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Your data matches 151 different statistics following compositions of up to 3 maps.
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Matching statistic: St000384
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 3
[2,1,1]
=> 3
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 3
[3,1,1]
=> 3
[2,2,1]
=> 3
[2,1,1,1]
=> 4
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 5
[4,2]
=> 4
[4,1,1]
=> 4
[3,3]
=> 4
[3,2,1]
=> 3
[3,1,1,1]
=> 4
[2,2,2]
=> 4
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 5
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 6
[5,2]
=> 5
[5,1,1]
=> 5
[4,3]
=> 4
[4,2,1]
=> 4
[4,1,1,1]
=> 4
[3,3,1]
=> 4
[3,2,2]
=> 4
[3,2,1,1]
=> 4
[3,1,1,1,1]
=> 5
[2,2,2,1]
=> 4
[2,2,1,1,1]
=> 5
[2,1,1,1,1,1]
=> 6
[1,1,1,1,1,1,1]
=> 7
[8]
=> 8
[7,1]
=> 7
[6,2]
=> 6
[6,1,1]
=> 6
[5,3]
=> 5
[5,2,1]
=> 5
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St001176
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 96%●distinct values known / distinct values provided: 85%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 96%●distinct values known / distinct values provided: 85%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 3 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1 = 2 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 3 = 4 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 3 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3 = 4 - 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]
=> 4 = 5 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 3 = 4 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2 = 3 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 3 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2 = 3 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 3 = 4 - 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]
=> 4 = 5 - 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]
=> 5 = 6 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [2,1,1,1,1]
=> 4 = 5 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 3 = 4 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 3 = 4 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 3 = 4 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> 3 = 4 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 3 = 4 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 3 = 4 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [2,1,1,1,1]
=> 4 = 5 - 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]
=> 5 = 6 - 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]
=> 6 = 7 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [2,1,1,1,1,1]
=> 5 = 6 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [2,2,1,1]
=> 4 = 5 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [2,1,1,1,1]
=> 4 = 5 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 3 = 4 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> 3 = 4 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 3 = 4 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 3 = 4 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [2,1,1,1,1]
=> 4 = 5 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 3 = 4 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [2,1,1,1,1]
=> 4 = 5 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [2,1,1,1,1,1]
=> 5 = 6 - 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]
=> 6 = 7 - 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]
=> 7 = 8 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [2,1,1,1,1,1,1]
=> 6 = 7 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [2,2,1,1,1]
=> 5 = 6 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [2,1,1,1,1,1]
=> 5 = 6 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [2,2,1,1]
=> 4 = 5 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [2,2,1,1]
=> 4 = 5 - 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]
=> ? => ?
=> ? = 11 - 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]
=> ? => ?
=> ? = 12 - 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]
=> [11,1,2,3,4,5,6,7,8,9,10,12] => ?
=> ? = 11 - 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]
=> [9,1,2,3,4,5,6,7,8,10,11] => ?
=> ? = 10 - 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ?
=> ? = 8 - 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]
=> [2,3,4,5,6,1,7,9,8] => ?
=> ? = 8 - 1
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ?
=> ? = 7 - 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]
=> [2,3,4,5,6,7,8,9,1,10,11] => ?
=> ? = 10 - 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]
=> [2,3,4,5,6,7,8,9,10,11,1,12] => ?
=> ? = 11 - 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]
=> ? => ?
