- FindStat

Your data matches 21 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 1 = 0 + 1
[1,2] => [1,2] => [1,1]
 => 2 = 1 + 1
[1,-2] => [1,-2] => [1]
 => 1 = 0 + 1
[2,1] => [2,1] => [2]
 => 1 = 0 + 1
[2,-1] => [-1,2] => [1]
 => 1 = 0 + 1
[-2,1] => [-2,-1] => [2]
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => 3 = 2 + 1
[1,2,-3] => [1,2,-3] => [1,1]
 => 2 = 1 + 1
[1,-2,3] => [1,-2,-3] => [1]
 => 1 = 0 + 1
[1,-2,-3] => [1,-2,-3] => [1]
 => 1 = 0 + 1
[-1,2,3] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
 => 2 = 1 + 1
[1,3,-2] => [1,-2,3] => [1,1]
 => 2 = 1 + 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => 2 = 1 + 1
[1,-3,-2] => [1,-2,-3] => [1]
 => 1 = 0 + 1
[-1,3,2] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1]
 => 2 = 1 + 1
[2,1,-3] => [2,1,-3] => [2]
 => 1 = 0 + 1
[2,-1,3] => [-1,2,-3] => [1]
 => 1 = 0 + 1
[2,-1,-3] => [-1,2,-3] => [1]
 => 1 = 0 + 1
[-2,1,3] => [-2,-1,3] => [2,1]
 => 2 = 1 + 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => 1 = 0 + 1
[2,3,1] => [3,2,1] => [2,1]
 => 2 = 1 + 1
[2,3,-1] => [-1,2,3] => [1,1]
 => 2 = 1 + 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => 2 = 1 + 1
[2,-3,-1] => [-1,2,-3] => [1]
 => 1 = 0 + 1
[-2,3,1] => [-2,-1,3] => [2,1]
 => 2 = 1 + 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => 1 = 0 + 1
[3,1,2] => [3,2,1] => [2,1]
 => 2 = 1 + 1
[3,1,-2] => [3,-2,1] => [2]
 => 1 = 0 + 1
[3,-1,2] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[3,-1,-2] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[-3,1,2] => [-3,2,-1] => [2,1]
 => 2 = 1 + 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
 => 2 = 1 + 1
[3,2,-1] => [-1,3,2] => [2]
 => 1 = 0 + 1
[3,-2,1] => [-2,-1,3] => [2,1]
 => 2 = 1 + 1
[3,-2,-1] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[-3,2,1] => [-3,2,-1] => [2,1]
 => 2 = 1 + 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => 1 = 0 + 1
[-3,-2,1] => [-2,-1,-3] => [2]
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => 3 = 2 + 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => 2 = 1 + 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => 2 = 1 + 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => 2 = 1 + 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => 1 = 0 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => 1 = 0 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => 1 = 0 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => 2 = 1 + 1

Description

The length of the partition.

