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Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000053

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
St000053: 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]
 => 0
[1,-2] => [1,-2] => [1]
 => [1,0]
 => 0
[2,1] => [2,1] => [2]
 => [1,0,1,0]
 => 1
[2,-1] => [-1,2] => [1]
 => [1,0]
 => 0
[-2,1] => [-2,-1] => [2]
 => [1,0,1,0]
 => 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,-3] => [1,2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[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]
 => 0
[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]
 => 1
[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]
 => 1
[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]
 => 0
[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]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 1
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 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]
 => [1,0]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 0

Description

The number of valleys of the Dyck path.

Matching statistic: St000331

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
St000331: 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]
 => 0
[1,-2] => [1,-2] => [1]
 => [1,0]
 => 0
[2,1] => [2,1] => [2]
 => [1,0,1,0]
 => 1
[2,-1] => [-1,2] => [1]
 => [1,0]
 => 0
[-2,1] => [-2,-1] => [2]
 => [1,0,1,0]
 => 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,-3] => [1,2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[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]
 => 0
[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]
 => 1
[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]
 => 1
[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]
 => 0
[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]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 1
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 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]
 => [1,0]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 0

Description

The number of upper interactions of a Dyck path. An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''

$A^{k}$

$B^{k}$ ''' for any '''

${k ≥ 1}$ ''', where '''

${A}$ ''' is a down-step and '''

${B}$ ''' is a up-step.

Matching statistic: St001169

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
St001169: 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]
 => 0
[1,-2] => [1,-2] => [1]
 => [1,0]
 => 0
[2,1] => [2,1] => [2]
 => [1,0,1,0]
 => 1
[2,-1] => [-1,2] => [1]
 => [1,0]
 => 0
[-2,1] => [-2,-1] => [2]
 => [1,0,1,0]
 => 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,-3] => [1,2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[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]
 => 0
[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]
 => 1
[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]
 => 1
[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]
 => 0
[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]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 1
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 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]
 => [1,0]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 0

Description

Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.

Matching statistic: St001509

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
St001509: 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]
 => 0
[1,-2] => [1,-2] => [1]
 => [1,0]
 => 0
[2,1] => [2,1] => [2]
 => [1,0,1,0]
 => 1
[2,-1] => [-1,2] => [1]
 => [1,0]
 => 0
[-2,1] => [-2,-1] => [2]
 => [1,0,1,0]
 => 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,-3] => [1,2,-3] => [1,1]
 => [1,1,0,0]
 => 0
[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]
 => 0
[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]
 => 1
[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]
 => 1
[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]
 => 0
[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]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 1
[-3,-2,1] => [-2,-1,-3] => [2]
 => [1,0,1,0]
 => 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,2,3,-4] => [1,2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => 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]
 => [1,0]
 => 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => 0

Description

The degree of the standard monomial associated to a Dyck path relative to the trivial lower 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 below the diagonal and relative to the trivial lower boundary.

Matching statistic: St000015

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
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of peaks of a Dyck path.

Matching statistic: St001068

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
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St001200

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
St001200: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of simple modules in

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

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001604

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
St001604: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%

Values

[1] => [1] => []
 => ?
 => ? = 0
[1,2] => [1,2] => []
 => ?
 => ? = 0
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0
[2,1] => [2,1] => []
 => ?
 => ? = 1
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0
[-2,1] => [-2,-1] => []
 => ?
 => ? = 1
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 1
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 1
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 1
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 1
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 1
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 1
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 1
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 1
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 1
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 1
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 1
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 1
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 1
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 1
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 1
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 1
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 1
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 1
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 1
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 1
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 1
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 2
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 1
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0
[1,-2,3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,3,-4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,-3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,3,-5,-4] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,-3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,4,-3,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,4,-3,-5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,4,3,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,-5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,4,-5,-3] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,4,5,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,5,-3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,5,-3,-4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,5,3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-2,-5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,5,-4,-3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,5,4,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,-3,-2,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 0

Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St001603

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
St001603: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%