=> ? = 12 - 1
[]
=> []
=> [] => []
=> ? = 0 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 95%●distinct values known / distinct values provided: 69%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 95%●distinct values known / distinct values provided: 69%
Values
[1]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 4 = 3 + 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]
=> 4 = 3 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 4 = 3 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 4 = 3 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 4 = 3 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 5 = 4 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [5,1]
=> 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 5 = 4 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 4 = 3 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 4 = 3 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 4 = 3 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 5 = 4 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [5,1]
=> 6 = 5 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [6,1]
=> 7 = 6 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [4,1,1]
=> 6 = 5 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 5 = 4 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 5 = 4 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 5 = 4 + 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]
=> 5 = 4 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 5 = 4 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 5 = 4 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [4,1,1]
=> 6 = 5 + 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]
=> 7 = 6 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [7,1]
=> 8 = 7 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => [5,1,1]
=> 7 = 6 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => [4,1,1]
=> 6 = 5 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => [3,2,1]
=> 6 = 5 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 5 = 4 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 5 = 4 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 5 = 4 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 5 = 4 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 5 = 4 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 5 = 4 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [4,1,1]
=> 6 = 5 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 5 = 4 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [3,2,1]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 8 = 7 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [8,1]
=> 9 = 8 + 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]
=> 8 = 7 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => [5,1,1]
=> 7 = 6 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => [4,2,1]
=> 7 = 6 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => [4,1,1]
=> 6 = 5 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => [3,1,1,1]
=> 6 = 5 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [10,1]
=> ? = 10 + 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]
=> [1,10] => [10,1]
=> ? = 10 + 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]
=> [11,1] => [11,1]
=> ? = 11 + 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] => ?
=> ? = 10 + 1
[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]
=> [1,9,1] => [9,1,1]
=> ? = 10 + 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]
=> ? => ?
=> ? = 11 + 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]
=> ? => ?
=> ? = 12 + 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] => ?
=> ? = 11 + 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] => ?
=> ? = 10 + 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] => ?
=> ? = 10 + 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]
=> [1,9,1] => [9,1,1]
=> ? = 10 + 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]
=> [1,8,2] => [8,2,1]
=> ? = 10 + 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]
=> [1,10,1] => [10,1,1]
=> ? = 11 + 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]
=> ? => ?
=> ? = 12 + 1
[]
=> []
=> [] => ?
=> ? = 0 + 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001382
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 92%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 92%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 3
[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]
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 4
[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]
=> 5
[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]
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [3,1,1,1]
=> 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 4
[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]
=> 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 4
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 4
[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]
=> 5
[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]
=> 6
[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]
=> 7
[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]
=> 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [3,1,1,1]
=> 5
[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]
=> 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> 4
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 4
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 4
[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]
=> 5
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 4
[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]
=> 5
[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]
=> 6
[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]
=> 7
[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]
=> 8
[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]
=> 7
[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]
=> 6
[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]
=> 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,1,1,1]
=> 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [4,1,1]
=> 5
[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] => ?
=> ? = 8
[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] => ?
=> ? = 7
[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] => ?
=> ? = 8
[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] => ?
=> ? = 9
[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] => ?
=> ? = 8
[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] => ?
=> ? = 9
[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]
=> ? => ?
=> ? = 11
[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]
=> ? => ?
=> ? = 12
[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] => ?
=> ? = 10
[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] => ?
=> ? = 9
[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] => ?
=> ? = 8
[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] => ?
=> ? = 8
[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] => ?
=> ? = 8
[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] => ?
=> ? = 7
[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] => ?
=> ? = 7
[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] => ?
=> ? = 7
[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] => ?
=> ? = 8
[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] => ?
=> ? = 7
[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] => ?
=> ? = 8
[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] => ?
=> ? = 9
[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] => ?
=> ? = 8
[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] => ?
=> ? = 10
[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] => ?
=> ? = 8
[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] => ?
=> ? = 9
[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] => ?
=> ? = 10
[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] => ?
=> ? = 11
[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]
=> ? => ?
=> ? = 12
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Matching statistic: St000460
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 90%●distinct values known / distinct values provided: 85%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 90%●distinct values known / distinct values provided: 85%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 4 = 3 + 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]
=> 4 = 3 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 4 = 3 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 4 = 3 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 4 = 3 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 5 = 4 + 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]
=> 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> 5 = 4 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 4 = 3 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 4 = 3 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 4 = 3 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 5 = 4 + 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]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 6 = 5 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> 5 = 4 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> 5 = 4 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 5 = 4 + 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]
=> 5 = 4 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 5 = 4 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 5 = 4 + 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]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 8 = 7 + 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]
=> 7 = 6 + 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]
=> 6 = 5 + 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]
=> 6 = 5 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> 5 = 4 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> 5 = 4 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 5 = 4 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> 5 = 4 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 5 = 4 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 5 = 4 + 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]
=> 6 = 5 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 5 = 4 + 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]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 8 = 7 + 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]
=> 9 = 8 + 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]
=> 8 = 7 + 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]
=> 7 = 6 + 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]
=> 7 = 6 + 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]
=> 6 = 5 + 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]
=> 6 = 5 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 9 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 9 + 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]
=> ? => ?
