Your data matches 31 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000148
(load all 3 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => 0
[1,1,0,0]
 => [1,2] => [1,1]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 3
Description
The number of odd parts of a partition.
Matching statistic: St000475
(load all 2 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => 0
[1,1,0,0]
 => [1,2] => [1,1]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 3
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001247
(load all 2 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => 0
[1,1,0,0]
 => [1,2] => [1,1]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 3
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001249
(load all 2 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => 0
[1,1,0,0]
 => [1,2] => [1,1]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 3
Description
Sum of the odd parts of a partition.
Matching statistic: St000835
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000835: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => [1]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => [1,1]
 => 0
[1,1,0,0]
 => [1,2] => [1,1]
 => [2]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [2,1]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [2,1]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [2,1]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [2,1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [3]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => [3,1]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => [3,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [2,2]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => [3,1]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [2,2]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [4,1]
 => 3
Description
The minimal difference in size when partitioning the integer partition into two subpartitions. This is the optimal value of the optimisation version of the partition problem [1].
Matching statistic: St000992
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => [1]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => [1,1]
 => 0
[1,1,0,0]
 => [1,2] => [1,1]
 => [2]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [2,1]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [2,1]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [2,1]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [2,1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [3]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => [3,1]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => [3,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [2,2]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => [3,1]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [2,2]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [4,1]
 => 3
Description
The alternating sum of the parts of an integer partition. For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St001055
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001055: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => [1]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => [1,1]
 => 0
[1,1,0,0]
 => [1,2] => [1,1]
 => [2]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [2,1]
 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [2,1]
 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [2,1]
 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [2,1]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [3]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => [3,1]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => [3,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [2,2]
 => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => [3,1]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [2,2]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [3,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => [3,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => [3,2]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => [3,2]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => [4,1]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [4,1]
 => 3
Description
The Grundy value for the game of removing cells of a row in an integer partition. Two players alternately remove any positive number of cells in a row of the Ferrers diagram of an integer partition, such that the result is still a Ferrers diagram. The player facing the empty partition looses.
Matching statistic: St000714
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => [1]
 => ? = 1 + 1
[1,0,1,0]
 => [2,1] => [2]
 => [1,1]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => [1,1]
 => [2]
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [2,1]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [3]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [2,2]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [2,2]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => [3,1]
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => [3,2]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => [4,1]
 => 4 = 3 + 1
Description
The number of semistandard Young tableau of given shape, with entries at most 2. This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St000288
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Values
[1,0]
 => [1] => [1]
 => 1 => 1
[1,0,1,0]
 => [2,1] => [2]
 => 0 => 0
[1,1,0,0]
 => [1,2] => [1,1]
 => 11 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 01 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 01 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 01 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 01 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 111 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 011 => 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 011 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 00 => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 011 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 011 => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 011 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 011 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 00 => 0
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 011 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 011 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 011 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 011 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 011 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1111 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 0111 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 0111 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 001 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 0111 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 0111 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 001 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 001 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 001 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 0111 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 0111 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 001 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 001 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 0111 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 0111 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 0111 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 0111 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 001 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 0111 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 0111 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 001 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 001 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 001 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 0111 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 0111 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 001 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 001 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 0111 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 0111 => 3
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,12,10] => [2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => ? = 8
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1]
 => 1111111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
 => 11111111111 => ? = 11
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,5,6,7,8,9,10,11,12,3] => [2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => ? = 8
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [7,8,9,10,11,1,2,3,4,5,12,6] => [2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,6,7,8,9,10,1,12,11] => [2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => ? = 8
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000392
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 100%●distinct values known / distinct values provided: 92%
Values
[1,0]
 => [1] => [1]
 => 1 => 1
[1,0,1,0]
 => [2,1] => [2]
 => 0 => 0
[1,1,0,0]
 => [1,2] => [1,1]
 => 11 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 01 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 01 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 01 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 01 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 111 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 011 => 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 011 => 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 00 => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 011 => 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 011 => 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 011 => 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 011 => 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 00 => 0
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 011 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 011 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 011 => 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 011 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 011 => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1111 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 0111 => 3
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 0111 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 001 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 0111 => 3
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 0111 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 001 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 001 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 001 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 0111 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 0111 => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 001 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 001 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 0111 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 0111 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 0111 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 0111 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 001 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 0111 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 0111 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 001 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 001 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 001 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 0111 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 0111 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 001 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 001 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 0111 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 0111 => 3
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,12,10] => [2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => ? = 8
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1]
 => 1111111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
 => 11111111111 => ? = 11
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,5,6,7,8,9,10,11,12,3] => [2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => ? = 8
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [7,8,9,10,11,1,2,3,4,5,12,6] => [2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,6,7,8,9,10,1,12,11] => [2,2,1,1,1,1,1,1,1,1]
 => 0011111111 => ? = 8
Description
The length of the longest run of ones in a binary word.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001372The length of a longest cyclic run of ones of a binary word. St000878The number of ones minus the number of zeros of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000445The number of rises of length 1 of a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000248The number of anti-singletons of a set partition. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000160The multiplicity of the smallest part of a partition. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St000247The number of singleton blocks of a set partition. St000215The number of adjacencies of a permutation, zero appended. St000385The number of vertices with out-degree 1 in a binary tree. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001948The number of augmented double ascents of a permutation. St001524The degree of symmetry of a binary word.