Your data matches 315 different statistics following compositions of up to 3 maps.
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Matching statistic: St000929
(load all 28 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3]
 => [1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => 0
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [3,1,1]
 => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [2,1,1,1,1]
 => 0
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => 0
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => 0
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [4,1,1]
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
 => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1,1,1]
 => 0
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,3] => [3,1,1,1,1]
 => 0
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => [4,1,1]
 => 0
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,4] => [4,1,1,1]
 => 0
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,5] => [5,1,1]
 => 0
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
 => 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
 => 0
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,3] => [3,1,1,1,1]
 => 0
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,3] => [3,1,1,1,1,1]
 => 0
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => [4,1,1]
 => 0
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,4] => [4,1,1,1]
 => 0
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,4] => [4,1,1,1,1]
 => 0
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => [4,1,1]
 => 0
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,5] => [5,1,1]
 => 0
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,5] => [5,1,1,1]
 => 0
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,6] => [6,1,1]
 => 0
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
 => 0
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,3] => [3,1,1,1,1,1]
 => 0
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,3] => [3,1,1,1,1,1,1]
 => 0
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,4] => [4,1,1,1]
 => 0
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,4] => [4,1,1,1,1]
 => 0
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,4] => [4,1,1,1,1,1]
 => 0
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,5] => [5,1,1]
 => 0
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,4] => [4,1,1,1]
 => 0
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,5] => [5,1,1,1]
 => 0
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,5] => [5,1,1,1,1]
 => 0
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,5] => [5,1,1]
 => 0
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,6] => [6,1,1]
 => 0
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,6] => [6,1,1,1]
 => 0
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,7] => [7,1,1]
 => 0
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,3] => [3,1,1,1,1,1,1]
 => 0
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1]
 => 0
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,4] => [4,1,1,1,1]
 => 0
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,4] => [4,1,1,1,1,1]
 => 0
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,4] => [4,1,1,1,1,1,1]
 => 0
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,5] => [5,1,1]
 => 0
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,5] => [5,1,1,1]
 => 0
Description
The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001629
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3]
 => [1,0,1,0,1,0]
 => [1,1,1] => [3] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1] => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [3,1] => 0
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [2,1] => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [6] => 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [4,1] => 0
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [2,1] => 0
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1] => 0
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [2,1] => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1] => [7] => 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,2] => [5,1] => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1] => 0
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,3] => [4,1] => 0
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => [2,1] => 0
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,4] => [3,1] => 0
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1] => [8] => 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,2] => [6,1] => 0
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,3] => [4,1] => 0
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,3] => [5,1] => 0
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => [2,1] => 0
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,4] => [3,1] => 0
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,4] => [4,1] => 0
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => [2,1] => 0
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,5] => [3,1] => 0
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,6] => [2,1] => 0
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,2] => [7,1] => 0
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,3] => [5,1] => 0
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,3] => [6,1] => 0
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,4] => [3,1] => 0
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,4] => [4,1] => 0
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,4] => [5,1] => 0
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,4] => [3,1] => 0
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,5] => [3,1] => 0
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,5] => [4,1] => 0
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,6] => [2,1] => 0
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,6] => [3,1] => 0
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,7] => [2,1] => 0
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,3] => [6,1] => 0
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,3] => [7,1] => 0
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,4] => [4,1] => 0
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,4] => [5,1] => 0
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,4] => [6,1] => 0
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,5] => [2,1] => 0
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,5] => [3,1] => 0
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001675
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001675: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3]
 => [1,0,1,0,1,0]
 => [1,1,1] => [3] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1] => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [3,1] => 0
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [2,1] => 0
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => [6] => 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => [4,1] => 0
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [2,1] => 0
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => [3,1] => 0
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => [2,1] => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1] => [7] => 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,2] => [5,1] => 0
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => [3,1] => 0
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,3] => [4,1] => 0
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => [2,1] => 0
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,4] => [3,1] => 0
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1] => [8] => 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,2] => [6,1] => 0
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,3] => [4,1] => 0
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,3] => [5,1] => 0
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => [2,1] => 0
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,4] => [3,1] => 0
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,4] => [4,1] => 0
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => [2,1] => 0
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,5] => [3,1] => 0
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,6] => [2,1] => 0
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,2] => [7,1] => 0
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,3] => [5,1] => 0
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,3] => [6,1] => 0
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,4] => [3,1] => 0
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,4] => [4,1] => 0
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,4] => [5,1] => 0
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,4] => [3,1] => 0
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,5] => [3,1] => 0
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,5] => [4,1] => 0
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,5] => [2,1] => 0
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,6] => [2,1] => 0
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,6] => [3,1] => 0
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,7] => [2,1] => 0
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,3] => [6,1] => 0
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,3] => [7,1] => 0
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,4] => [4,1] => 0
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,4] => [5,1] => 0
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,4] => [6,1] => 0
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,5] => [2,1] => 0
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,5] => [3,1] => 0
Description
The number of parts equal to the part in the reversed composition.