Matching statistic: St001250
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 1 = 0 + 1
[1,2] => [1,2] => [1,1]
 => 2 = 1 + 1
[1,-2] => [1,-2] => [1]
 => 1 = 0 + 1
[2,1] => [2,1] => [2]
 => 1 = 0 + 1
[2,-1] => [-1,2] => [1]
 => 1 = 0 + 1
[-2,1] => [-2,-1] => [2]
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => 3 = 2 + 1
[1,2,-3] => [1,2,-3] => [1,1]
 => 2 = 1 + 1
[1,-2,3] => [1,-2,-3] => [1]
 => 1 = 0 + 1
[1,-2,-3] => [1,-2,-3] => [1]
 => 1 = 0 + 1
[-1,2,3] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
 => 2 = 1 + 1
[1,3,-2] => [1,-2,3] => [1,1]
 => 2 = 1 + 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => 2 = 1 + 1
[1,-3,-2] => [1,-2,-3] => [1]
 => 1 = 0 + 1
[-1,3,2] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1]
 => 2 = 1 + 1
[2,1,-3] => [2,1,-3] => [2]
 => 1 = 0 + 1
[2,-1,3] => [-1,2,-3] => [1]
 => 1 = 0 + 1
[2,-1,-3] => [-1,2,-3] => [1]
 => 1 = 0 + 1
[-2,1,3] => [-2,-1,3] => [2,1]
 => 2 = 1 + 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => 1 = 0 + 1
[2,3,1] => [3,2,1] => [2,1]
 => 2 = 1 + 1
[2,3,-1] => [-1,2,3] => [1,1]
 => 2 = 1 + 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => 2 = 1 + 1
[2,-3,-1] => [-1,2,-3] => [1]
 => 1 = 0 + 1
[-2,3,1] => [-2,-1,3] => [2,1]
 => 2 = 1 + 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => 1 = 0 + 1
[3,1,2] => [3,2,1] => [2,1]
 => 2 = 1 + 1
[3,1,-2] => [3,-2,1] => [2]
 => 1 = 0 + 1
[3,-1,2] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[3,-1,-2] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[-3,1,2] => [-3,2,-1] => [2,1]
 => 2 = 1 + 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
 => 2 = 1 + 1
[3,2,-1] => [-1,3,2] => [2]
 => 1 = 0 + 1
[3,-2,1] => [-2,-1,3] => [2,1]
 => 2 = 1 + 1
[3,-2,-1] => [-1,-2,3] => [1]
 => 1 = 0 + 1
[-3,2,1] => [-3,2,-1] => [2,1]
 => 2 = 1 + 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => 1 = 0 + 1
[-3,-2,1] => [-2,-1,-3] => [2]
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 4 = 3 + 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => 3 = 2 + 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => 2 = 1 + 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => 2 = 1 + 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => 2 = 1 + 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => 1 = 0 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => 1 = 0 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => 1 = 0 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => 2 = 1 + 1

Description

The number of parts of a partition that are not congruent 0 modulo 3.

Matching statistic: St000157
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => [[1]]
 => 0
[1,2] => [1,2] => [1,1]
 => [[1],[2]]
 => 1
[1,-2] => [1,-2] => [1]
 => [[1]]
 => 0
[2,1] => [2,1] => [2]
 => [[1,2]]
 => 0
[2,-1] => [-1,2] => [1]
 => [[1]]
 => 0
[-2,1] => [-2,-1] => [2]
 => [[1,2]]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => 2
[1,2,-3] => [1,2,-3] => [1,1]
 => [[1],[2]]
 => 1
[1,-2,3] => [1,-2,-3] => [1]
 => [[1]]
 => 0
[1,-2,-3] => [1,-2,-3] => [1]
 => [[1]]
 => 0
[-1,2,3] => [-1,-2,3] => [1]
 => [[1]]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [[1,2],[3]]
 => 1
[1,3,-2] => [1,-2,3] => [1,1]
 => [[1],[2]]
 => 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => [[1,2],[3]]
 => 1
[1,-3,-2] => [1,-2,-3] => [1]
 => [[1]]
 => 0
[-1,3,2] => [-1,-2,3] => [1]
 => [[1]]
 => 0
[2,1,3] => [2,1,3] => [2,1]
 => [[1,2],[3]]
 => 1
[2,1,-3] => [2,1,-3] => [2]
 => [[1,2]]
 => 0
[2,-1,3] => [-1,2,-3] => [1]
 => [[1]]
 => 0
[2,-1,-3] => [-1,2,-3] => [1]
 => [[1]]
 => 0
[-2,1,3] => [-2,-1,3] => [2,1]
 => [[1,2],[3]]
 => 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => [[1,2]]
 => 0
[2,3,1] => [3,2,1] => [2,1]
 => [[1,2],[3]]
 => 1
[2,3,-1] => [-1,2,3] => [1,1]
 => [[1],[2]]
 => 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => [[1,2],[3]]
 => 1
[2,-3,-1] => [-1,2,-3] => [1]
 => [[1]]
 => 0
[-2,3,1] => [-2,-1,3] => [2,1]
 => [[1,2],[3]]
 => 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => [[1,2]]
 => 0
[3,1,2] => [3,2,1] => [2,1]
 => [[1,2],[3]]
 => 1
[3,1,-2] => [3,-2,1] => [2]
 => [[1,2]]
 => 0
[3,-1,2] => [-1,-2,3] => [1]
 => [[1]]
 => 0
[3,-1,-2] => [-1,-2,3] => [1]
 => [[1]]
 => 0
[-3,1,2] => [-3,2,-1] => [2,1]
 => [[1,2],[3]]
 => 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => [[1,2]]
 => 0
[3,2,1] => [3,2,1] => [2,1]
 => [[1,2],[3]]
 => 1
[3,2,-1] => [-1,3,2] => [2]
 => [[1,2]]
 => 0
[3,-2,1] => [-2,-1,3] => [2,1]
 => [[1,2],[3]]
 => 1
[3,-2,-1] => [-1,-2,3] => [1]
 => [[1]]
 => 0
[-3,2,1] => [-3,2,-1] => [2,1]
 => [[1,2],[3]]
 => 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => [[1,2]]
 => 0
[-3,-2,1] => [-2,-1,-3] => [2]
 => [[1,2]]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => [[1]]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => [[1]]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => [[1]]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [[1],[2]]
 => 1