=> ? = 11 + 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]
=> ? => ?
=> ? = 12 + 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] => ?
=> ? = 10 + 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] => ?
=> ? = 9 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 9 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 10 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 9 + 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] => ?
=> ? = 10 + 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] => ?
=> ? = 11 + 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]
=> ? => ?
=> ? = 12 + 1
[]
=> []
=> [] => []
=> ? = 0 + 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 90%●distinct values known / distinct values provided: 85%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 90%●distinct values known / distinct values provided: 85%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 4 = 3 + 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]
=> 4 = 3 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 4 = 3 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 4 = 3 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 4 = 3 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 5 = 4 + 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]
=> 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> 5 = 4 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 4 = 3 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 4 = 3 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 4 = 3 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 5 = 4 + 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]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 6 = 5 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> 5 = 4 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> 5 = 4 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 5 = 4 + 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]
=> 5 = 4 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 5 = 4 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 5 = 4 + 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]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 8 = 7 + 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]
=> 7 = 6 + 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]
=> 6 = 5 + 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]
=> 6 = 5 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> 5 = 4 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> 5 = 4 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 5 = 4 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> 5 = 4 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 5 = 4 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 5 = 4 + 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]
=> 6 = 5 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 5 = 4 + 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]
=> 6 = 5 + 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]
=> 7 = 6 + 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]
=> 8 = 7 + 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]
=> 9 = 8 + 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]
=> 8 = 7 + 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]
=> 7 = 6 + 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]
=> 7 = 6 + 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]
=> 6 = 5 + 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]
=> 6 = 5 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 9 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 9 + 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]
=> ? => ?
=> ? = 11 + 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]
=> ? => ?
=> ? = 12 + 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] => ?
=> ? = 10 + 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] => ?
=> ? = 9 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 7 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 9 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 10 + 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] => ?
=> ? = 8 + 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] => ?
=> ? = 9 + 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] => ?
=> ? = 10 + 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] => ?
=> ? = 11 + 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]
=> ? => ?
=> ? = 12 + 1
[]
=> []
=> [] => []
=> ? = 0 + 1
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001004
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 86%●distinct values known / distinct values provided: 77%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 86%●distinct values known / distinct values provided: 77%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1] => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,2] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,3] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1] => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,1] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,4] => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,1,2] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,2] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,3,1] => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,1] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1,2,3] => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,1,2] => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,3] => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,1] => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [5,1,2,3,4] => 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,3] => 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4] => 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,3] => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 4
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,1,6] => [2,3,4,5,1] => 5
[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] => [2,3,4,5,6,1] => 6
[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] => [1,2,3,4,5,6,7] => 7
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => [6,1,2,3,4,5] => 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => [5,1,2,3,4] => 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5,6] => [4,1,2,3,5] => 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,1,2,3] => 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 4
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,4,1,2] => 4
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,3,1,4] => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,5,6] => [2,3,4,1,5] => 5
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => [2,3,4,1,5] => 5
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [2,3,4,5,6,1] => 6
[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] => [2,3,4,5,6,7,1] => 7
[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] => [1,2,3,4,5,6,7,8] => 8
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => 7
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => [6,1,2,3,4,5] => 6
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7] => [5,1,2,3,4,6] => 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,4] => [5,1,2,3,4] => 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => [4,1,2,3,5] => 5
[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] => ? => ? = 11
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9,10] => ? => ? = 9
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,9,6,7] => ? => ? = 8
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,8,9] => ? => ? = 8
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,1,2,3,8,4,5,6] => ? => ? = 7
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => ? => ? = 7
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,8,7] => ? => ? = 7
[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]