Matching statistic: St001353
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001353: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[3]
 => 1 => [1] => ([],1)
 => 1
[4]
 => 0 => [1] => ([],1)
 => 1
[3,1]
 => 11 => [2] => ([],2)
 => 0
[5]
 => 1 => [1] => ([],1)
 => 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 0
[3,1,1]
 => 111 => [3] => ([],3)
 => 0
[6]
 => 0 => [1] => ([],1)
 => 1
[5,1]
 => 11 => [2] => ([],2)
 => 0
[4,2]
 => 00 => [2] => ([],2)
 => 0
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 0
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => 0
[7]
 => 1 => [1] => ([],1)
 => 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 0
[5,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 0
[5,1,1]
 => 111 => [3] => ([],3)
 => 0
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 0
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 0
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0
[8]
 => 0 => [1] => ([],1)
 => 1
[7,1]
 => 11 => [2] => ([],2)
 => 0
[6,2]
 => 00 => [2] => ([],2)
 => 0
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 0
[5,3]
 => 11 => [2] => ([],2)
 => 0
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => 0
[4,2,2]
 => 000 => [3] => ([],3)
 => 0
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 0
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 0
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 0
[8,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 0
[7,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 0
[7,1,1]
 => 111 => [3] => ([],3)
 => 0
[6,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 0
[6,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 0
[6,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 0
[5,3,1]
 => 111 => [3] => ([],3)
 => 0
[5,2,2]
 => 100 => [1,2] => ([(1,2)],3)
 => 0
[5,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[5,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0
[4,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[4,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => 0
[4,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => 0
[3,1,1,1,1,1,1]
 => 1111111 => [7] => ([],7)
 => 0
[8,2]
 => 00 => [2] => ([],2)
 => 0
[8,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 0
[7,3]
 => 11 => [2] => ([],2)
 => 0
[7,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[7,1,1,1]
 => 1111 => [4] => ([],4)
 => 0
[6,4]
 => 00 => [2] => ([],2)
 => 0
[6,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 0
[3,1,1,1,1,1,1,1]
 => 11111111 => [8] => ([],8)
 => ? = 0
[3,1,1,1,1,1,1,1,1]
 => 111111111 => [9] => ([],9)
 => ? = 0
Description
The number of prime nodes in the modular decomposition of a graph.