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Matching statistic: St000319

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: 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] => [1,2] => [1,1]
 => [2]
 => 1
[1,-2] => [1,-2] => [1]
 => [1]
 => 0
[2,1] => [2,1] => [2]
 => [1,1]
 => 0
[2,-1] => [-1,2] => [1]
 => [1]
 => 0
[-2,1] => [-2,-1] => [2]
 => [1,1]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2
[1,2,-3] => [1,2,-3] => [1,1]
 => [2]
 => 1
[1,-2,3] => [1,-2,-3] => [1]
 => [1]
 => 0
[1,-2,-3] => [1,-2,-3] => [1]
 => [1]
 => 0
[-1,2,3] => [-1,-2,3] => [1]
 => [1]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1
[1,3,-2] => [1,-2,3] => [1,1]
 => [2]
 => 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => [2,1]
 => 1
[1,-3,-2] => [1,-2,-3] => [1]
 => [1]
 => 0
[-1,3,2] => [-1,-2,3] => [1]
 => [1]
 => 0
[2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1
[2,1,-3] => [2,1,-3] => [2]
 => [1,1]
 => 0
[2,-1,3] => [-1,2,-3] => [1]
 => [1]
 => 0
[2,-1,-3] => [-1,2,-3] => [1]
 => [1]
 => 0
[-2,1,3] => [-2,-1,3] => [2,1]
 => [2,1]
 => 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => [1,1]
 => 0
[2,3,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1
[2,3,-1] => [-1,2,3] => [1,1]
 => [2]
 => 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => [2,1]
 => 1
[2,-3,-1] => [-1,2,-3] => [1]
 => [1]
 => 0
[-2,3,1] => [-2,-1,3] => [2,1]
 => [2,1]
 => 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => [1,1]
 => 0
[3,1,2] => [3,2,1] => [2,1]
 => [2,1]
 => 1
[3,1,-2] => [3,-2,1] => [2]
 => [1,1]
 => 0
[3,-1,2] => [-1,-2,3] => [1]
 => [1]
 => 0
[3,-1,-2] => [-1,-2,3] => [1]
 => [1]
 => 0
[-3,1,2] => [-3,2,-1] => [2,1]
 => [2,1]
 => 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => [1,1]
 => 0
[3,2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1
[3,2,-1] => [-1,3,2] => [2]
 => [1,1]
 => 0
[3,-2,1] => [-2,-1,3] => [2,1]
 => [2,1]
 => 1
[3,-2,-1] => [-1,-2,3] => [1]
 => [1]
 => 0
[-3,2,1] => [-3,2,-1] => [2,1]
 => [2,1]
 => 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => [1,1]
 => 0
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,1]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [3]
 => 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [2]
 => 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [2]
 => 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [2]
 => 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => [1]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => [1]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => [1]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [2]
 => 1

