Your data matches 87 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000010
(load all 4 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: 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]
 => 1
[1,1,0,0]
 => [1,2] => [1,1]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 2
[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]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 3
[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]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 4
Description
The length of the partition.
Matching statistic: St001250
(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
St001250: 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]
 => 1
[1,1,0,0]
 => [1,2] => [1,1]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 2
[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]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 3
[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]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 4
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St000147
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => [1]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => [1,1]
 => 1
[1,1,0,0]
 => [1,2] => [1,1]
 => [2]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [2,1]
 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [2,1]
 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [2,1]
 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [2,1]
 => 2
[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]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => [3,1]
 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [2,2]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => [3,1]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [3,1]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => [3,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => [3,1]
 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [2,2]
 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [3,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => [3,1]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => [3,1]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => [3,1]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => [3,1]
 => 3
[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]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => [3,2]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => [4,1]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => [3,2]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => [3,2]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => [3,2]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => [4,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => [3,2]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => [3,2]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => [4,1]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [4,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => [4,1]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => [3,2]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => [3,2]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => [3,2]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [4,1]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [4,1]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => [3,2]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => [3,2]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => [4,1]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [4,1]
 => 4
Description
The largest part of an integer partition.
Matching statistic: St000378
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: 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]
 => 1
[1,1,0,0]
 => [1,2] => [1,1]
 => [2]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [3]
 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [3]
 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [3]
 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [3]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [2,1]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => [2,2]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => [2,2]
 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [4]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => [2,2]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [2,2]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => [2,2]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => [2,2]
 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [4]
 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [2,2]
 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => [2,2]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => [2,2]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => [2,2]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => [2,2]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [3,1]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => [2,2,1]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => [2,2,1]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => [2,2,1]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => [2,2,1]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => [2,2,1]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => [2,2,1]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => [2,2,1]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => [2,2,1]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => [2,2,1]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => [2,2,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => [2,2,1]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => [2,2,1]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => [3,1,1]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [3,1,1]
 => 4
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000876
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000876: Binary words ⟶ 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]
 => 0 => 1
[1,1,0,0]
 => [1,2] => [1,1]
 => 11 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 01 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 01 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 01 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 01 => 2
[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 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 011 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 00 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 011 => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 011 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 011 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 011 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 00 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 011 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 011 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 011 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 011 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 011 => 3
[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 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 0111 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 001 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 0111 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 0111 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 001 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 001 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 001 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 0111 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 0111 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 001 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 001 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 0111 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 0111 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 0111 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 0111 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 001 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 0111 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 0111 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 001 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 001 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 001 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 0111 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 0111 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 001 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 001 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 0111 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 0111 => 4