Matching statistic: St001356
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001356: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[3]
 => 1 => [1] => ([],1)
 => 1
[4]
 => 0 => [1] => ([],1)
 => 1
[3,1]
 => 11 => [2] => ([],2)
 => 0
[5]
 => 1 => [1] => ([],1)
 => 1
[4,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 0
[3,1,1]
 => 111 => [3] => ([],3)
 => 0
[6]
 => 0 => [1] => ([],1)
 => 1
[5,1]
 => 11 => [2] => ([],2)
 => 0
[4,2]
 => 00 => [2] => ([],2)
 => 0
[4,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 0
[3,1,1,1]
 => 1111 => [4] => ([],4)
 => 0
[7]
 => 1 => [1] => ([],1)
 => 1
[6,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 0
[5,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 0
[5,1,1]
 => 111 => [3] => ([],3)
 => 0
[4,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 0
[4,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 0
[3,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0
[8]
 => 0 => [1] => ([],1)
 => 1
[7,1]
 => 11 => [2] => ([],2)
 => 0
[6,2]
 => 00 => [2] => ([],2)
 => 0
[6,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 0
[5,3]
 => 11 => [2] => ([],2)
 => 0
[5,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[5,1,1,1]
 => 1111 => [4] => ([],4)
 => 0
[4,2,2]
 => 000 => [3] => ([],3)
 => 0
[4,2,1,1]
 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 0
[4,1,1,1,1]
 => 01111 => [1,4] => ([(3,4)],5)
 => 0
[3,1,1,1,1,1]
 => 111111 => [6] => ([],6)
 => 0
[8,1]
 => 01 => [1,1] => ([(0,1)],2)
 => 0
[7,2]
 => 10 => [1,1] => ([(0,1)],2)
 => 0
[7,1,1]
 => 111 => [3] => ([],3)
 => 0
[6,3]
 => 01 => [1,1] => ([(0,1)],2)
 => 0
[6,2,1]
 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 0
[6,1,1,1]
 => 0111 => [1,3] => ([(2,3)],4)
 => 0
[5,3,1]
 => 111 => [3] => ([],3)
 => 0
[5,2,2]
 => 100 => [1,2] => ([(1,2)],3)
 => 0
[5,2,1,1]
 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[5,1,1,1,1]
 => 11111 => [5] => ([],5)
 => 0
[4,2,2,1]
 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[4,2,1,1,1]
 => 00111 => [2,3] => ([(2,4),(3,4)],5)
 => 0
[4,1,1,1,1,1]
 => 011111 => [1,5] => ([(4,5)],6)
 => 0
[3,1,1,1,1,1,1]
 => 1111111 => [7] => ([],7)
 => 0
[8,2]
 => 00 => [2] => ([],2)
 => 0
[8,1,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 0
[7,3]
 => 11 => [2] => ([],2)
 => 0
[7,2,1]
 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[7,1,1,1]
 => 1111 => [4] => ([],4)
 => 0
[6,4]
 => 00 => [2] => ([],2)
 => 0
[6,3,1]
 => 011 => [1,2] => ([(1,2)],3)
 => 0
[3,1,1,1,1,1,1,1]
 => 11111111 => [8] => ([],8)
 => ? = 0
[3,1,1,1,1,1,1,1,1]
 => 111111111 => [9] => ([],9)
 => ? = 0
Description
The number of vertices in prime modules of a graph.
Matching statistic: St000296
(load all 12 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[3]
 => []
 => => ? = 1
[4]
 => []
 => => ? = 1
[3,1]
 => [1]
 => 10 => 0
[5]
 => []
 => => ? = 1
[4,1]
 => [1]
 => 10 => 0
[3,1,1]
 => [1,1]
 => 110 => 0
[6]
 => []
 => => ? = 1
[5,1]
 => [1]
 => 10 => 0
[4,2]
 => [2]
 => 100 => 0
[4,1,1]
 => [1,1]
 => 110 => 0
[3,1,1,1]
 => [1,1,1]
 => 1110 => 0
[7]
 => []
 => => ? = 1
[6,1]
 => [1]
 => 10 => 0
[5,2]
 => [2]
 => 100 => 0
[5,1,1]
 => [1,1]
 => 110 => 0
[4,2,1]
 => [2,1]
 => 1010 => 0
[4,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[8]
 => []
 => => ? = 1
[7,1]
 => [1]
 => 10 => 0
[6,2]
 => [2]
 => 100 => 0
[6,1,1]
 => [1,1]
 => 110 => 0
[5,3]
 => [3]
 => 1000 => 0
[5,2,1]
 => [2,1]
 => 1010 => 0
[5,1,1,1]
 => [1,1,1]
 => 1110 => 0
[4,2,2]
 => [2,2]
 => 1100 => 0
[4,2,1,1]
 => [2,1,1]
 => 10110 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[8,1]
 => [1]
 => 10 => 0
[7,2]
 => [2]
 => 100 => 0
[7,1,1]
 => [1,1]
 => 110 => 0
[6,3]
 => [3]
 => 1000 => 0
[6,2,1]
 => [2,1]
 => 1010 => 0
[6,1,1,1]
 => [1,1,1]
 => 1110 => 0
[5,3,1]
 => [3,1]
 => 10010 => 0
[5,2,2]
 => [2,2]
 => 1100 => 0
[5,2,1,1]
 => [2,1,1]
 => 10110 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[4,2,2,1]
 => [2,2,1]
 => 11010 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0
[8,2]
 => [2]
 => 100 => 0
[8,1,1]
 => [1,1]
 => 110 => 0
[7,3]
 => [3]
 => 1000 => 0
[7,2,1]
 => [2,1]
 => 1010 => 0
[7,1,1,1]
 => [1,1,1]
 => 1110 => 0
[6,4]
 => [4]
 => 10000 => 0
[6,3,1]
 => [3,1]
