Your data matches 335 different statistics following compositions of up to 3 maps.
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Matching statistic: St000159
(load all 5 compositions to match this statistic)
Mapping
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 1
[1,1]
 => 1
[3]
 => 1
[2,1]
 => 2
[1,1,1]
 => 1
[4]
 => 1
[3,1]
 => 2
[2,2]
 => 1
[2,1,1]
 => 2
[1,1,1,1]
 => 1
[5]
 => 1
[4,1]
 => 2
[3,2]
 => 2
[3,1,1]
 => 2
[2,2,1]
 => 2
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 1
[6]
 => 1
[5,1]
 => 2
[4,2]
 => 2
[4,1,1]
 => 2
[3,3]
 => 1
[3,2,1]
 => 3
[3,1,1,1]
 => 2
[2,2,2]
 => 1
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 1
[6,1]
 => 2
[5,2]
 => 2
[5,1,1]
 => 2
[4,3]
 => 2
[4,2,1]
 => 3
[4,1,1,1]
 => 2
[3,3,1]
 => 2
[3,2,2]
 => 2
[3,2,1,1]
 => 3
[3,1,1,1,1]
 => 2
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 1
[8]
 => 1
[7,1]
 => 2
[6,2]
 => 2
[6,1,1]
 => 2
[5,3]
 => 2
[5,2,1]
 => 3
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1] => [1,1]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [2,1] => [2,1]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,2] => [2,1]
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [4,1]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1] => [5,1]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [3,1,1]
 => 3 = 2 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => [5,1]
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,1] => [6,1]
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,1,1] => [4,1,1]
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,4,1] => [4,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,6] => [6,1]
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1] => [7,1]
 => 2 = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,1,1] => [5,1,1]
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,1,1] => [4,1,1]
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,2,1] => [3,2,1]
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => 3 = 2 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [2,1,1,1]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,4,1] => [4,1,1]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,3,2] => [3,2,1]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,5,1] => [5,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,7] => [7,1]
 => 2 = 1 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => [8,1]
 => 2 = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [6,1,1] => [6,1,1]
 => 3 = 2 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [5,1,1] => [5,1,1]
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [4,2,1] => [4,2,1]
 => 3 = 2 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,1,1] => [4,1,1]
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [3,1,1,1] => [3,1,1,1]
 => 4 = 3 + 1
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [9,1,1] => ?
 => ? ∊ {1,2} + 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,2} + 1
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2} + 1
[11,1]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,1] => ?
 => ? ∊ {1,1,2,2,2} + 1
[10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,1] => ?
 => ? ∊ {1,1,2,2,2} + 1
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [8,2,1] => ?
 => ? ∊ {1,1,2,2,2} + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2} + 1
Description
The length of the partition.
Matching statistic: St001918
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,1,1,1,1]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [3,1,1,1]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [3,1,1,1]
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,1,1,1,1]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [3,1,1,1]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [3,1,1,1]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [3,1,1,1]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,1,1,1]
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [3,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1]
 => 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
 => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => [3,1,1,1,1,1]
 => 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [3,1,1,1,1]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [3,1,1,1,1]
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,1,1,1]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [4,1,1]
 => 3
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ?
 => ? ∊ {1,2,2,2,2,3,3}
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? ∊ {1,2,2,2,2,3,3}
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3}
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3}
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3}
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,2,2,2,2,3,3}
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,3,1,4,5,6,7,8,9,10] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,2,1,4,5,6,7,8,9] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [9,5,1,2,3,4,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,1,3,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,4,1,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [8,4,3,1,2,5,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[4,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [5,3,2,4,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[4,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [4,3,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,12,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition. Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$. The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is $$ \sum_{p\in\lambda} [p]_{q^{N/p}}, $$ where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer. This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals $$ \left(1 - \frac{1}{\lambda_1}\right) N, $$ where $\lambda_1$ is the largest part of $\lambda$. The statistic is undefined for the empty partition.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3]
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,1,1,1,1]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => 3 = 2 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => 3 = 2 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => 3 = 2 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [3,1,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
 => 2 = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => [3,1,1,1,1,1]
 => 3 = 2 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [4,1,1]
 => 4 = 3 + 1
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,3,1,4,5,6,7,8,9,10] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,2,1,4,5,6,7,8,9] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [9,5,1,2,3,4,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,1,3,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,4,1,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [8,4,3,1,2,5,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [5,3,2,4,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [4,3,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,12,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
Description
The largest part of an integer partition.
Matching statistic: St000668
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3]
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,1,1,1,1]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => 3 = 2 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => 3 = 2 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => 3 = 2 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [3,1,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
 => 2 = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => [3,1,1,1,1,1]
 => 3 = 2 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [4,1,1]
 => 4 = 3 + 1
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,3,1,4,5,6,7,8,9,10] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,2,1,4,5,6,7,8,9] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [9,5,1,2,3,4,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,1,3,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,4,1,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [8,4,3,1,2,5,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [5,3,2,4,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [4,3,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,12,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3]
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,1,1,1,1]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => 3 = 2 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => 3 = 2 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => 3 = 2 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [3,1,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
 => 2 = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => [3,1,1,1,1,1]
 => 3 = 2 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [4,1,1]
 => 4 = 3 + 1
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,3,1,4,5,6,7,8,9,10] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,2,1,4,5,6,7,8,9] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [9,5,1,2,3,4,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,1,3,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,4,1,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [8,4,3,1,2,5,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [5,3,2,4,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [4,3,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,12,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
Description
The product of the parts of an integer partition.
Matching statistic: St001279
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3]
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,1,1,1,1]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => 3 = 2 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => 3 = 2 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => 3 = 2 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [3,1,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
 => 2 = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => [3,1,1,1,1,1]
 => 3 = 2 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [4,1,1]
 => 4 = 3 + 1
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,3,1,4,5,6,7,8,9,10] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,2,1,4,5,6,7,8,9] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [9,5,1,2,3,4,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,1,3,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,4,1,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [8,4,3,1,2,5,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [5,3,2,4,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [4,3,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,12,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001389
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3]
 => 3 = 2 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 2 = 1 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 2 = 1 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,1,1,1,1]
 => 2 = 1 + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => 3 = 2 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => 3 = 2 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => 3 = 2 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 2 = 1 + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => 3 = 2 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => 3 = 2 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [3,1,1,1]
 => 3 = 2 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => 3 = 2 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => 3 = 2 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => 3 = 2 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [3,1,1,1,1]
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
 => 2 = 1 + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => [3,1,1,1,1,1]
 => 3 = 2 + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [3,1,1,1,1]
 => 3 = 2 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,1,1,1]
 => 3 = 2 + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [4,1,1]
 => 4 = 3 + 1
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,2,2,2,2,3,3} + 1
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,3,1,4,5,6,7,8,9,10] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,2,1,4,5,6,7,8,9] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [9,5,1,2,3,4,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,1,3,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,4,1,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [8,4,3,1,2,5,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [5,3,2,4,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[4,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [4,3,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,12,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3} + 1
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000288
(load all 5 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 1 => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 01 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 01 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 001 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 11 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 001 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 0001 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 101 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 001 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 011 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0001 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => 00001 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 1001 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 011 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 011 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 101 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 0101 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 00001 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => 000001 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => 10001 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 0101 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => 0101 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 0001 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 111 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => 0011 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 0001 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => 0011 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => 01001 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 000001 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => 0000001 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => 100001 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => 01001 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => 01001 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 0011 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 1101 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => 0011 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 1001 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 0011 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 0111 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => 00101 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => 1001 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => 00101 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => 010001 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => 0000001 => 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => 00000001 => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => 1000001 => 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => 010001 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => 010001 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 00101 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => 11001 => 3
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => 100000001 => ? = 2
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,1,3,4,5,6,7,8,9,10] => 1000000001 => ? ∊ {1,2,2,2,2,3,3}
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ? => ? ∊ {1,2,2,2,2,3,3}
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ? => ? ∊ {1,2,2,2,2,3,3}
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ? => ? ∊ {1,2,2,2,2,3,3}
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ? => ? ∊ {1,2,2,2,2,3,3}
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ? => ? ∊ {1,2,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? => ? ∊ {1,2,2,2,2,3,3}
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[11,1]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,2,1,3,4,5,6,7,8,9,10,11] => 10000000001 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,3,1,4,5,6,7,8,9,10] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,2,1,4,5,6,7,8,9] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [9,5,1,2,3,4,6,7,8] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,1,3,5,6,7,8] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,4,1,5,6,7,8] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [8,4,3,1,2,5,6,7] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [5,3,2,4,6,7,8,9,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,10,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [4,3,5,2,6,7,8,9,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,11,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,11,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,12,1] => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [2]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,1]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => [2,1]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,1,1]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3]
 => 2
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,1]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,1,1,1]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,1]
 => 2
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,1]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,1,1]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,5] => [2,1,1,1,1]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [3,1,1]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,1]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [3,1,1]
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,1,1,1,1]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [6,2,1,3,4,5] => [3,1,1,1]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,1]
 => 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [3,1,1]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [2,1,1,1]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4]
 => 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [3,1,1]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,1,1]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,1]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,6,1] => [3,1,1,1]
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,1,1,1,1,1]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
 => 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [7,2,1,3,4,5,6] => [3,1,1,1,1]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [6,3,1,2,4,5] => [3,1,1,1]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [6,2,3,1,4,5] => [3,1,1,1]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,1,1]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,1]
 => 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [3,1,1]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,1,1]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [3,1,1]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [4,1]
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,6,1] => [3,1,1,1]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,1,1]
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,6,1] => [3,1,1,1]
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [3,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [2,1,1,1,1,1,1]
 => 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
 => 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [8,2,1,3,4,5,6,7] => [3,1,1,1,1,1]
 => 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [7,3,1,2,4,5,6] => [3,1,1,1,1]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [7,2,3,1,4,5,6] => [3,1,1,1,1]
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [3,1,1,1]
 => 2
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1,4,5] => [4,1,1]
 => 3
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1}
[1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [2,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1}
[11]
 => [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,1,2,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[10,1]
 => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,1,3,4,5,6,7,8,9,10] => [3,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,1,4,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,1,5,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[3,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,1] => [3,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,3,3}
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[11,1]
 => [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [12,2,1,3,4,5,6,7,8,9,10,11] => [3,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => [11,3,1,2,4,5,6,7,8,9,10] => [3,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[10,1,1]
 => [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [11,2,3,1,4,5,6,7,8,9,10] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[9,2,1]
 => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
 => [10,3,2,1,4,5,6,7,8,9] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => [9,5,1,2,3,4,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => [9,4,2,1,3,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[8,2,1,1]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
 => [9,3,2,4,1,5,6,7,8] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [8,4,3,1,2,5,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[7,2,1,1,1]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,2,4,5,1,6,7] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[5,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,3,2,4,5,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[5,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [6,2,3,4,5,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[4,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [5,3,4,2,6,7,8,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[4,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [5,3,2,4,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[4,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [4,3,5,2,6,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,11,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,12,1] => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ?
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3}
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
The following 325 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000320The dinv adjustment of an integer partition. St000011The number of touch points (or returns) of a Dyck path. St000507The number of ascents of a standard tableau. St000157The number of descents of a standard tableau. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St000098The chromatic number of a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000676The number of odd rises of a Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000306The bounce count of a Dyck path. St001280The number of parts of an integer partition that are at least two. St000925The number of topologically connected components of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St000053The number of valleys of the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000024The number of double up and double down steps of a Dyck path. 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. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000172The Grundy number of a graph. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001029The size of the core of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000167The number of leaves of an ordered tree. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000912The number of maximal antichains in a poset. St000672The number of minimal elements in Bruhat order not less than the permutation. St000528The height of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000164The number of short pairs. St001083The number of boxed occurrences of 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000527The width of the poset. St000662The staircase size of the code of a permutation. St000386The number of factors DDU in a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000052The number of valleys of a Dyck path not on the x-axis. St001732The number of peaks visible from the left. St001461The number of topologically connected components of the chord diagram of a permutation. St000141The maximum drop size of a permutation. St000069The number of maximal elements of a poset. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000546The number of global descents of a permutation. St000632The jump number of the poset. St000007The number of saliances of the permutation. St000354The number of recoils of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000702The number of weak deficiencies of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000312The number of leaves in a graph. St000619The number of cyclic descents of a permutation. St000646The number of big ascents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000201The number of leaf nodes in a binary tree. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. 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 $eA$. St000153The number of adjacent cycles of a permutation. St000308The height of the tree associated to a permutation. St000843The decomposition number of a perfect matching. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000661The number of rises of length 3 of a Dyck path. St000884The number of isolated descents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001427The number of descents of a signed permutation. St000647The number of big descents of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000065The number of entries equal to -1 in an alternating sign matrix. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000482The (zero)-forcing number of a graph. St000778The metric dimension of a graph. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001480The number of simple summands of the module J^2/J^3. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001874Lusztig's a-function for the symmetric group. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000171The degree of the graph. St000237The number of small exceedances. St000328The maximum number of child nodes in a tree. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000389The number of runs of ones of odd length in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000636The hull number of a graph. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000742The number of big ascents of a permutation after prepending zero. St000822The Hadwiger number of the graph. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001530The depth of a Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001883The mutual visibility number of a graph. St000094The depth of an ordered tree. 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. St001674The number of vertices of the largest induced star graph in the graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000758The length of the longest staircase fitting into an integer composition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000761The number of ascents in an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001812The biclique partition number of a graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000260The radius of a connected graph. St000353The number of inner valleys of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000659The number of rises of length at least 2 of a Dyck path. St000862The number of parts of the shifted shape of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000023The number of inner peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000710The number of big deficiencies of a permutation. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001469The holeyness of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000264The girth of a graph, which is not a tree. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000451The length of the longest pattern of the form k 1 2. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000388The number of orbits of vertices of a graph under automorphisms. St000744The length of the path to the largest entry in a standard Young tableau. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000317The cycle descent number of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000358The number of occurrences of the pattern 31-2. St000654The first descent of a permutation. St000732The number of double deficiencies of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001569The maximal modular displacement of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001665The number of pure excedances of a permutation. St001727The number of invisible inversions of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000365The number of double ascents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001520The number of strict 3-descents. St001578The minimal number of edges to add or remove to make a graph a line graph. St001589The nesting number of a perfect matching. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001728The number of invisible descents of a permutation. St001948The number of augmented double ascents of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000640The rank of the largest boolean interval in a poset. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001964The interval resolution global dimension of a poset. St000307The number of rowmotion orbits of a poset. St000872The number of very big descents of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001399The distinguishing number of a poset. St000824The sum of the number of descents and the number of recoils of a permutation. St001093The detour number of a graph. St001470The cyclic holeyness of a permutation. St001871The number of triconnected components of a graph. St000633The size of the automorphism group of a poset. St001354The number of series nodes in the modular decomposition of a graph. St000850The number of 1/2-balanced pairs in a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001863The number of weak excedances of a signed permutation. St001487The number of inner corners of a skew partition. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice.