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Your data matches 165 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001599
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001599: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees.

Matching statistic: St000346

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000346: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of coarsenings of a partition. A partition

$\mu$ coarsens a partition

$\lambda$ if the parts of

$\mu$ can be subdivided to obtain the parts of

$\lambda$ .

Matching statistic: St000939

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of characters of the symmetric group whose value on the partition is positive.

Matching statistic: St001380
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of monomer-dimer tilings of a Ferrers diagram. For a hook of length

$n$ , this is the

$n$ -th Fibonacci number.

Matching statistic: St001914
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001914: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of the orbit of an integer partition in Bulgarian solitaire. Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row. This statistic returns the number of partitions that can be obtained from the given partition.

Matching statistic: St000005

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000005: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The bounce statistic of a Dyck path. The '''bounce path'''

$D'$ of a Dyck path

$D$ is the Dyck path obtained from

$D$ by starting at the end point

$(2n,0)$ , traveling north-west until hitting

$D$ , then bouncing back south-west to the

$x$ -axis, and repeating this procedure until finally reaching the point

$(0,0)$ . The points where

$D'$ touches the

$x$ -axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points. This statistic is given by the sum of all

$i$ for which the bounce path

$D'$ of

$D$ touches the

$x$ -axis at

$(2i,0)$ . In particular, the bounce statistics of

$D$ and

$D'$ coincide.

Matching statistic: St000913

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000913: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ways to refine the partition into singletons. For example there is only one way to refine

$[2,2]$ :

$[2,2] > [2,1,1] > [1,1,1,1]$ . However, there are two ways to refine

$[3,2]$ :

$[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and

$[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$ . In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition. The sequence of values on the partitions with only one part is [[A002846]].

Matching statistic: St000998

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001020

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001033

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The normalized area of the parallelogram polyomino associated with the Dyck path. The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path. The area itself is equidistributed with [[St001034]] and with [[St000395]].

The following 155 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000264The girth of a graph, which is not a tree. St000478Another weight of a partition according to Alladi. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition. St001819The flag Denert index of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001060The distinguishing index of a graph. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000259The diameter of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000456The monochromatic index of a connected graph. St000302The determinant of the distance matrix of a connected graph. St001545The second Elser number of a connected graph. St000467The hyper-Wiener index of a connected graph. St000464The Schultz index of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph.