-2,3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,-1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-3,1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,2,-1] => [2]
 => [[1,2]]
 => [1,2] => 0
[3,-2,1] => [1]
 => [[1]]
 => [1] => 0
[3,-2,-1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[-3,2,1] => [2]
 => [[1,2]]
 => [1,2] => 0
[-3,-2,1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[-3,-2,-1] => [1]
 => [[1]]
 => [1] => 0
[1,2,3,-4] => [1]
 => [[1]]
 => [1] => 0
[1,2,-3,4] => [1]
 => [[1]]
 => [1] => 0
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,3,4] => [1]
 => [[1]]
 => [1] => 0
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[-1,2,3,4] => [1]
 => [[1]]
 => [1] => 0
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000293
(load all 3 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => 10 => 01 => 0
[1,-2] => [1]
 => 10 => 01 => 0
[-1,2] => [1]
 => 10 => 01 => 0
[-1,-2] => [1,1]
 => 110 => 101 => 1
[2,-1] => [2]
 => 100 => 001 => 0
[-2,1] => [2]
 => 100 => 001 => 0
[1,2,-3] => [1]
 => 10 => 01 => 0
[1,-2,3] => [1]
 => 10 => 01 => 0
[1,-2,-3] => [1,1]
 => 110 => 101 => 1
[-1,2,3] => [1]
 => 10 => 01 => 0
[-1,2,-3] => [1,1]
 => 110 => 101 => 1
[-1,-2,3] => [1,1]
 => 110 => 101 => 1
[-1,-2,-3] => [1,1,1]
 => 1110 => 1101 => 2
[1,3,-2] => [2]
 => 100 => 001 => 0
[1,-3,2] => [2]
 => 100 => 001 => 0
[-1,3,2] => [1]
 => 10 => 01 => 0
[-1,3,-2] => [2,1]
 => 1010 => 0101 => 1
[-1,-3,2] => [2,1]
 => 1010 => 0101 => 1
[-1,-3,-2] => [1]
 => 10 => 01 => 0
[2,1,-3] => [1]
 => 10 => 01 => 0
[2,-1,3] => [2]
 => 100 => 001 => 0
[2,-1,-3] => [2,1]
 => 1010 => 0101 => 1
[-2,1,3] => [2]
 => 100 => 001 => 0
[-2,1,-3] => [2,1]
 => 1010 => 0101 => 1
[-2,-1,-3] => [1]
 => 10 => 01 => 0
[2,3,-1] => [3]
 => 1000 => 0001 => 0
[2,-3,1] => [3]
 => 1000 => 0001 => 0
[-2,3,1] => [3]
 => 1000 => 0001 => 0
[-2,-3,-1] => [3]
 => 1000 => 0001 => 0
[3,1,-2] => [3]
 => 1000 => 0001 => 0
[3,-1,2] => [3]
 => 1000 => 0001 => 0
[-3,1,2] => [3]
 => 1000 => 0001 => 0
[-3,-1,-2] => [3]
 => 1000 => 0001 => 0
[3,2,-1] => [2]
 => 100 => 001 => 0
[3,-2,1] => [1]
 => 10 => 01 => 0
[3,-2,-1] => [2,1]
 => 1010 => 0101 => 1
[-3,2,1] => [2]
 => 100 => 001 => 0
[-3,-2,1] => [2,1]
 => 1010 => 0101 => 1
[-3,-2,-1] => [1]
 => 10 => 01 => 0
[1,2,3,-4] => [1]
 => 10 => 01 => 0
[1,2,-3,4] => [1]
 => 10 => 01 => 0
[1,2,-3,-4] => [1,1]
 => 110 => 101 => 1
[1,-2,3,4] => [1]
 => 10 => 01 => 0
[1,-2,3,-4] => [1,1]
 => 110 => 101 => 1
[1,-2,-3,4] => [1,1]
 => 110 => 101 => 1
[1,-2,-3,-4] => [1,1,1]
 => 1110 => 1101 => 2
[-1,2,3,4] => [1]
 => 10 => 01 => 0
[-1,2,3,-4] => [1,1]
 => 110 => 101 => 1
[-1,2,-3,4] => [1,1]
 => 110 => 101 => 1
[-1,2,-3,-4] => [1,1,1]
 => 1110 => 1101 => 2
Description
The number of inversions of a binary word.
