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Your data matches 114 different statistics following compositions of up to 3 maps.
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Matching statistic: St000480
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,1] => [1,1]
=> 0
[2] => [2]
=> 1
[1,1,1] => [1,1,1]
=> 0
[1,2] => [2,1]
=> 1
[2,1] => [2,1]
=> 1
[3] => [3]
=> 1
[1,1,1,1] => [1,1,1,1]
=> 0
[1,1,2] => [2,1,1]
=> 1
[1,2,1] => [2,1,1]
=> 1
[1,3] => [3,1]
=> 1
[2,1,1] => [2,1,1]
=> 1
[2,2] => [2,2]
=> 1
[3,1] => [3,1]
=> 1
[4] => [4]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,1,1,2] => [2,1,1,1]
=> 1
[1,1,2,1] => [2,1,1,1]
=> 1
[1,1,3] => [3,1,1]
=> 1
[1,2,1,1] => [2,1,1,1]
=> 1
[1,2,2] => [2,2,1]
=> 1
[1,3,1] => [3,1,1]
=> 1
[1,4] => [4,1]
=> 1
[2,1,1,1] => [2,1,1,1]
=> 1
[2,1,2] => [2,2,1]
=> 1
[2,2,1] => [2,2,1]
=> 1
[2,3] => [3,2]
=> 1
[3,1,1] => [3,1,1]
=> 1
[3,2] => [3,2]
=> 1
[4,1] => [4,1]
=> 1
[5] => [5]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
[1,1,1,1,2] => [2,1,1,1,1]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 1
[1,1,1,3] => [3,1,1,1]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> 1
[1,1,2,2] => [2,2,1,1]
=> 1
[1,1,3,1] => [3,1,1,1]
=> 1
[1,1,4] => [4,1,1]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> 1
[1,2,1,2] => [2,2,1,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> 1
[1,2,3] => [3,2,1]
=> 2
[1,3,1,1] => [3,1,1,1]
=> 1
[1,3,2] => [3,2,1]
=> 2
[1,4,1] => [4,1,1]
=> 1
[1,5] => [5,1]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 1
[2,1,1,2] => [2,2,1,1]
=> 1
[2,1,2,1] => [2,2,1,1]
=> 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000481
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 0
[1,1] => [1,1]
=> [2]
=> 0
[2] => [2]
=> [1,1]
=> 1
[1,1,1] => [1,1,1]
=> [3]
=> 0
[1,2] => [2,1]
=> [2,1]
=> 1
[2,1] => [2,1]
=> [2,1]
=> 1
[3] => [3]
=> [1,1,1]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [4]
=> 0
[1,1,2] => [2,1,1]
=> [3,1]
=> 1
[1,2,1] => [2,1,1]
=> [3,1]
=> 1
[1,3] => [3,1]
=> [2,1,1]
=> 1
[2,1,1] => [2,1,1]
=> [3,1]
=> 1
[2,2] => [2,2]
=> [2,2]
=> 1
[3,1] => [3,1]
=> [2,1,1]
=> 1
[4] => [4]
=> [1,1,1,1]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 0
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,1,3] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[1,2,2] => [2,2,1]
=> [3,2]
=> 1
[1,3,1] => [3,1,1]
=> [3,1,1]
=> 1
[1,4] => [4,1]
=> [2,1,1,1]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[2,1,2] => [2,2,1]
=> [3,2]
=> 1
[2,2,1] => [2,2,1]
=> [3,2]
=> 1
[2,3] => [3,2]
=> [2,2,1]
=> 1
[3,1,1] => [3,1,1]
=> [3,1,1]
=> 1
[3,2] => [3,2]
=> [2,2,1]
=> 1
[4,1] => [4,1]
=> [2,1,1,1]
=> 1
[5] => [5]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> 0
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> 1
[1,2,3] => [3,2,1]
=> [3,2,1]
=> 2
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> 1
[1,3,2] => [3,2,1]
=> [3,2,1]
=> 2
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St001188
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St001188: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St001188: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2] => [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1}
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1}
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,6,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,7] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path.
Also the number of simple modules that are isolated vertices in the sense of 4.5. (4) in the reference.
Matching statistic: St001244
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St001244: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St001244: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2] => [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1}
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1}
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,6,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,7] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ?
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path.
For the projective dimension see [[St001007]] and for 1-regularity see [[St001126]]. After applying the inverse zeta map [[Mp00032]], this statistic matches the number of rises of length at least 2 [[St000659]].