=> [2,3,4,5,6,1,7,8,9] => ? => ? = 8
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,9,8] => ? => ? = 8
[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]
=> [2,3,4,5,6,7,8,1,9,10] => ? => ? = 9
[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]
=> [2,3,4,5,6,7,8,9,10,1,11] => ? => ? = 10
[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]
=> ? => ? => ? = 11
[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]
=> ? => ? => ? = 12
[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]
=> [11,1,2,3,4,5,6,7,8,9,10,12] => ? => ? = 11
[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]
=> [9,1,2,3,4,5,6,7,8,10,11] => ? => ? = 10
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,10,7,8] => ? => ? = 9
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,10,9] => ? => ? = 9
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8,9,10] => ? => ? = 9
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [8,1,2,3,4,9,5,6,7] => ? => ? = 8
[8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,9,6,8] => ? => ? = 8
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7,9,8] => ? => ? = 8
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6,7,8,9] => ? => ? = 8
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [6,1,2,3,8,4,5,7] => ? => ? = 7
[7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5,8] => ? => ? = 7
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,8,6,7] => ? => ? = 7
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,7,8,6] => ? => ? = 7
[6,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,3,1,4,5,6,7,8] => ? => ? = 7
[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]
=> [2,3,4,5,1,6,7,8,9] => ? => ? = 8
[4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,7,8,6] => ? => ? = 7
[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]
=> [2,3,4,5,6,1,7,9,8] => ? => ? = 8
[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]
=> [2,3,4,5,6,7,1,8,9,10] => ? => ? = 9
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6,8] => ? => ? = 7
[3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,9,1,7,8] => ? => ? = 8
[3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ? => ? = 7
[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]
=> [2,3,4,5,6,8,1,9,7] => ? => ? = 8
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1,10,9] => ? => ? = 9
[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]
=> [2,3,4,5,6,7,8,9,1,10,11] => ? => ? = 10
[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]
=> [2,3,4,5,6,7,8,9,10,11,1,12] => ? => ? = 11
[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]
=> ? => ? => ? = 12
[]
=> []
=> [] => ? => ? = 0
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St000393
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 85%●distinct values known / distinct values provided: 77%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 85%●distinct values known / distinct values provided: 77%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 0 => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 00 => 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 00 => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 000 => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 00 => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0000 => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 000 => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 000 => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 00000 => 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0000 => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 000 => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 000 => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 000 => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0000 => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 00000 => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 000000 => 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 00000 => 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0000 => 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 0000 => 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0000 => 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 000 => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 0000 => 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 4
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0000 => 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 00000 => 5
[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] => 000000 => 6
[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] => 0000000 => 7
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 000000 => 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 00000 => 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 00000 => 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0000 => 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0000 => 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0000 => 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0000 => 4
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0000 => 4
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0000 => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 00000 => 5
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0000 => 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 00000 => 5
[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] => 000000 => 6
[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] => 0000000 => 7
[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] => 00000000 => 8
[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] => 0000000 => 7
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => 000000 => 6
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => 000000 => 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 00000 => 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 00000 => 5
[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] => ? => ? = 11
[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] => ? => ? = 10
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => ? => ? = 9
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,1,4,5,6,7,8,9] => ? => ? = 9
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => ? => ? = 8
[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] => ? => ? = 8
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,1,5,6,7,8] => ? => ? = 8
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [8,4,2,1,3,5,6,7] => ? => ? = 7
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,1,2,5,6,7] => ? => ? = 7
[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] => ? => ? = 7
[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] => ? => ? = 8
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? => ? = 7
[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] => ? => ? = 8
[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] => ? => ? = 9
[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] => ? => ? = 10
[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]
=> ? => ? => ? = 11
[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]
=> ? => ? => ? = 12
[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] => ? => ? = 11
[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] => ? => ? = 10
[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] => ? => ? = 10
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [10,4,1,2,3,5,6,7,8,9] => ? => ? = 9
[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] => ? => ? = 9
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,1,5,6,7,8,9] => ? => ? = 9
[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] => ? => ? = 8
[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] => ? => ? = 8
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,3,4,1,2,5,6,7,8] => ? => ? = 8
[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] => ? => ? = 8