 => 10010 => 0
[6,2,2]
 => [2,2]
 => 1100 => 0
[6,2,1,1]
 => [2,1,1]
 => 10110 => 0
[6,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[5,3,2]
 => [3,2]
 => 10100 => 0
[5,3,1,1]
 => [3,1,1]
 => 100110 => 0
[5,2,2,1]
 => [2,2,1]
 => 11010 => 0
Description
The length of the symmetric border of a binary word. The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix. The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000629
(load all 33 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[3]
 => []
 => => ? = 1
[4]
 => []
 => => ? = 1
[3,1]
 => [1]
 => 10 => 0
[5]
 => []
 => => ? = 1
[4,1]
 => [1]
 => 10 => 0
[3,1,1]
 => [1,1]
 => 110 => 0
[6]
 => []
 => => ? = 1
[5,1]
 => [1]
 => 10 => 0
[4,2]
 => [2]
 => 100 => 0
[4,1,1]
 => [1,1]
 => 110 => 0
[3,1,1,1]
 => [1,1,1]
 => 1110 => 0
[7]
 => []
 => => ? = 1
[6,1]
 => [1]
 => 10 => 0
[5,2]
 => [2]
 => 100 => 0
[5,1,1]
 => [1,1]
 => 110 => 0
[4,2,1]
 => [2,1]
 => 1010 => 0
[4,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[8]
 => []
 => => ? = 1
[7,1]
 => [1]
 => 10 => 0
[6,2]
 => [2]
 => 100 => 0
[6,1,1]
 => [1,1]
 => 110 => 0
[5,3]
 => [3]
 => 1000 => 0
[5,2,1]
 => [2,1]
 => 1010 => 0
[5,1,1,1]
 => [1,1,1]
 => 1110 => 0
[4,2,2]
 => [2,2]
 => 1100 => 0
[4,2,1,1]
 => [2,1,1]
 => 10110 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[8,1]
 => [1]
 => 10 => 0
[7,2]
 => [2]
 => 100 => 0
[7,1,1]
 => [1,1]
 => 110 => 0
[6,3]
 => [3]
 => 1000 => 0
[6,2,1]
 => [2,1]
 => 1010 => 0
[6,1,1,1]
 => [1,1,1]
 => 1110 => 0
[5,3,1]
 => [3,1]
 => 10010 => 0
[5,2,2]
 => [2,2]
 => 1100 => 0
[5,2,1,1]
 => [2,1,1]
 => 10110 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[4,2,2,1]
 => [2,2,1]
 => 11010 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0
[8,2]
 => [2]
 => 100 => 0
[8,1,1]
 => [1,1]
 => 110 => 0
[7,3]
 => [3]
 => 1000 => 0
[7,2,1]
 => [2,1]
 => 1010 => 0
[7,1,1,1]
 => [1,1,1]
 => 1110 => 0
[6,4]
 => [4]
 => 10000 => 0
[6,3,1]
 => [3,1]
 => 10010 => 0
[6,2,2]
 => [2,2]
 => 1100 => 0
[6,2,1,1]
 => [2,1,1]
 => 10110 => 0
[6,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[5,3,2]
 => [3,2]
 => 10100 => 0
[5,3,1,1]
 => [3,1,1]
 => 100110 => 0
[5,2,2,1]
 => [2,2,1]
 => 11010 => 0
Description
The defect of a binary word. The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
Matching statistic: St001371
(load all 11 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[3]
 => []
 => => ? = 1
[4]
 => []
 => => ? = 1
[3,1]
 => [1]
 => 10 => 0
[5]
 => []
 => => ? = 1
[4,1]
 => [1]
 => 10 => 0
[3,1,1]
 => [1,1]
 => 110 => 0
[6]
 => []
 => => ? = 1
[5,1]
 => [1]
 => 10 => 0
[4,2]
 => [2]
 => 100 => 0
[4,1,1]
 => [1,1]
 => 110 => 0
[3,1,1,1]
 => [1,1,1]
 => 1110 => 0
[7]
 => []
 => => ? = 1
[6,1]
 => [1]
 => 10 => 0
[5,2]
 => [2]
 => 100 => 0
[5,1,1]
 => [1,1]
 => 110 => 0
[4,2,1]
 => [2,1]
 => 1010 => 0
[4,1,1,1]
 => [1,1,1]
 => 1110 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[8]
 => []
 => => ? = 1
[7,1]
 => [1]
 => 10 => 0
[6,2]
 => [2]
 => 100 => 0
[6,1,1]
 => [1,1]
 => 110 => 0
[5,3]
 => [3]
 => 1000 => 0
[5,2,1]
 => [2,1]
 => 1010 => 0
[5,1,1,1]
 => [1,1,1]
 => 1110 => 0
[4,2,2]
 => [2,2]
 => 1100 => 0
[4,2,1,1]
 => [2,1,1]
 => 10110 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[8,1]
 => [1]
 => 10 => 0
[7,2]
 => [2]
 => 100 => 0
[7,1,1]
 => [1,1]
 => 110 => 0
[6,3]
 => [3]
 => 1000 => 0
[6,2,1]
 => [2,1]
 => 1010 => 0
[6,1,1,1]
 => [1,1,1]
 => 1110 => 0
[5,3,1]
 => [3,1]
 => 10010 => 0
[5,2,2]
 => [2,2]
 => 1100 => 0
[5,2,1,1]
 => [2,1,1]
 => 10110 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[4,2,2,1]
 => [2,2,1]
 => 11010 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0
[8,2]
 => [2]
 => 100 => 0
[8,1,1]
 => [1,1]
 => 110 => 0
[7,3]
 => [3]
 => 1000 => 0
[7,2,1]
 => [2,1]
 => 1010 => 0
[7,1,1,1]
 => [1,1,1]
 => 1110 => 0
[6,4]
 => [4]
 => 10000 => 0
[6,3,1]
 => [3,1]
 => 10010 => 0
[6,2,2]
 => [2,2]
 => 1100 => 0
[6,2,1,1]
 => [2,1,1]
 => 10110 => 0
[6,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 0
[5,3,2]
 => [3,2]
 => 10100 => 0
[5,3,1,1]
 => [3,1,1]
 => 100110 => 0
[5,2,2,1]
 => [2,2,1]
 => 11010 => 0
Description
The length of the longest Yamanouchi prefix of a binary word. This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St000326
(load all 12 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[3]
 => []
 => => ? = 1 + 1
[4]
 => []
 => => ? = 1 + 1
[3,1]
 => [1]
 => 10 => 1 = 0 + 1
[5]
 => []
 => => ? = 1 + 1
[4,1]
 => [1]
 => 10 => 1 = 0 + 1
[3,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[6]
 => []
 => => ? = 1 + 1
[5,1]
 => [1]
 => 10 => 1 = 0 + 1
[4,2]
 => [2]
 => 100 => 1 = 0 + 1
[4,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[3,1,1,1]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[7]
 => []
 => => ? = 1 + 1
[6,1]
 => [1]
 => 10 => 1 = 0 + 1
[5,2]
 => [2]
 => 100 => 1 = 0 + 1
[5,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[4,2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[4,1,1,1]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[8]
 => []
 => => ? = 1 + 1
[7,1]
 => [1]
 => 10 => 1 = 0 + 1
[6,2]
 => [2]
 => 100 => 1 = 0 + 1
[6,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[5,3]
 => [3]
 => 1000 => 1 = 0 + 1
[5,2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[5,1,1,1]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[4,2,2]
 => [2,2]
 => 1100 => 1 = 0 + 1
[4,2,1,1]
 => [2,1,1]
 => 10110 => 1 = 0 + 1
[4,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 1 = 0 + 1
[8,1]
 => [1]
 => 10 => 1 = 0 + 1
[7,2]
 => [2]
 => 100 => 1 = 0 + 1
[7,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[6,3]
 => [3]
 => 1000 => 1 = 0 + 1
[6,2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[6,1,1,1]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[5,3,1]
 => [3,1]
 => 10010 => 1 = 0 + 1
[5,2,2]
 => [2,2]
 => 1100 => 1 = 0 + 1
[5,2,1,1]
 => [2,1,1]
 => 10110 => 1 = 0 + 1
[5,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[4,2,2,1]
 => [2,2,1]
 => 11010 => 1 = 0 + 1
[4,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 1 = 0 + 1
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 1 = 0 + 1
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 1 = 0 + 1
[8,2]
 => [2]
 => 100 => 1 = 0 + 1
[8,1,1]
 => [1,1]
 => 110 => 1 = 0 + 1
[7,3]
 => [3]
 => 1000 => 1 = 0 + 1
[7,2,1]
 => [2,1]
 => 1010 => 1 = 0 + 1
[7,1,1,1]
 => [1,1,1]
 => 1110 => 1 = 0 + 1
[6,4]
 => [4]
 => 10000 => 1 = 0 + 1
[6,3,1]
 => [3,1]
 => 10010 => 1 = 0 + 1
[6,2,2]
 => [2,2]
 => 1100 => 1 = 0 + 1
[6,2,1,1]
 => [2,1,1]
 => 10110 => 1 = 0 + 1
[6,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 1 = 0 + 1
[5,3,2]
 => [3,2]
 => 10100 => 1 = 0 + 1
[5,3,1,1]
 => [3,1,1]
 => 100110 => 1 = 0 + 1
[5,2,2,1]
 => [2,2,1]
 => 11010 => 1 = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000232
(load all 3 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 50% ●values known / values provided: 96%●distinct values known / distinct values provided: 50%
Values
[3]
 => []
 => []
 => {}
 => ? = 1
[4]
 => []
 => []
 => {}
 => ? = 1
[3,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[5]
 => []
 => []
 => {}
 => ? = 1
[4,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[6]
 => []
 => []
 => {}
 => ? = 1
[5,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[4,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[7]
 => []
 => []
 => {}
 => ? = 1
[6,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[5,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
[8]
 => []
 => []
 => {}
 => ? = 1
[7,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[6,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[5,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 0
[8,1]
 => [1]
 => [1,0,1,0]
 => {{1},{2}}
 => 0
[7,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[7,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[6,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
[6,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[5,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 0
[5,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
[4,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1},{2,3,4,5,6,7}}
 => 0
[8,2]
 => [2]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
[8,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 0
[7,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
[7,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 0
[7,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 0