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: 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] => [1,2] => [1,1]
 => [2]
 => 1
[1,-2] => [1,-2] => [1]
 => [1]
 => 0
[2,1] => [2,1] => [2]
 => [1,1]
 => 0
[2,-1] => [-1,2] => [1]
 => [1]
 => 0
[-2,1] => [-2,-1] => [2]
 => [1,1]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 2
[1,2,-3] => [1,2,-3] => [1,1]
 => [2]
 => 1
[1,-2,3] => [1,-2,-3] => [1]
 => [1]
 => 0
[1,-2,-3] => [1,-2,-3] => [1]
 => [1]
 => 0
[-1,2,3] => [-1,-2,3] => [1]
 => [1]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1
[1,3,-2] => [1,-2,3] => [1,1]
 => [2]
 => 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => [2,1]
 => 1
[1,-3,-2] => [1,-2,-3] => [1]
 => [1]
 => 0
[-1,3,2] => [-1,-2,3] => [1]
 => [1]
 => 0
[2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1
[2,1,-3] => [2,1,-3] => [2]
 => [1,1]
 => 0
[2,-1,3] => [-1,2,-3] => [1]
 => [1]
 => 0
[2,-1,-3] => [-1,2,-3] => [1]
 => [1]
 => 0
[-2,1,3] => [-2,-1,3] => [2,1]
 => [2,1]
 => 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => [1,1]
 => 0
[2,3,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1
[2,3,-1] => [-1,2,3] => [1,1]
 => [2]
 => 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => [2,1]
 => 1
[2,-3,-1] => [-1,2,-3] => [1]
 => [1]
 => 0
[-2,3,1] => [-2,-1,3] => [2,1]
 => [2,1]
 => 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => [1,1]
 => 0
[3,1,2] => [3,2,1] => [2,1]
 => [2,1]
 => 1
[3,1,-2] => [3,-2,1] => [2]
 => [1,1]
 => 0
[3,-1,2] => [-1,-2,3] => [1]
 => [1]
 => 0
[3,-1,-2] => [-1,-2,3] => [1]
 => [1]
 => 0
[-3,1,2] => [-3,2,-1] => [2,1]
 => [2,1]
 => 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => [1,1]
 => 0
[3,2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 1
[3,2,-1] => [-1,3,2] => [2]
 => [1,1]
 => 0
[3,-2,1] => [-2,-1,3] => [2,1]
 => [2,1]
 => 1
[3,-2,-1] => [-1,-2,3] => [1]
 => [1]
 => 0
[-3,2,1] => [-3,2,-1] => [2,1]
 => [2,1]
 => 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => [1,1]
 => 0
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,1]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [3]
 => 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [2]
 => 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [2]
 => 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [2]
 => 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => [1]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => [1]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => [1]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [2]
 => 1

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

Matching statistic: St000329

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => [1,0]
 => 0
[1,2] => [1,2] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2] => [1,-2] => [1]
 => [1,0]
 => 0
[2,1] => [2,1] => [2]
 => [1,0,1,0]
 => 0
[2,-1] => [-1,2] => [1]
 => [1,0]
 => 0
[-2,1] => [-2,-1] => [2]
 => [1,0,1,0]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,-3] => [1,2,-3] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3] => [1,-2,-3] => [1]
 => [1,0]
 => 0
[1,-2,-3] => [1,-2,-3] => [1]
 => [1,0]
 => 0
[-1,2,3] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,3,-2] => [1,-2,3] => [1,1]
 => [1,1,0,0]
 => 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,-3,-2] => [1,-2,-3] => [1]
 => [1,0]
 => 0
[-1,3,2] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[2,1,3] => [2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,1,-3] => [2,1,-3] => [2]
 => [1,0,1,0]
 => 0
[2,-1,3] => [-1,2,-3] => [1]
 => [1,0]
 => 0
[2,-1,-3] => [-1,2,-3] => [1]
 => [1,0]
 => 0
[-2,1,3] => [-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 0
[2,3,1] => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,-1] => [-1,2,3] => [1,1]
 => [1,1,0,0]
 => 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,-3,-1] => [-1,2,-3] => [1]
 => [1,0]
 => 0
[-2,3,1] => [-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 0
[3,1,2] => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,1,-2] => [3,-2,1] => [2]
 => [1,0,1,0]
 => 0
[3,-1,2] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[3,-1,-2] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[-3,1,2] => [-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => [1,0,1,0]
 => 0
[3,2,1] => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,2,-1] => [-1,3,2] => [2]
 => [1,0,1,0]
 => 0
[3,-2,1] => [-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,-2,-1] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[-3,2,1] => [-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => [1,0,1,0]
 => 0
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => [1,0]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => [1,0]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => [1,0]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 1

Description

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.