Matching statistic: St000377
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [1]
 => [1]
 => 0
[1,-2] => [1]
 => [1]
 => [1]
 => 0
[-1,2] => [1]
 => [1]
 => [1]
 => 0
[-1,-2] => [1,1]
 => [2]
 => [1,1]
 => 1
[2,-1] => [2]
 => [1,1]
 => [2]
 => 0
[-2,1] => [2]
 => [1,1]
 => [2]
 => 0
[1,2,-3] => [1]
 => [1]
 => [1]
 => 0
[1,-2,3] => [1]
 => [1]
 => [1]
 => 0
[1,-2,-3] => [1,1]
 => [2]
 => [1,1]
 => 1
[-1,2,3] => [1]
 => [1]
 => [1]
 => 0
[-1,2,-3] => [1,1]
 => [2]
 => [1,1]
 => 1
[-1,-2,3] => [1,1]
 => [2]
 => [1,1]
 => 1
[-1,-2,-3] => [1,1,1]
 => [3]
 => [1,1,1]
 => 2
[1,3,-2] => [2]
 => [1,1]
 => [2]
 => 0
[1,-3,2] => [2]
 => [1,1]
 => [2]
 => 0
[-1,3,2] => [1]
 => [1]
 => [1]
 => 0
[-1,3,-2] => [2,1]
 => [2,1]
 => [3]
 => 1
[-1,-3,2] => [2,1]
 => [2,1]
 => [3]
 => 1
[-1,-3,-2] => [1]
 => [1]
 => [1]
 => 0
[2,1,-3] => [1]
 => [1]
 => [1]
 => 0
[2,-1,3] => [2]
 => [1,1]
 => [2]
 => 0
[2,-1,-3] => [2,1]
 => [2,1]
 => [3]
 => 1
[-2,1,3] => [2]
 => [1,1]
 => [2]
 => 0
[-2,1,-3] => [2,1]
 => [2,1]
 => [3]
 => 1
[-2,-1,-3] => [1]
 => [1]
 => [1]
 => 0
[2,3,-1] => [3]
 => [1,1,1]
 => [2,1]
 => 0
[2,-3,1] => [3]
 => [1,1,1]
 => [2,1]
 => 0
[-2,3,1] => [3]
 => [1,1,1]
 => [2,1]
 => 0
[-2,-3,-1] => [3]
 => [1,1,1]
 => [2,1]
 => 0
[3,1,-2] => [3]
 => [1,1,1]
 => [2,1]
 => 0
[3,-1,2] => [3]
 => [1,1,1]
 => [2,1]
 => 0
[-3,1,2] => [3]
 => [1,1,1]
 => [2,1]
 => 0
[-3,-1,-2] => [3]
 => [1,1,1]
 => [2,1]
 => 0
[3,2,-1] => [2]
 => [1,1]
 => [2]
 => 0
[3,-2,1] => [1]
 => [1]
 => [1]
 => 0
[3,-2,-1] => [2,1]
 => [2,1]
 => [3]
 => 1
[-3,2,1] => [2]
 => [1,1]
 => [2]
 => 0
[-3,-2,1] => [2,1]
 => [2,1]
 => [3]
 => 1
[-3,-2,-1] => [1]
 => [1]
 => [1]
 => 0
[1,2,3,-4] => [1]
 => [1]
 => [1]
 => 0
[1,2,-3,4] => [1]
 => [1]
 => [1]
 => 0
[1,2,-3,-4] => [1,1]
 => [2]
 => [1,1]
 => 1
[1,-2,3,4] => [1]
 => [1]
 => [1]
 => 0
[1,-2,3,-4] => [1,1]
 => [2]
 => [1,1]
 => 1
[1,-2,-3,4] => [1,1]
 => [2]
 => [1,1]
 => 1
[1,-2,-3,-4] => [1,1,1]
 => [3]
 => [1,1,1]
 => 2
[-1,2,3,4] => [1]
 => [1]
 => [1]
 => 0
[-1,2,3,-4] => [1,1]
 => [2]
 => [1,1]
 => 1
[-1,2,-3,4] => [1,1]
 => [2]
 => [1,1]
 => 1
[-1,2,-3,-4] => [1,1,1]
 => [3]
 => [1,1,1]
 => 2
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000662
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [[1]]
 => [1] => 0
[1,-2] => [1]
 => [[1]]
 => [1] => 0
[-1,2] => [1]
 => [[1]]
 => [1] => 0
[-1,-2] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[2,-1] => [2]
 => [[1,2]]
 => [1,2] => 0
[-2,1] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,2,-3] => [1]
 => [[1]]
 => [1] => 0
[1,-2,3] => [1]
 => [[1]]
 => [1] => 0
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,3] => [1]
 => [[1]]
 => [1] => 0
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[1,3,-2] => [2]
 => [[1,2]]
 => [1,2] => 0
[1,-3,2] => [2]
 => [[1,2]]
 => [1,2] => 0
[-1,3,2] => [1]
 => [[1]]
 => [1] => 0
[-1,3,-2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[-1,-3,2] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[-1,-3,-2] => [1]
 => [[1]]
 => [1] => 0
[2,1,-3] => [1]
 => [[1]]
 => [1] => 0
[2,-1,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[2,-1,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[-2,1,3] => [2]
 => [[1,2]]
 => [1,2] => 0
[-2,1,-3] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[-2,-1,-3] => [1]
 => [[1]]
 => [1] => 0
[2,3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[2,-3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-2,3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,-1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-3,1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[3,2,-1] => [2]
 => [[1,2]]
 => [1,2] => 0
[3,-2,1] => [1]
 => [[1]]
 => [1] => 0
[3,-2,-1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[-3,2,1] => [2]
 => [[1,2]]
 => [1,2] => 0
[-3,-2,1] => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 1
[-3,-2,-1] => [1]
 => [[1]]
 => [1] => 0
[1,2,3,-4] => [1]
 => [[1]]
 => [1] => 0
[1,2,-3,4] => [1]
 => [[1]]
 => [1] => 0
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,3,4] => [1]
 => [[1]]
 => [1] => 0
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[-1,2,3,4] => [1]
 => [[1]]
 => [1] => 0
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St001034
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => []
 => []
 => 0
[1,-2] => [1]
 => []
 => []
 => 0
[-1,2] => [1]
 => []
 => []
 => 0
[-1,-2] => [1,1]
 => [1]
 => [1,0]
 => 1
[2,-1] => [2]
 => []
 => []
 => 0
[-2,1] => [2]