Matching statistic: St000260
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 50%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1,1] => ([(0,1)],2)
=> 1
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1}
[2,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1}
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1}
[1,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1}
[1,3] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1}
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,2] => [2] => [1,1] => ([(0,1)],2)
=> 1
[3,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1}
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[1,1,2,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[1,1,3] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[1,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[1,4] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[2,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[2,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[2,3] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[4,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1}
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,1,2] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,3] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,3,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,4] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,1,2] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,2,2] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,3] => [2] => [1,1] => ([(0,1)],2)
=> 1
[4,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[4,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[5,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,2,2] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,3,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,4] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1,1,1] => [2,1,3] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,1,2] => [2,1,1,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,3] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,3,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,3,2] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,5] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1,1,1] => [1,1,4] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,1,1,2] => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,2,1] => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,3] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,2,2] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,4,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,1,1,1,1] => [1,5] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,1,2,1,1] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,2,2] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,2,1,1,1] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[4,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[7] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001044
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001044: Perfect matchings ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 75%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001044: Perfect matchings ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 1 = 0 + 1
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1 = 0 + 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2 = 1 + 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2 = 1 + 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2 = 1 + 1
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 0 + 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2 = 1 + 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2 = 1 + 1
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2 = 1 + 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2 = 1 + 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? ∊ {0,1} + 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2 = 1 + 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2 = 1 + 1
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 2 = 1 + 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2 = 1 + 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 2 = 1 + 1
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 2 = 1 + 1
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 2 = 1 + 1
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2 = 1 + 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 2 = 1 + 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 2 = 1 + 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 2 = 1 + 1
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 2 = 1 + 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 2 = 1 + 1
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 2 = 1 + 1
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? ∊ {0,1} + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? ∊ {0,1,2,2,2,2,2,2,2} + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? ∊ {0,1,2,2,2,2,2,2,2} + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? ∊ {0,1,2,2,2,2,2,2,2} + 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? ∊ {0,1,2,2,2,2,2,2,2} + 1
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 2 = 1 + 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 2 = 1 + 1
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 2 = 1 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? ∊ {0,1,2,2,2,2,2,2,2} + 1
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 2 = 1 + 1
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 2 = 1 + 1
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2 = 1 + 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 2 = 1 + 1
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2 = 1 + 1
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 2 = 1 + 1
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? ∊ {0,1,2,2,2,2,2,2,2} + 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? ∊ {0,1,2,2,2,2,2,2,2} + 1
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 2 = 1 + 1
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 2 = 1 + 1
[2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2 = 1 + 1
[2,2,1,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 2 = 1 + 1
[2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 2 = 1 + 1
[2,3,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2 = 1 + 1
[2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> 2 = 1 + 1
[3,1,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 2 = 1 + 1
[3,1,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2 = 1 + 1
[3,2,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2 = 1 + 1
[3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 3 = 2 + 1
[5,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> ? ∊ {0,1,2,2,2,2,2,2,2} + 1
[6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? ∊ {0,1,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,3] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,2] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,3,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,1,2] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,2,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,3,1,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,5] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,2,1,1,2] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,2,1,2,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,2,2,1,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,3,1,1,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,5,1] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[2,1,1,1,2] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[2,1,1,2,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[2,1,2,1,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[2,2,1,1,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[2,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[3,1,1,1,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[5,1,1] => [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[5,2] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,2),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,4] => [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
Description
The number of pairs whose larger element is at most one more than half the size of the perfect matching.
Under the bijection between perfect matchings of $\{1,\dots,2n\}$ and rooted unordered binary trees with $n+1$ labelled leaves described in Example 5.2.6 of [1], this is the number of nodes having two leaves as children.
The number of perfect matchings of $\{1,\dots,2n\}$ with $j$ such pairs, computed in [2], is
$$
\frac{(2j-3)!! (n+1)!}{2^j j!}\binom{n-1}{2j-2}.
$$
Matching statistic: St001165
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,3] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,3,1] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,4] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,3,1,1] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,4,1] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,3,1,1,1] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,4,1,1] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,5,1] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[3,1,1,1,1] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[4,1,1,1] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[5,1,1] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[6,1] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[7] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2} + 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,4] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,1,2] => [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,3] => [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,3,2] => [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,4,1] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,5] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,1,3] => [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,3,1] => [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,2,4] => [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,3,1,2] => [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,3,2,1] => [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,4,1,1] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,4,2] => [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2} + 1
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001568
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Values
[1] => [1,0]
=> [1,0]
=> []
=> ? = 0
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1}
[2] => [1,1,0,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1}
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1}
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1}
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,1}
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1}
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {0,1}
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1}
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? ∊ {0,2,2,2,2,2,2,2,2}
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> ? ∊ {0,2,2,2,2,2,2,2,2}
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> ? ∊ {0,2,2,2,2,2,2,2,2}
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {0,2,2,2,2,2,2,2,2}
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {0,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? ∊ {0,2,2,2,2,2,2,2,2}
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> ? ∊ {0,2,2,2,2,2,2,2,2}
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? ∊ {0,2,2,2,2,2,2,2,2}
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1]
=> 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1]
=> 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,2,1]
=> 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,2,1]
=> 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,2,1]
=> 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [6,4,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [6,5,1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [6,3,2]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [5,4,2]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [6,4,2]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [5,4,3]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,1,1] => [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]
=> [6,5,4]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001630
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 0
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,1}
[2] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1}
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1}
[3] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[2,2] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[4] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,2,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,4] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 0
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,1}
[2] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1}
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1}
[3] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[2,2] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[4] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,2,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[5] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,4] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
The following 104 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001890The maximum magnitude of the Möbius function of a poset. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000834The number of right outer peaks of a permutation. St000862The number of parts of the shifted shape of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001471The magnitude of a Dyck path. St000035The number of left outer peaks of a permutation. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001981The size of the largest square of zeros in the top left corner of an alternating sign matrix. St001877Number of indecomposable injective modules with projective dimension 2. St001333The cardinality of a minimal edge-isolating set of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000526The number of posets with combinatorially isomorphic order polytopes. St000647The number of big descents of a permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001393The induced matching number of a graph. St000662The staircase size of the code of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St000286The number of connected components of the complement of a graph. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001665The number of pure excedances of a permutation. St000486The number of cycles of length at least 3 of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000646The number of big ascents of a permutation. St000741The Colin de Verdière graph invariant. St001737The number of descents of type 2 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000456The monochromatic index of a connected graph. St001734The lettericity of a graph. St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000779The tier of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000314The number of left-to-right-maxima of a permutation. St000638The number of up-down runs of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001589The nesting number of a perfect matching. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001555The order of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral.
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