Matching statistic: St001508

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001508: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => [1,0]
 => 0
[1,2] => [1,2] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2] => [1,-2] => [1]
 => [1,0]
 => 0
[2,1] => [2,1] => [2]
 => [1,0,1,0]
 => 0
[2,-1] => [-1,2] => [1]
 => [1,0]
 => 0
[-2,1] => [-2,-1] => [2]
 => [1,0,1,0]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,-3] => [1,2,-3] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3] => [1,-2,-3] => [1]
 => [1,0]
 => 0
[1,-2,-3] => [1,-2,-3] => [1]
 => [1,0]
 => 0
[-1,2,3] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,3,-2] => [1,-2,3] => [1,1]
 => [1,1,0,0]
 => 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,-3,-2] => [1,-2,-3] => [1]
 => [1,0]
 => 0
[-1,3,2] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[2,1,3] => [2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,1,-3] => [2,1,-3] => [2]
 => [1,0,1,0]
 => 0
[2,-1,3] => [-1,2,-3] => [1]
 => [1,0]
 => 0
[2,-1,-3] => [-1,2,-3] => [1]
 => [1,0]
 => 0
[-2,1,3] => [-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 0
[2,3,1] => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,-1] => [-1,2,3] => [1,1]
 => [1,1,0,0]
 => 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,-3,-1] => [-1,2,-3] => [1]
 => [1,0]
 => 0
[-2,3,1] => [-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 0
[3,1,2] => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,1,-2] => [3,-2,1] => [2]
 => [1,0,1,0]
 => 0
[3,-1,2] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[3,-1,-2] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[-3,1,2] => [-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => [1,0,1,0]
 => 0
[3,2,1] => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,2,-1] => [-1,3,2] => [2]
 => [1,0,1,0]
 => 0
[3,-2,1] => [-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,-2,-1] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[-3,2,1] => [-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => [1,0,1,0]
 => 0
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => [1,0]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => [1,0]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => [1,0]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 1

Description

The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. Given two lattice paths

$U,L$ from

$(0,0)$ to

$(d,n-d)$ , [1] describes a bijection between lattice paths weakly between

$U$ and

$L$ and subsets of

$\{1,\dots,n\}$ such that the set of all such subsets gives the standard complex of the lattice path matroid

$M[U,L]$ . This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly above the diagonal and relative to the diagonal boundary.

Matching statistic: St001553

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001553: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => [1,0]
 => 0
[1,2] => [1,2] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2] => [1,-2] => [1]
 => [1,0]
 => 0
[2,1] => [2,1] => [2]
 => [1,0,1,0]
 => 0
[2,-1] => [-1,2] => [1]
 => [1,0]
 => 0
[-2,1] => [-2,-1] => [2]
 => [1,0,1,0]
 => 0
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,-3] => [1,2,-3] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3] => [1,-2,-3] => [1]
 => [1,0]
 => 0
[1,-2,-3] => [1,-2,-3] => [1]
 => [1,0]
 => 0
[-1,2,3] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,3,-2] => [1,-2,3] => [1,1]
 => [1,1,0,0]
 => 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[1,-3,-2] => [1,-2,-3] => [1]
 => [1,0]
 => 0
[-1,3,2] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[2,1,3] => [2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,1,-3] => [2,1,-3] => [2]
 => [1,0,1,0]
 => 0
[2,-1,3] => [-1,2,-3] => [1]
 => [1,0]
 => 0
[2,-1,-3] => [-1,2,-3] => [1]
 => [1,0]
 => 0
[-2,1,3] => [-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 0
[2,3,1] => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,3,-1] => [-1,2,3] => [1,1]
 => [1,1,0,0]
 => 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,-3,-1] => [-1,2,-3] => [1]
 => [1,0]
 => 0
[-2,3,1] => [-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 0
[3,1,2] => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,1,-2] => [3,-2,1] => [2]
 => [1,0,1,0]
 => 0
[3,-1,2] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[3,-1,-2] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[-3,1,2] => [-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => [1,0,1,0]
 => 0
[3,2,1] => [3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,2,-1] => [-1,3,2] => [2]
 => [1,0,1,0]
 => 0
[3,-2,1] => [-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,-2,-1] => [-1,-2,3] => [1]
 => [1,0]
 => 0
[-3,2,1] => [-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => [1,0,1,0]
 => 0
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => [1,0]
 => 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => [1,0]
 => 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => [1,0]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 1