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,5,1,6,7,8] => ? => ? = 8
[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] => ? => ? = 7
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,2,1,5,6,7] => ? => ? = 7
[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] => ? => ? = 7
[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] => ? => ? = 7
[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] => ? => ? = 8
[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] => ? => ? = 8
[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] => ? => ? = 9
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,6,7,8,1] => ? => ? = 7
[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] => ? => ? = 8
[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] => ? => ? = 10
[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] => ? => ? = 8
[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] => ? => ? = 9
[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] => ? => ? = 10
[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] => ? => ? = 11
[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]
=> ? => ? => ? = 12
[]
=> []
=> [] => => ? = 0
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000627
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000627: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 85%●distinct values known / distinct values provided: 77%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000627: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 85%●distinct values known / distinct values provided: 77%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 0 => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 00 => 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 00 => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 000 => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 00 => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0000 => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 000 => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 000 => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 00000 => 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0000 => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 000 => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 000 => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 000 => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0000 => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 00000 => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 000000 => 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 00000 => 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0000 => 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 0000 => 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0000 => 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 000 => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 0000 => 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 4
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0000 => 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 00000 => 5
[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] => 000000 => 6
[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] => 0000000 => 7
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 000000 => 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 00000 => 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 00000 => 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0000 => 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0000 => 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0000 => 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0000 => 4
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0000 => 4
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0000 => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 00000 => 5
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0000 => 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 00000 => 5
[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] => 000000 => 6
[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] => 0000000 => 7
[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] => 00000000 => 8
[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] => 0000000 => 7
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => 000000 => 6
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => 000000 => 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 00000 => 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 00000 => 5
[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] => ? => ? = 11
[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] => ? => ? = 10
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => ? => ? = 9
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,1,4,5,6,7,8,9] => ? => ? = 9
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => ? => ? = 8
[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] => ? => ? = 8
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,1,5,6,7,8] => ? => ? = 8
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [8,4,2,1,3,5,6,7] => ? => ? = 7
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,1,2,5,6,7] => ? => ? = 7
[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] => ? => ? = 7
[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] => ? => ? = 8
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? => ? = 7
[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] => ? => ? = 8
[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] => ? => ? = 9
[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] => ? => ? = 10
[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]
=> ? => ? => ? = 11
[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]
=> ? => ? => ? = 12
[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] => ? => ? = 11
[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] => ? => ? = 10
[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] => ? => ? = 10
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [10,4,1,2,3,5,6,7,8,9] => ? => ? = 9
[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] => ? => ? = 9
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,1,5,6,7,8,9] => ? => ? = 9
[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] => ? => ? = 8
[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] => ? => ? = 8
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,3,4,1,2,5,6,7,8] => ? => ? = 8
[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] => ? => ? = 8
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,5,1,6,7,8] => ? => ? = 8
[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] => ? => ? = 7
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,2,1,5,6,7] => ? => ? = 7
[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] => ? => ? = 7
[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] => ? => ? = 7
[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] => ? => ? = 8
[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] => ? => ? = 8
[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] => ? => ? = 9
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,6,7,8,1] => ? => ? = 7
[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] => ? => ? = 8
[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] => ? => ? = 10
[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] => ? => ? = 8
[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] => ? => ? = 9
[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] => ? => ? = 10
[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] => ? => ? = 11
[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]
=> ? => ? => ? = 12
[]
=> []
=> [] => => ? = 0
Description
The exponent of a binary word.
This is the largest number $e$ such that $w$ is the concatenation of $e$ identical factors. This statistic is also called '''frequency'''.