Values

[1] => [1] => []
 => ?
 => ? = 0 + 1
[1,2] => [1,2] => []
 => ?
 => ? = 0 + 1
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0 + 1
[2,1] => [2,1] => []
 => ?
 => ? = 1 + 1
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0 + 1
[-2,1] => [-2,-1] => []
 => ?
 => ? = 1 + 1
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 1 + 1
[1,2,-3] => [1,2,-3] => [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
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 1
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 1 + 1
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0 + 1
[1,-3,2] => [1,-3,-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] => []
 => ?
 => ? = 1 + 1
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 1 + 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] => []
 => ?
 => ? = 1 + 1
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 1 + 1
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 1 + 1
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0 + 1
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 1 + 1
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0 + 1
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 1 + 1
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 1 + 1
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 1 + 1
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 1 + 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] => []
 => ?
 => ? = 1 + 1
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 1 + 1
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 1 + 1
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 1 + 1
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 1 + 1
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 1
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 1 + 1
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 1 + 1
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 2 + 1
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 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] => [1,1,1]
 => [1,1]
 => ? = 0 + 1
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0 + 1
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0 + 1
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0 + 1
[1,-2,3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,3,-4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,-3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,3,-5,-4] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,-3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,4,-3,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,4,-3,-5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,4,3,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,-5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,4,-5,-3] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,4,5,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,5,-3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,5,-3,-4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,5,3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-2,-5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,2,5,-4,-3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[-1,-2,5,4,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,-3,-2,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1

Description

The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St001605

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
St001605: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 25%

Values

[1] => [1] => []
 => ?
 => ? = 0 + 2
[1,2] => [1,2] => []
 => ?
 => ? = 0 + 2
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0 + 2
[2,1] => [2,1] => []
 => ?
 => ? = 1 + 2
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0 + 2
[-2,1] => [-2,-1] => []
 => ?
 => ? = 1 + 2
[1,2,3] => [1,2,3] => []
 => ?
 => ? = 1 + 2
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0 + 2
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0 + 2
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0 + 2
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 2
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 1 + 2
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0 + 2
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 1 + 2
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0 + 2
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 2
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 1 + 2
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 1 + 2
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0 + 2
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0 + 2
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 1 + 2
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 1 + 2
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 1 + 2
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0 + 2
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 1 + 2
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0 + 2
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 1 + 2
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 1 + 2
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 1 + 2
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 1 + 2
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 2
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 2
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 1 + 2
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 1 + 2
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 1 + 2
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 1 + 2
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 1 + 2
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0 + 2
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 1 + 2
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 1 + 2
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 1 + 2
[1,2,3,4] => [1,2,3,4] => []
 => ?
 => ? = 2 + 2
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 1 + 2
[1,2,-3,4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0 + 2
[1,2,-3,-4] => [1,2,-3,-4] => [1,1]
 => [1]
 => ? = 0 + 2
[1,-2,3,4] => [1,-2,-3,4] => [1,1]
 => [1]
 => ? = 0 + 2
[1,-2,3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0 + 2
[1,-2,-3,4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0 + 2
[1,-2,-3,-4] => [1,-2,-3,-4] => [1,1,1]
 => [1,1]
 => ? = 0 + 2
[-1,2,3,4] => [-1,-2,3,4] => [1,1]
 => [1]
 => ? = 0 + 2
[1,-2,3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-3,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-3,4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-3,-4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-3,-4,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,3,-4,5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,3,-4,-5] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,-3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,-2,3,4,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-3,5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-3,5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-3,-5,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-3,-5,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,3,-5,-4] => [-1,-2,3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,-3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,-2,3,5,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-4,3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-4,3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-4,-3,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-4,-3,-5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,4,-3,5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,4,-3,-5] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,-2,4,3,5] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-4,5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-4,5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-4,-5,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-4,-5,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,4,-5,-3] => [-1,-2,-3,4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,-2,4,5,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-5,3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-5,-3,4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-5,-3,-4] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,5,-3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,5,-3,-4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,-2,5,3,4] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-5,4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-5,-4,3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-2,-5,-4,-3] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,2,5,-4,-3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[-1,-2,5,4,3] => [-1,-2,-3,-4,5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2
[1,-3,-2,4,5] => [1,-2,-3,-4,-5] => [1,1,1,1]
 => [1,1,1]
 => 2 = 0 + 2

Description

The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.