 => []
 => []
 => 0
[1,2,-3] => [1]
 => []
 => []
 => 0
[1,-2,3] => [1]
 => []
 => []
 => 0
[1,-2,-3] => [1,1]
 => [1]
 => [1,0]
 => 1
[-1,2,3] => [1]
 => []
 => []
 => 0
[-1,2,-3] => [1,1]
 => [1]
 => [1,0]
 => 1
[-1,-2,3] => [1,1]
 => [1]
 => [1,0]
 => 1
[-1,-2,-3] => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[1,3,-2] => [2]
 => []
 => []
 => 0
[1,-3,2] => [2]
 => []
 => []
 => 0
[-1,3,2] => [1]
 => []
 => []
 => 0
[-1,3,-2] => [2,1]
 => [1]
 => [1,0]
 => 1
[-1,-3,2] => [2,1]
 => [1]
 => [1,0]
 => 1
[-1,-3,-2] => [1]
 => []
 => []
 => 0
[2,1,-3] => [1]
 => []
 => []
 => 0
[2,-1,3] => [2]
 => []
 => []
 => 0
[2,-1,-3] => [2,1]
 => [1]
 => [1,0]
 => 1
[-2,1,3] => [2]
 => []
 => []
 => 0
[-2,1,-3] => [2,1]
 => [1]
 => [1,0]
 => 1
[-2,-1,-3] => [1]
 => []
 => []
 => 0
[2,3,-1] => [3]
 => []
 => []
 => 0
[2,-3,1] => [3]
 => []
 => []
 => 0
[-2,3,1] => [3]
 => []
 => []
 => 0
[-2,-3,-1] => [3]
 => []
 => []
 => 0
[3,1,-2] => [3]
 => []
 => []
 => 0
[3,-1,2] => [3]
 => []
 => []
 => 0
[-3,1,2] => [3]
 => []
 => []
 => 0
[-3,-1,-2] => [3]
 => []
 => []
 => 0
[3,2,-1] => [2]
 => []
 => []
 => 0
[3,-2,1] => [1]
 => []
 => []
 => 0
[3,-2,-1] => [2,1]
 => [1]
 => [1,0]
 => 1
[-3,2,1] => [2]
 => []
 => []
 => 0
[-3,-2,1] => [2,1]
 => [1]
 => [1,0]
 => 1
[-3,-2,-1] => [1]
 => []
 => []
 => 0
[1,2,3,-4] => [1]
 => []
 => []
 => 0
[1,2,-3,4] => [1]
 => []
 => []
 => 0
[1,2,-3,-4] => [1,1]
 => [1]
 => [1,0]
 => 1
[1,-2,3,4] => [1]
 => []
 => []
 => 0
[1,-2,3,-4] => [1,1]
 => [1]
 => [1,0]
 => 1
[1,-2,-3,4] => [1,1]
 => [1]
 => [1,0]
 => 1
[1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
[-1,2,3,4] => [1]
 => []
 => []
 => 0
[-1,2,3,-4] => [1,1]
 => [1]
 => [1,0]
 => 1
[-1,2,-3,4] => [1,1]
 => [1]
 => [1,0]
 => 1
[-1,2,-3,-4] => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 2
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Matching statistic: St000054
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[1,-2] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[-1,2] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[-1,-2] => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 1 + 1
[2,-1] => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[-2,1] => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[1,2,-3] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[1,-2,3] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[1,-2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 1 + 1
[-1,2,3] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[-1,2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 1 + 1
[-1,-2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 1 + 1
[-1,-2,-3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3 = 2 + 1
[1,3,-2] => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[1,-3,2] => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[-1,3,2] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[-1,3,-2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2 = 1 + 1
[-1,-3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2 = 1 + 1
[-1,-3,-2] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[2,1,-3] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[2,-1,3] => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[2,-1,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2 = 1 + 1
[-2,1,3] => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[-2,1,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2 = 1 + 1
[-2,-1,-3] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[2,3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[2,-3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[-2,3,1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[-2,-3,-1] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[3,1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[3,-1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[-3,1,2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[-3,-1,-2] => [3]
 => [[1,2,3]]
 => [1,2,3] => 1 = 0 + 1
[3,2,-1] => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[3,-2,1] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[3,-2,-1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2 = 1 + 1
[-3,2,1] => [2]
 => [[1,2]]
 => [1,2] => 1 = 0 + 1
[-3,-2,1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 2 = 1 + 1
[-3,-2,-1] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[1,2,3,-4] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[1,2,-3,4] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 1 + 1