Matching statistic: St000826
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000826: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 85%●distinct values known / distinct values provided: 77%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000826: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 85%●distinct values known / distinct values provided: 77%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 0 => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 00 => 2
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 00 => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 000 => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 00 => 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0000 => 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 000 => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 000 => 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 00000 => 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0000 => 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 000 => 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 000 => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 000 => 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0000 => 4
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 00000 => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 000000 => 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 00000 => 5
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0000 => 4
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 0000 => 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0000 => 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 000 => 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 0000 => 4
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 4
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0000 => 4
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 00000 => 5
[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] => 000000 => 6
[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] => 0000000 => 7
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 000000 => 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 00000 => 5
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 00000 => 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0000 => 4
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0000 => 4
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 0000 => 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 0000 => 4
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0000 => 4
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0000 => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 00000 => 5
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0000 => 4
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 00000 => 5
[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] => 000000 => 6
[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] => 0000000 => 7
[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] => 00000000 => 8
[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] => 0000000 => 7
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => 000000 => 6
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => 000000 => 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 00000 => 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 00000 => 5
[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] => ? => ? = 11
[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] => ? => ? = 10
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => ? => ? = 9
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,1,4,5,6,7,8,9] => ? => ? = 9
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => ? => ? = 8
[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] => ? => ? = 8
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,1,5,6,7,8] => ? => ? = 8
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [8,4,2,1,3,5,6,7] => ? => ? = 7
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,1,2,5,6,7] => ? => ? = 7
[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] => ? => ? = 7
[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] => ? => ? = 8
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => ? => ? = 7
[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] => ? => ? = 8
[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] => ? => ? = 9
[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] => ? => ? = 10
[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]
=> ? => ? => ? = 11
[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]
=> ? => ? => ? = 12
[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] => ? => ? = 11
[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] => ? => ? = 10
[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] => ? => ? = 10
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [10,4,1,2,3,5,6,7,8,9] => ? => ? = 9
[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] => ? => ? = 9
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,4,1,5,6,7,8,9] => ? => ? = 9
[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] => ? => ? = 8
[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] => ? => ? = 8
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,3,4,1,2,5,6,7,8] => ? => ? = 8
[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] => ? => ? = 8
[8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,5,1,6,7,8] => ? => ? = 8
[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] => ? => ? = 7
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,2,1,5,6,7] => ? => ? = 7
[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] => ? => ? = 7
[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] => ? => ? = 7
[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] => ? => ? = 8
[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] => ? => ? = 8
[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] => ? => ? = 9
[3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,3,2,6,7,8,1] => ? => ? = 7
[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] => ? => ? = 8
[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] => ? => ? = 10
[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] => ? => ? = 8
[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] => ? => ? = 9
[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] => ? => ? = 10
[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] => ? => ? = 11
[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]
=> ? => ? => ? = 12
[]
=> []
=> [] => => ? = 0
Description
The stopping time of the decimal representation of the binary word for the 3x+1 problem.
Prepend $1$ to the binary word, interpret it as a positive number, and count the number of halving and tripling steps needed to reach $1$.
The following 141 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000518The number of distinct subsequences in a binary word. St000519The largest length of a factor maximising the subword complexity. St000296The length of the symmetric border of a binary word. St001267The length of the Lyndon factorization of the binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001437The flex of a binary word. St001884The number of borders of a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000081The number of edges of a graph. St000553The number of blocks of a graph. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001479The number of bridges of a graph. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000026The position of the first return of a Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001746The coalition number of a graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000019The cardinality of the support of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001917The order of toric promotion on the set of labellings of a graph. St000459The hook length of the base cell of a partition. St001622The number of join-irreducible elements of a lattice. St001268The size of the largest ordinal summand in the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000656The number of cuts of a poset. St000653The last descent of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001725The harmonious chromatic number of a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001430The number of positive entries in a signed permutation. St000147The largest part of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000503The maximal difference between two elements in a common block. 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. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St000501The size of the first part in the decomposition of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000209Maximum difference of elements in cycles. St000316The number of non-left-to-right-maxima of a permutation. St000956The maximal displacement of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001480The number of simple summands of the module J^2/J^3. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000354The number of recoils of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000829The Ulam distance of a permutation to the identity permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000167The number of leaves of an ordered tree. St000451The length of the longest pattern of the form k 1 2. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001925The minimal number of zeros in a row of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000080The rank of the poset. St000863The length of the first row of the shifted shape of a permutation. St000924The number of topologically connected components of a perfect matching. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000093The cardinality of a maximal independent set of vertices of a graph. St000050The depth or height of a binary tree. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000673The number of non-fixed points of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000327The number of cover relations in a poset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000197The number of entries equal to positive one in the alternating sign matrix. St000216The absolute length 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). St001077The prefix exchange distance of a permutation. St001668The number of points of the poset minus the width of the poset. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001468The smallest fixpoint of a permutation. St001637The number of (upper) dissectors of a poset. St001090The number of pop-stack-sorts needed to sort a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001720The minimal length of a chain of small intervals in a lattice.
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