Description

The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. The statistic returns zero in case that bimodule is the zero module.

Matching statistic: St000147

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ 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] => [1,2] => [1,1]
 => [2]
 => 2 = 1 + 1
[1,-2] => [1,-2] => [1]
 => [1]
 => 1 = 0 + 1
[2,1] => [2,1] => [2]
 => [1,1]
 => 1 = 0 + 1
[2,-1] => [-1,2] => [1]
 => [1]
 => 1 = 0 + 1
[-2,1] => [-2,-1] => [2]
 => [1,1]
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 3 = 2 + 1
[1,2,-3] => [1,2,-3] => [1,1]
 => [2]
 => 2 = 1 + 1
[1,-2,3] => [1,-2,-3] => [1]
 => [1]
 => 1 = 0 + 1
[1,-2,-3] => [1,-2,-3] => [1]
 => [1]
 => 1 = 0 + 1
[-1,2,3] => [-1,-2,3] => [1]
 => [1]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,3,-2] => [1,-2,3] => [1,1]
 => [2]
 => 2 = 1 + 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,-3,-2] => [1,-2,-3] => [1]
 => [1]
 => 1 = 0 + 1
[-1,3,2] => [-1,-2,3] => [1]
 => [1]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[2,1,-3] => [2,1,-3] => [2]
 => [1,1]
 => 1 = 0 + 1
[2,-1,3] => [-1,2,-3] => [1]
 => [1]
 => 1 = 0 + 1
[2,-1,-3] => [-1,2,-3] => [1]
 => [1]
 => 1 = 0 + 1
[-2,1,3] => [-2,-1,3] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => [1,1]
 => 1 = 0 + 1
[2,3,1] => [3,2,1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[2,3,-1] => [-1,2,3] => [1,1]
 => [2]
 => 2 = 1 + 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[2,-3,-1] => [-1,2,-3] => [1]
 => [1]
 => 1 = 0 + 1
[-2,3,1] => [-2,-1,3] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => [1,1]
 => 1 = 0 + 1
[3,1,2] => [3,2,1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[3,1,-2] => [3,-2,1] => [2]
 => [1,1]
 => 1 = 0 + 1
[3,-1,2] => [-1,-2,3] => [1]
 => [1]
 => 1 = 0 + 1
[3,-1,-2] => [-1,-2,3] => [1]
 => [1]
 => 1 = 0 + 1
[-3,1,2] => [-3,2,-1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => [1,1]
 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[3,2,-1] => [-1,3,2] => [2]
 => [1,1]
 => 1 = 0 + 1
[3,-2,1] => [-2,-1,3] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[3,-2,-1] => [-1,-2,3] => [1]
 => [1]
 => 1 = 0 + 1
[-3,2,1] => [-3,2,-1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => [1,1]
 => 1 = 0 + 1
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,1]
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 4 = 3 + 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [3]
 => 3 = 2 + 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [2]
 => 2 = 1 + 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [2]
 => 2 = 1 + 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [2]
 => 2 = 1 + 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => [1]
 => 1 = 0 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => [1]
 => 1 = 0 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => [1]
 => 1 = 0 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [2]
 => 2 = 1 + 1

Description

The largest part of an integer partition.