[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 => ? = 10
[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 => ? = 10
[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 => ? = 10
[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 => ? = 10
Description
The number of factors in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000885
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000885: Binary words ⟶ 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]
 => 0 => 1
[1,1,0,0]
 => [1,2] => [1,1]
 => 11 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 01 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 01 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 01 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 01 => 2
[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 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 011 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 00 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 011 => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 011 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 011 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 011 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 00 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 011 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 011 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 011 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 011 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 011 => 3
[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 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 0111 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 001 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 0111 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 0111 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 001 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 001 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 001 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 0111 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 0111 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 001 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 001 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 0111 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 0111 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 0111 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 0111 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 001 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 0111 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 0111 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 001 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 001 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 001 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 0111 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 0111 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 001 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 001 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 0111 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 0111 => 4
[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 => ? = 10
[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 => ? = 10
[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 => ? = 10
[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 => ? = 10
Description
The number of critical steps in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of critical steps $\ell + m$ in the Catalan factorisation. The distribution of this statistic on words of length $n$ is $$ (n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}. $$
Matching statistic: St000288
(load all 9 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 98%●distinct values known / distinct values provided: 91%
Values
[1,0]
 => [1] => [1]
 => 10 => 1
[1,0,1,0]
 => [2,1] => [2]
 => 100 => 1
[1,1,0,0]
 => [1,2] => [1,1]
 => 110 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => 1010 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => 1010 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => 1010 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => 1010 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => 1110 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => 10110 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => 10110 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => 1100 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => 10110 => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => 10110 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => 10110 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => 10110 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => 1100 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 10110 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => 10110 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => 10110 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => 10110 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => 10110 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => 11110 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 101110 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => 101110 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => 11010 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => 101110 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => 101110 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => 11010 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => 11010 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => 11010 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => 101110 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => 101110 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => 11010 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => 11010 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => 101110 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 101110 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 101110 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 101110 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => 11010 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => 101110 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 101110 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => 11010 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => 11010 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => 11010 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 101110 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => 101110 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => 11010 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => 11010 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => 101110 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => 101110 => 4
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,11,1,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[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]
 => 110111111110 => ? = 10
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [2,12,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => 1011111111110 => ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,7,8,9,10,1,11] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,10,11,1,9] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,7,8,9,10,11,1,3] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,8,9,10,11,1,2] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,5,6,7,8,11,1,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,6,7,11,1,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,3,4,5,6,11,1,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [6,7,8,9,10,1,2,3,4,5,11] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,7,8,9,10,11,2,3,4,5,6] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,7,8,9,10,11,2] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => [10,1,2,3,4,5,6,7,8,9,11] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [1,11,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => [1,2,3,4,5,6,7,8,11,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,7,8,9,10,1,2,11] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,9,10,11,1,2,8] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,6,7,8,9,10,11,1,2,5] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,4,6,8,10,12,1,3,5,7,9,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => 1011111111110 => ? = 11
[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]
 => 111111111110 => ? = 11
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
 => [1,2,3,4,5,6,7,8,9,12,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => 1011111111110 => ? = 11
[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8,9,10,11,12] => [2,1,1,1,1,1,1,1,1,1,1]
 => 1011111111110 => ? = 11
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,2,3,4,5,6,7,8,10,9,11] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
 => [10,11,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,3,4,5,11,1,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,10,11,9] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [2,8,9,10,11,1,3,4,5,6,7] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [1,8,2,9,3,10,4,11,5,12,6,7] => [2,2,2,2,2,1,1]
 => 111110110 => ? = 7
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,4,2,6,3,8,5,10,7,12,9,11] => [2,2,2,2,2,1,1]
 => 111110110 => ? = 7
[1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,1,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => 1011111111110 => ? = 11
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0]
 => [2,1,5,6,3,4,9,10,7,8,12,11] => [2,2,2,2,1,1,1,1]
 => 1111011110 => ? = 8
[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]
 => 110111111110 => ? = 10
[1,0,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,3,11,1,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,7,4,9,6,11,8,10,12] => [2,2,2,2,2,1,1]
 => 111110110 => ? = 7
[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]
 => 110111111110 => ? = 10
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [2,8,9,10,11,12,1,3,4,5,6,7] => [2,1,1,1,1,1,1,1,1,1,1]
 => 1011111111110 => ? = 11
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,7,8,9,10,11,12,2] => [2,1,1,1,1,1,1,1,1,1,1]
 => 1011111111110 => ? = 11
[1,0,1,0,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [2,3,4,11,1,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => 101111111110 => ? = 10
[1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [1,8,9,10,11,12,2,3,4,5,6,7] => [2,1,1,1,1,1,1,1,1,1,1]
 => 1011111111110 => ? = 11
[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]
 => 110111111110 => ? = 10
[1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,1,0,0]
 => [7,1,8,2,9,3,10,4,11,12,5,6] => [2,2,2,2,2,1,1]
 => 111110110 => ? = 7
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000733
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 98%●distinct values known / distinct values provided: 91%
Values
[1,0]
 => [1] => [1]
 => [[1]]
 => 1
[1,0,1,0]
 => [2,1] => [2]
 => [[1,2]]
 => 1
[1,1,0,0]
 => [1,2] => [1,1]
 => [[1],[2]]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [[1,2],[3]]
 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [[1,2],[3]]
 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [[1,2],[3]]
 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [[1,2],[3]]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,11,9] => [2,2,1,1,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 9
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,11,1,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[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]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 10
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [2,12,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 11
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,7,8,9,10,1,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,10,11,1,9] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,7,8,9,10,11,1,3] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,8,9,10,11,1,2] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,5,6,7,8,11,1,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,6,7,11,1,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,3,4,5,6,11,1,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [6,7,8,9,10,1,2,3,4,5,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,7,8,9,10,11,2,3,4,5,6] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,7,8,9,10,11,2] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => [10,1,2,3,4,5,6,7,8,9,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [1,11,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => [1,2,3,4,5,6,7,8,11,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0]
 => [10,1,2,3,4,5,6,7,11,8,9] => [2,2,1,1,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,7,8,9,10,1,2,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,9,10,11,1,2,8] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,6,7,8,9,10,11,1,2,5] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,4,6,8,10,12,1,3,5,7,9,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 11
[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]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 11
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
 => [1,2,3,4,5,6,7,8,9,12,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 11
[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8,9,10,11,12] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 11
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [3,1,4,5,6,7,8,9,10,11,2] => [2,2,1,1,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 9
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,2,3,4,5,6,7,8,10,9,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
 => [10,11,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,6,7,8,9,1,11,10] => [2,2,1,1,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 9
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,6,7,8,9,10,1,11,2] => [2,2,1,1,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 9
[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,3,4,5,11,1,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,11,9] => [2,2,1,1,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 9
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,10,11,9] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [7,8,9,10,1,2,3,4,5,11,6] => [2,2,1,1,1,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 9
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [2,8,9,10,11,1,3,4,5,6,7] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [1,8,2,9,3,10,4,11,5,12,6,7] => [2,2,2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
 => ? = 7
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,4,2,6,3,8,5,10,7,12,9,11] => [2,2,2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
 => ? = 7
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [3,4,5,6,7,8,1,9,2,11,10] => [2,2,2,1,1,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
 => ? = 8
[1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,1,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 11
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0]
 => [2,1,5,6,3,4,9,10,7,8,12,11] => [2,2,2,2,1,1,1,1]
 => [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
 => ? = 8
[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]
 => [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 10
[1,0,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,3,11,1,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 10
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,7,4,9,6,11,8,10,12] => [2,2,2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
 => ? = 7
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000157
(load all 4 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 98%●distinct values known / distinct values provided: 91%
Values
[1,0]
 => [1] => [1]
 => [[1]]
 => 0 = 1 - 1
[1,0,1,0]
 => [2,1] => [2]
 => [[1,2]]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => [1,1]
 => [[1],[2]]
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [2,1]
 => [[1,3],[2]]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1]
 => [[1,3],[2]]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [2,1]
 => [[1,3],[2]]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [2,1]
 => [[1,3],[2]]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [2,2]
 => [[1,2],[3,4]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,11,9] => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ? = 9 - 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,11,1,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[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]
 => [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
 => ? = 10 - 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [2,12,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 11 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,7,8,9,10,1,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,11,1,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,10,11,1,9] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,7,8,9,10,11,1,3] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,8,9,10,11,1,2] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,5,6,7,8,11,1,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,4,5,6,7,11,1,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,3,4,5,6,11,1,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
 => [6,7,8,9,10,1,2,3,4,5,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,7,8,9,10,11,2,3,4,5,6] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,7,8,9,10,11,2] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
 => [10,1,2,3,4,5,6,7,8,9,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
 => [1,11,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
 => [1,2,3,4,5,6,7,8,11,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0]
 => [10,1,2,3,4,5,6,7,11,8,9] => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ? = 9 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,7,8,9,10,1,2,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,9,10,11,1,2,8] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,6,7,8,9,10,11,1,2,5] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,4,6,8,10,12,1,3,5,7,9,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 11 - 1
[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]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 11 - 1
[1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0]
 => [1,2,3,4,5,6,7,8,9,12,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 11 - 1
[1,1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7,8,9,10,11,12] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 11 - 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [3,1,4,5,6,7,8,9,10,11,2] => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,2,3,4,5,6,7,8,10,9,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
 => [10,11,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,6,7,8,9,1,11,10] => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ? = 9 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,6,7,8,9,10,1,11,2] => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,3,4,5,11,1,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,4,5,6,7,8,10,1,11,9] => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ? = 9 - 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,10,11,9] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [7,8,9,10,1,2,3,4,5,11,6] => [2,2,1,1,1,1,1,1,1]
 => [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
 => ? = 9 - 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [2,8,9,10,11,1,3,4,5,6,7] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [1,8,2,9,3,10,4,11,5,12,6,7] => [2,2,2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
 => ? = 7 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,4,2,6,3,8,5,10,7,12,9,11] => [2,2,2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
 => ? = 7 - 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [3,4,5,6,7,8,1,9,2,11,10] => [2,2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
 => ? = 8 - 1
[1,0,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,1,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 11 - 1
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0]
 => [2,1,5,6,3,4,9,10,7,8,12,11] => [2,2,2,2,1,1,1,1]
 => [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
 => ? = 8 - 1
[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]
 => [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
 => ? = 10 - 1
[1,0,1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,3,11,1,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 10 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,7,4,9,6,11,8,10,12] => [2,2,2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
 => ? = 7 - 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000507
(load all 7 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [[1]]
 => 1
[1,0,1,0]
 => [2,1] => [[1],[2]]
 => 1
[1,1,0,0]
 => [1,2] => [[1,2]]
 => 2
[1,0,1,0,1,0]
 => [2,3,1] => [[1,2],[3]]
 => 2
[1,0,1,1,0,0]
 => [2,1,3] => [[1,3],[2]]
 => 2
[1,1,0,0,1,0]
 => [1,3,2] => [[1,2],[3]]
 => 2
[1,1,0,1,0,0]
 => [3,1,2] => [[1,3],[2]]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [[1,2,3]]
 => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [[1,2,3],[4]]
 => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [[1,2],[3,4]]
 => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [[1,3],[2,4]]
 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [[1,3,4],[2]]
 => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [[1,2,4],[3]]
 => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [[1,3,4],[2]]
 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [[1,2,3,4]]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [[1,2,3,4],[5]]
 => 4
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [[1,2,3,5],[4]]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [[1,2,4],[3,5]]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [[1,2,3],[4,5]]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [[1,2,4,5],[3]]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [[1,3,4],[2,5]]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [[1,3,5],[2,4]]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [[1,2,4],[3,5]]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [[1,2,3],[4,5]]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [[1,2,5],[3,4]]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [[1,3,5],[2,4]]
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [[1,2,5],[3,4]]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [[1,2,3,4],[5]]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [[1,2,3,5],[4]]
 => 4
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [[1,2,3],[4,5]]
 => 4
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 4
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [[1,3,4],[2,5]]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [[1,3,5],[2,4]]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [[1,2,4],[3,5]]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [[1,2,3],[4,5]]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [[1,3,4],[2,5]]
 => 3
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [[1,3,5],[2,4]]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [[1,2,5],[3,4]]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [[1,3,4,5],[2]]
 => 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [2,3,5,6,1,7,4,8] => ?
 => ? = 6
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [2,3,6,7,1,4,8,5] => ?
 => ? = 6
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,3,6,7,1,4,5,8] => ?
 => ? = 7
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,3,6,8,1,4,5,7] => ?
 => ? = 7
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,4,6,7,3,5,8] => ?
 => ? = 6
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [2,4,5,1,6,7,3,8] => ?
 => ? = 6
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [2,4,5,7,1,3,8,6] => ?
 => ? = 6
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [2,4,1,6,7,3,5,8] => ?
 => ? = 6
[1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [2,4,6,1,3,7,8,5] => ?
 => ? = 6
[1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,4,6,1,7,8,3,5] => ?
 => ? = 6
[1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [2,4,6,7,1,8,3,5] => ?
 => ? = 6
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [2,4,7,1,3,8,5,6] => ?
 => ? = 6
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [2,5,1,6,3,8,4,7] => ?
 => ? = 5
[1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [2,5,1,7,3,8,4,6] => ?
 => ? = 5
[1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [2,5,7,1,3,8,4,6] => ?
 => ? = 6
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [2,1,3,4,6,7,5,8] => ?
 => ? = 6
[1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [2,1,6,7,3,8,4,5] => ?
 => ? = 5
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,5,7,2,8,6] => ?
 => ? = 6
[1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,3,4,5,2,6,8,7] => ?
 => ? = 6
[1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,2,5,7,4,8,6] => ?
 => ? = 5
[1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,3,5,2,6,7,8,4] => ?
 => ? = 6
[1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,3,5,2,6,4,7,8] => ?
 => ? = 6
[1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,3,5,2,4,7,8,6] => ?
 => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,3,2,6,4,7,8,5] => ?
 => ? = 5
[1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [3,1,4,7,2,8,5,6] => ?
 => ? = 5
[1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [3,4,5,1,7,2,6,8] => ?
 => ? = 6
[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [3,4,1,7,2,5,8,6] => ?
 => ? = 5
[1,1,0,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,5,6,4,7,8] => ?
 => ? = 6
[1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [3,1,2,5,7,8,4,6] => ?
 => ? = 6
[1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => [3,1,5,2,4,7,8,6] => ?
 => ? = 5
[1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [3,1,5,2,4,6,8,7] => ?
 => ? = 5
[1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,1,5,2,4,6,7,8] => ?
 => ? = 6
[1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [3,5,7,1,2,8,4,6] => ?
 => ? = 6
[1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,2,4,6,3,7,8,5] => ?
 => ? = 6
[1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,2,4,6,7,3,8,5] => ?
 => ? = 6
[1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,2,4,6,7,3,5,8] => ?
 => ? = 7
[1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,4,2,5,3,7,8,6] => ?
 => ? = 5
[1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,4,5,6,2,3,8,7] => ?
 => ? = 6
[1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,4,5,6,2,8,3,7] => ?
 => ? = 6
[1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
 => [4,1,5,7,2,3,6,8] => ?
 => ? = 6
[1,1,1,0,1,0,1,1,0,0,1,1,0,0,0,0]
 => [4,5,1,7,2,3,6,8] => ?
 => ? = 6
[1,1,1,0,1,0,1,1,0,1,0,0,0,0,1,0]
 => [4,5,7,1,2,3,8,6] => ?
 => ? = 6
[1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
 => [4,6,1,7,2,3,8,5] => ?
 => ? = 5
[1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
 => [1,5,6,2,7,3,4,8] => ?
 => ? = 6
[1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
 => [1,5,7,2,3,4,6,8] => ?
 => ? = 7
[1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
 => [5,1,6,2,7,8,3,4] => ?
 => ? = 5
[1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
 => [5,6,1,2,7,3,4,8] => ?
 => ? = 6
[1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
 => [5,6,1,7,2,8,3,4] => ?
 => ? = 5
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,9,7] => [[1,3,4,5,6,7,8],[2,9]]
 => ? = 7
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 10
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
The following 77 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000676The number of odd rises of a Dyck path. St000393The number of strictly increasing runs in a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000245The number of ascents of a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000925The number of topologically connected components of a set partition. St000702The number of weak deficiencies of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000167The number of leaves of an ordered tree. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000308The height of the tree associated to a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000021The number of descents of a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001480The number of simple summands of the module J^2/J^3. St001427The number of descents of a signed permutation. 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. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001863The number of weak excedances of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001935The number of ascents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001712The number of natural descents of a standard Young tableau. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001626The number of maximal proper sublattices of a lattice.