[6,4]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0
[6,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 0
[6,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 0
[6,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 0
[6,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 0
[5,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0
[5,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 0
[5,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 0
Description
The number of crossings of a set partition. This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
The following 305 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000295The length of the border of a binary word. St000297The number of leading ones in a binary word. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000974The length of the trunk of an ordered tree. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001584The area statistic between a Dyck path and its bounce path. St001696The natural major index of a standard Young tableau. St000047The number of standard immaculate tableaux of a given shape. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000701The protection number of a binary tree. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000805The number of peaks of the associated bargraph. St000913The number of ways to refine the partition into singletons. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001732The number of peaks visible from the left. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001884The number of borders of a binary word. St000397The Strahler number of a rooted tree. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000042The number of crossings of a perfect matching. St000119The number of occurrences of the pattern 321 in a permutation. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001394The genus of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001722The number of minimal chains with small intervals between a binary word and the top element. St000733The row containing the largest entry of a standard tableau. St000842The breadth of a permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000068The number of minimal elements in a poset. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000234The number of global ascents of a permutation. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000115The single entry in the last row. St001501The dominant dimension of magnitude 1 Nakayama algebras. St000769The major index of a composition regarded as a word. St000764The number of strong records in an integer composition. St000761The number of ascents in an integer composition. St000763The sum of the positions of the strong records of an integer composition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000317The cycle descent number of a permutation. St000475The number of parts equal to 1 in a partition. St000623The number of occurrences of the pattern 52341 in a permutation. St000674The number of hills of a Dyck path. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000759The smallest missing part in an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000889The number of alternating sign matrices with the same antidiagonal sums. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001657The number of twos in an integer partition. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000962The 3-shifted major index of a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001381The fertility of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001810The number of fixed points of a permutation smaller than its largest moved point. St000056The decomposition (or block) number of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000542The number of left-to-right-minima of a permutation. 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 $eA$. 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 $eA$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000091The descent variation of a composition. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000562The number of internal points of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000667The greatest common divisor of the parts of the partition. St001593This is the number of standard Young tableaux of the given shifted shape. St000355The number of occurrences of the pattern 21-3. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001735The number of permutations with the same set of runs. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001301The first Betti number of the order complex associated with the poset. St001811The Castelnuovo-Mumford regularity of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000908The length of the shortest maximal antichain in a poset. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000914The sum of the values of the Möbius function of a poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000658The number of rises of length 2 of a Dyck path. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St000478Another weight of a partition according to Alladi. St000011The number of touch points (or returns) of a Dyck path. St000546The number of global descents of a permutation. St000264The girth of a graph, which is not a tree. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001484The number of singletons of an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St001561The value of the elementary symmetric function evaluated at 1. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001933The largest multiplicity of a part in an integer partition. St000124The cardinality of the preimage of the Simion-Schmidt map. 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. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001730The number of times the path corresponding to a binary word crosses the base line. St001890The maximum magnitude of the Möbius function of a poset. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000390The number of runs of ones in a binary word. St000215The number of adjacencies of a permutation, zero appended. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000989The number of final rises of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000286The number of connected components of the complement of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000990The first ascent of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000352The Elizalde-Pak rank of a permutation. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000214The number of adjacencies of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000007The number of saliances of the permutation. St001271The competition number of a graph. St000823The number of unsplittable factors of the set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000699The toughness times the least common multiple of 1,. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000237The number of small exceedances. St000455The second largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000782The indicator function of whether a given perfect matching is an L & P matching. St000268The number of strongly connected orientations of a graph. St000344The number of strongly connected outdegree sequences of a graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001073The number of nowhere zero 3-flows of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001846The number of elements which do not have a complement in the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001820The size of the image of the pop stack sorting operator. St001964The interval resolution global dimension of a poset. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000241The number of cyclical small excedances. St000461The rix statistic of a permutation. St000873The aix statistic of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001354The number of series nodes in the modular decomposition of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001957The number of Hasse diagrams with a given underlying undirected graph. St000654The first descent of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001481The minimal height of a peak of a Dyck path. St000756The sum of the positions of the left to right maxima of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000259The diameter of a connected graph.