Matching statistic: St000288

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => 10 => 1 = 0 + 1
[1,2] => [1,2] => [1,1]
 => 110 => 2 = 1 + 1
[1,-2] => [1,-2] => [1]
 => 10 => 1 = 0 + 1
[2,1] => [2,1] => [2]
 => 100 => 1 = 0 + 1
[2,-1] => [-1,2] => [1]
 => 10 => 1 = 0 + 1
[-2,1] => [-2,-1] => [2]
 => 100 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => 1110 => 3 = 2 + 1
[1,2,-3] => [1,2,-3] => [1,1]
 => 110 => 2 = 1 + 1
[1,-2,3] => [1,-2,-3] => [1]
 => 10 => 1 = 0 + 1
[1,-2,-3] => [1,-2,-3] => [1]
 => 10 => 1 = 0 + 1
[-1,2,3] => [-1,-2,3] => [1]
 => 10 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1]
 => 1010 => 2 = 1 + 1
[1,3,-2] => [1,-2,3] => [1,1]
 => 110 => 2 = 1 + 1
[1,-3,2] => [1,-3,-2] => [2,1]
 => 1010 => 2 = 1 + 1
[1,-3,-2] => [1,-2,-3] => [1]
 => 10 => 1 = 0 + 1
[-1,3,2] => [-1,-2,3] => [1]
 => 10 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1]
 => 1010 => 2 = 1 + 1
[2,1,-3] => [2,1,-3] => [2]
 => 100 => 1 = 0 + 1
[2,-1,3] => [-1,2,-3] => [1]
 => 10 => 1 = 0 + 1
[2,-1,-3] => [-1,2,-3] => [1]
 => 10 => 1 = 0 + 1
[-2,1,3] => [-2,-1,3] => [2,1]
 => 1010 => 2 = 1 + 1
[-2,1,-3] => [-2,-1,-3] => [2]
 => 100 => 1 = 0 + 1
[2,3,1] => [3,2,1] => [2,1]
 => 1010 => 2 = 1 + 1
[2,3,-1] => [-1,2,3] => [1,1]
 => 110 => 2 = 1 + 1
[2,-3,1] => [-3,2,-1] => [2,1]
 => 1010 => 2 = 1 + 1
[2,-3,-1] => [-1,2,-3] => [1]
 => 10 => 1 = 0 + 1
[-2,3,1] => [-2,-1,3] => [2,1]
 => 1010 => 2 = 1 + 1
[-2,-3,1] => [-2,-1,-3] => [2]
 => 100 => 1 = 0 + 1
[3,1,2] => [3,2,1] => [2,1]
 => 1010 => 2 = 1 + 1
[3,1,-2] => [3,-2,1] => [2]
 => 100 => 1 = 0 + 1
[3,-1,2] => [-1,-2,3] => [1]
 => 10 => 1 = 0 + 1
[3,-1,-2] => [-1,-2,3] => [1]
 => 10 => 1 = 0 + 1
[-3,1,2] => [-3,2,-1] => [2,1]
 => 1010 => 2 = 1 + 1
[-3,1,-2] => [-3,-2,-1] => [2]
 => 100 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1]
 => 1010 => 2 = 1 + 1
[3,2,-1] => [-1,3,2] => [2]
 => 100 => 1 = 0 + 1
[3,-2,1] => [-2,-1,3] => [2,1]
 => 1010 => 2 = 1 + 1
[3,-2,-1] => [-1,-2,3] => [1]
 => 10 => 1 = 0 + 1
[-3,2,1] => [-3,2,-1] => [2,1]
 => 1010 => 2 = 1 + 1
[-3,2,-1] => [-1,-3,-2] => [2]
 => 100 => 1 = 0 + 1
[-3,-2,1] => [-2,-1,-3] => [2]
 => 100 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => 11110 => 4 = 3 + 1
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => 1110 => 3 = 2 + 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => 110 => 2 = 1 + 1
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => 110 => 2 = 1 + 1
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => 110 => 2 = 1 + 1
[1,-2,3,-4] => [1,-2,-3,-4] => [1]
 => 10 => 1 = 0 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1]
 => 10 => 1 = 0 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1]
 => 10 => 1 = 0 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => 110 => 2 = 1 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000733The row containing the largest entry of a standard tableau. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001176The size of a partition minus its first part. St001769The reflection length of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.