[1,-2,3,4] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 1 + 1
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 1 + 1
[1,-2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3 = 2 + 1
[-1,2,3,4] => [1]
 => [[1]]
 => [1] => 1 = 0 + 1
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 1 + 1
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 2 = 1 + 1
[-1,2,-3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3 = 2 + 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Matching statistic: St000507
(load all 2 compositions to match this statistic)
Mapping
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[1,-2] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,2] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,-2] => [1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[2,-1] => [2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[-2,1] => [2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[1,2,-3] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[1,-2,3] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[1,-2,-3] => [1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[-1,2,3] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,2,-3] => [1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[-1,-2,3] => [1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[-1,-2,-3] => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
[1,3,-2] => [2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[1,-3,2] => [2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[-1,3,2] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,3,-2] => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-1,-3,2] => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-1,-3,-2] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[2,1,-3] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[2,-1,3] => [2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[2,-1,-3] => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-2,1,3] => [2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[-2,1,-3] => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-2,-1,-3] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[2,3,-1] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[2,-3,1] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[-2,3,1] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[-2,-3,-1] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[3,1,-2] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[3,-1,2] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[-3,1,2] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[-3,-1,-2] => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[3,2,-1] => [2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[3,-2,1] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[3,-2,-1] => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-3,2,1] => [2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[-3,-2,1] => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[-3,-2,-1] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[1,2,3,-4] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[1,2,-3,4] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[1,2,-3,-4] => [1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[1,-2,3,4] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[1,-2,3,-4] => [1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[1,-2,-3,4] => [1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[1,-2,-3,-4] => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
[-1,2,3,4] => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[-1,2,3,-4] => [1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[-1,2,-3,4] => [1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[-1,2,-3,-4] => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000734The last entry in the first row of a standard tableau. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000839The largest opener of a set partition. St001726The number of visible inversions of a permutation. St000029The depth of a permutation. St000224The sorting index of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000209Maximum difference of elements in cycles. St000316The number of non-left-to-right-maxima of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000653The last descent of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000956The maximal displacement of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.