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Your data matches 87 different statistics following compositions of up to 3 maps.
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Matching statistic: St000369
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St000369: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> 2
Description
The dinv deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$
In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$.
See also [[St000376]] for the bounce deficit.
Matching statistic: St000376
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000376: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> 0
Description
The bounce deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).$$
The zeta map [[Mp00032]] sends this statistic to the dinv deficit [[St000369]], both are thus equidistributed.
Matching statistic: St001176
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1]
=> [1]
=> 0
[1,1,0,0]
=> []
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> [3]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> [2]
=> 0
[1,1,0,1,0,0]
=> [1]
=> [1]
=> [1]
=> 0
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,2,1]
=> [3,2]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,1,1,1]
=> [5]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2,1,1]
=> [3,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,1,1]
=> [4]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> [3]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [2,1]
=> [2,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> [2]
=> 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> [1]
=> 0
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [2,2,2,1,1,1]
=> [6,3]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1,1,1,1,1]
=> [7,2]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [3,2,2,1]
=> [4,3,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [3,2,2]
=> [3,3,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,2,1,1,1]
=> [5,2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [3,1,1,1,1,1]
=> [6,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [2,2,1,1,1,1]
=> [6,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,1,1,1,1,1,1]
=> [8]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,2,1]
=> [3,2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [2,2,1,1]
=> [4,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [3]
=> [1,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [2,2,1,1,1]
=> [5,2]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3,1,1,1]
=> [4,1,1]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,1,1,1,1]
=> [5,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,1,1,1]
=> [5]
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,8,8,8,9}
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: St000377
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
St000377: Integer partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
Values
[1,0,1,0]
=> [1]
=> 0
[1,1,0,0]
=> []
=> 0
[1,0,1,0,1,0]
=> [2,1]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1,0,0,1,0]
=> [2]
=> 0
[1,1,0,1,0,0]
=> [1]
=> 0
[1,1,1,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 0
[1,1,1,0,1,0,0,0]
=> [1]
=> 0
[1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10}
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000770
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 82%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 82%
Values
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1}
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 5
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 4
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 5
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 5
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 6
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,3] => [[5,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3] => [[5,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 3
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,2] => [[5,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => [[5,5],[4]]
=> [4]
=> 4
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St001175
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 82%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 82%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 3
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 3
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 5
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 3
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 3
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 5
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 3
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 3
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 3
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 3
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 3
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 3
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 3
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 5
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> 4
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 5
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 5
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> 4
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001498
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 36%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 36%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,0}
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,1}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 4
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000474
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 36%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 36%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 4
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 4
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 4
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 4
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> 4
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 3
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> 3
Description
Dyson's crank of a partition.
Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as
$$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
Matching statistic: St000698
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 45%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 45%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 5
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 5
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 4
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 4
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 5
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 6
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> 5
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 5
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 5
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 5
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> 5
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000502
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 55%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 55%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> {{1}}
=> ? ∊ {0,0}
[1,1,0,0]
=> []
=> []
=> {}
=> ? ∊ {0,0}
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> ? ∊ {0,0}
[1,1,1,0,0,0]
=> []
=> []
=> {}
=> ? ∊ {0,0}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> ? ∊ {0,1}
[1,1,1,1,0,0,0,0]
=> []
=> []
=> {}
=> ? ∊ {0,1}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> {{1,3,6,10},{2,5,9},{4,8},{7}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> {{1,3,6},{2,5,9},{4,8},{7}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> {{1,3,8,9},{2,5},{4,7},{6}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> {{1,4,5,9},{2,7,8},{3},{6}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> {{1,4,5},{2,7,8},{3},{6}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> {{1,2,5,9},{3,4,8},{6,7}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> {{1,2,7},{3,4},{5,6}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> {{1,3,4},{2,6,7},{5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> 4
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 4
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [[1]]
=> {{1}}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> {}
=> ? ∊ {1,1,1,2,2,2,2,2,3,3,3,3,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> {{1,3,6,10,15},{2,5,9,14},{4,8,13},{7,12},{11}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> {{1,3,6,10},{2,5,9,14},{4,8,13},{7,12},{11}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> {{1,3,6,13,14},{2,5,9},{4,8,12},{7,11},{10}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> {{1,3,6,13},{2,5,9},{4,8,12},{7,11},{10}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> {{1,3,6},{2,5,9},{4,8,12},{7,11},{10}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> {{1,3,8,9,14},{2,5,12,13},{4,7},{6,11},{10}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> {{1,3,8,9},{2,5,12,13},{4,7},{6,11},{10}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> {{1,3,8,12,13},{2,5,11},{4,7},{6,10},{9}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> {{1,3,8,12},{2,5,11},{4,7},{6,10},{9}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> {{1,3,8},{2,5,11},{4,7},{6,10},{9}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> {{1,3,10,11,12},{2,5},{4,7},{6,9},{8}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> {{1,3,10,11},{2,5},{4,7},{6,9},{8}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> {{1,3,10},{2,5},{4,7},{6,9},{8}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> {{1,3},{2,5},{4,7},{6,9},{8}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> {{1,4,5,9,14},{2,7,8,13},{3,11,12},{6},{10}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> {{1,4,5,9},{2,7,8,13},{3,11,12},{6},{10}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> {{1,4,5,12,13},{2,7,8},{3,10,11},{6},{9}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> {{1,4,5,12},{2,7,8},{3,10,11},{6},{9}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> {{1,4,5},{2,7,8},{3,10,11},{6},{9}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> {{1,4,7,8,13},{2,6,11,12},{3,10},{5},{9}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> {{1,4,7,8},{2,6,11,12},{3,10},{5},{9}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> {{1,4,7,11,12},{2,6,10},{3,9},{5},{8}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> {{1,4,7,11},{2,6,10},{3,9},{5},{8}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> {{1,4,7},{2,6,10},{3,9},{5},{8}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> {{1,4,9,10,11},{2,6},{3,8},{5},{7}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> {{1,4,9,10},{2,6},{3,8},{5},{7}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> {{1,4,9},{2,6},{3,8},{5},{7}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> {{1,5,6,7,12},{2,9,10,11},{3},{4},{8}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> {{1,5,6,7},{2,9,10,11},{3},{4},{8}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> {{1,5,6,10,11},{2,8,9},{3},{4},{7}}
=> ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> {{1,5},{2,7},{3},{4},{6}}
=> 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> {{1,6,7},{2},{3},{4},{5}}
=> 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> 0
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
The following 77 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001438The number of missing boxes of a skew partition. St000317The cycle descent number of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St000454The largest eigenvalue of a graph if it is integral. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000909The number of maximal chains of maximal size in a poset. St001435The number of missing boxes in the first row. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001423The number of distinct cubes in a binary word. St001556The number of inversions of the third entry of a permutation. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001902The number of potential covers of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001510The number of self-evacuating linear extensions of a finite poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000422The energy of a graph, if it is integral. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001964The interval resolution global dimension of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001060The distinguishing index of a graph. St001875The number of simple modules with projective dimension at most 1. St001644The dimension of a graph. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001488The number of corners of a skew partition. St000264The girth of a graph, which is not a tree. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000516The number of stretching pairs of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001330The hat guessing number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001487The number of inner corners of a skew partition. St000091The descent variation of a composition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000359The number of occurrences of the pattern 23-1. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000534The number of 2-rises of a permutation. St000731The number of double exceedences of a permutation. St000884The number of isolated descents of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001569The maximal modular displacement of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001705The number of occurrences of the pattern 2413 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000764The number of strong records in an integer composition.
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