searching the database
Your data matches 46 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000761
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000761: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,1] => 0 = 1 - 1
[2] => 0 = 1 - 1
[1,1,1] => 0 = 1 - 1
[1,2] => 1 = 2 - 1
[2,1] => 0 = 1 - 1
[3] => 0 = 1 - 1
[1,1,1,1] => 0 = 1 - 1
[1,1,2] => 1 = 2 - 1
[1,2,1] => 1 = 2 - 1
[1,3] => 1 = 2 - 1
[2,1,1] => 0 = 1 - 1
[2,2] => 0 = 1 - 1
[3,1] => 0 = 1 - 1
[4] => 0 = 1 - 1
[1,1,1,1,1] => 0 = 1 - 1
[1,1,1,2] => 1 = 2 - 1
[1,1,2,1] => 1 = 2 - 1
[1,1,3] => 1 = 2 - 1
[1,2,1,1] => 1 = 2 - 1
[1,2,2] => 1 = 2 - 1
[1,3,1] => 1 = 2 - 1
[1,4] => 1 = 2 - 1
[2,1,1,1] => 0 = 1 - 1
[2,1,2] => 1 = 2 - 1
[2,2,1] => 0 = 1 - 1
[2,3] => 1 = 2 - 1
[3,1,1] => 0 = 1 - 1
[3,2] => 0 = 1 - 1
[4,1] => 0 = 1 - 1
[5] => 0 = 1 - 1
[1,1,1,1,1,1] => 0 = 1 - 1
[1,1,1,1,2] => 1 = 2 - 1
[1,1,1,2,1] => 1 = 2 - 1
[1,1,1,3] => 1 = 2 - 1
[1,1,2,1,1] => 1 = 2 - 1
[1,1,2,2] => 1 = 2 - 1
[1,1,3,1] => 1 = 2 - 1
[1,1,4] => 1 = 2 - 1
[1,2,1,1,1] => 1 = 2 - 1
[1,2,1,2] => 2 = 3 - 1
[1,2,2,1] => 1 = 2 - 1
[1,2,3] => 2 = 3 - 1
[1,3,1,1] => 1 = 2 - 1
[1,3,2] => 1 = 2 - 1
[1,4,1] => 1 = 2 - 1
[1,5] => 1 = 2 - 1
[2,1,1,1,1] => 0 = 1 - 1
[2,1,1,2] => 1 = 2 - 1
[2,1,2,1] => 1 = 2 - 1
Description
The number of ascents in an integer composition.
A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.
Matching statistic: St001732
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [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
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
Description
The number of peaks visible from the left.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St000318
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 1
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {1,1}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {1,1}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> 1
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {1,1,2}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> 1
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> 1
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> 1
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 2
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> 1
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> 1
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> 1
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> 1
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 3
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> 2
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 2
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> 1
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> 1
[6] => [[6],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> []
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,1,5] => [[5,1,1],[]]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,3}
[1,6] => [[6,1],[]]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,3}
[7] => [[7],[]]
=> []
=> ?
=> ? ∊ {2,2,2,2,2,2,3}
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000668
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> []
=> ? = 1
[1,1] => [1,1]
=> [2]
=> []
=> ? ∊ {1,1}
[2] => [2]
=> [1,1]
=> [1]
=> ? ∊ {1,1}
[1,1,1] => [1,1,1]
=> [3]
=> []
=> ? ∊ {1,1,2}
[1,2] => [2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,2}
[2,1] => [2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,2}
[3] => [3]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1] => [1,1,1,1]
=> [4]
=> []
=> ? ∊ {1,1,2,2}
[1,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,2,2}
[1,2,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,2,2}
[1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,2,2}
[2,2] => [2,2]
=> [2,2]
=> [2]
=> 2
[3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {1,2,2,2,2}
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2}
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2}
[1,1,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2}
[1,2,2] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[1,3,1] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,4] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {1,2,2,2,2}
[2,1,2] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[2,2,1] => [2,2,1]
=> [3,2]
=> [2]
=> 2
[2,3] => [3,2]
=> [2,2,1]
=> [2,1]
=> 2
[3,1,1] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2] => [3,2]
=> [2,2,1]
=> [2,1]
=> 2
[4,1] => [4,1]
=> [2,1,1,1]
=> [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]
=> [6]
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,3}
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,3}
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2
[1,2,3] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
[1,3,2] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? ∊ {1,2,2,2,2,3}
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2
[2,1,3] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[2,2,1,1] => [2,2,1,1]
=> [4,2]
=> [2]
=> 2
[2,2,2] => [2,2,2]
=> [3,3]
=> [3]
=> 3
[2,3,1] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[2,4] => [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[3,1,1,1] => [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
[3,1,2] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[3,2,1] => [3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[3,3] => [3,3]
=> [2,2,2]
=> [2,2]
=> 2
[4,1,1] => [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[4,2] => [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[5,1] => [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[6] => [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [7]
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,1,3] => [3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,2,2] => [2,2,1,1,1]
=> [5,2]
=> [2]
=> 2
[1,1,1,3,1] => [3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 1
[1,1,1,4] => [4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,2,1,2] => [2,2,1,1,1]
=> [5,2]
=> [2]
=> 2
[1,1,2,2,1] => [2,2,1,1,1]
=> [5,2]
=> [2]
=> 2
[1,1,2,3] => [3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 2
[1,1,3,1,1] => [3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 1
[1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,3}
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? ∊ {2,2,2,2,2,2,3}
Description
The least common multiple of the parts of the partition.
Matching statistic: St001165
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 1
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {1,1}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {1,1}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {1,1,2}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001471
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 1
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {1,1}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {1,1}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {1,1,2}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {1,1,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,3}
Description
The magnitude of a Dyck path.
The magnitude of a finite dimensional algebra with invertible Cartan matrix C is defined as the sum of all entries of the inverse of C.
We define the magnitude of a Dyck path as the magnitude of the corresponding LNakayama algebra.
Matching statistic: St000836
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => ? = 1 - 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => 1 = 2 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 0 = 1 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 1 = 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]
=> [4,1,2,5,3] => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1 = 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]
=> [4,1,5,2,3] => 1 = 2 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1 = 2 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0 = 1 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,3,4,6,5] => 0 = 1 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,3,6,4,5] => 1 = 2 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,3,5,4,6] => 0 = 1 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => 1 = 2 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,2,5,3,4,6] => 1 = 2 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,2,5,3,6,4] => 0 = 1 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5] => 0 = 1 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5] => 1 = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,2,3,4,6] => 1 = 2 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,2,3,6,4] => 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5] => 1 = 2 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4] => 1 = 2 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => 1 = 2 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => 0 = 1 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => 0 = 1 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 1 = 2 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 1 = 2 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,6,4] => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5] => 1 = 2 - 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,7,2,3,4,5,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,4,7,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,5,2,3,4,7,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,6,2,3,7,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,2,3,7,4,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,3,6,4,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,2,7,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,5,2,7,3,4,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,2,6,3,7,4] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,1,2,3,7,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[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]
=> [5,1,2,3,6,4,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [6,1,2,7,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[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]
=> [5,1,2,6,3,4,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,1,2,6,3,5,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [5,1,2,6,3,7,4] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,2,6,3,7,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [4,1,2,5,3,7,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,7,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [5,1,6,2,3,4,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [4,1,6,2,3,5,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[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]
=> [5,1,6,2,3,7,4] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,6,2,3,7,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,5,2,3,7,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,7,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[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]
=> [5,1,6,2,7,3,4] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,6,2,7,3,5] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,5,2,7,3,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,7,4,6] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
[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]
=> [2,1,4,3,6,5,7] => ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3} - 1
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St001637
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 1
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[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] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[6] => [[6],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,4,1] => [[4,4,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,5] => [[5,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,6] => [[6,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 1
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2}
[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] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[6] => [[6],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,4,1] => [[4,4,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,5] => [[5,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 3
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,6] => [[6,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St001568
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[1,1,1] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,2}
[1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2}
[2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2}
[3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,2}
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[1,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[2,1,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2}
[2,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2}
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,2] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[2,2,1] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[3,1,1] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[3,2] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[4,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,4] => [4,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,2,3] => [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,3,2] => [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[1,4,1] => [4,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[1,5] => [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,3] => [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[2,2,1,1] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[2,2,2] => [2,2,2]
=> [2,2]
=> [2]
=> 1
[2,3,1] => [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[2,4] => [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[3,1,1,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[3,1,2] => [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[3,2,1] => [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[3,3] => [3,3]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[4,1,1] => [4,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[4,2] => [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[5,1] => [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[6] => [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,3,3}
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 2
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,1,1,1,3] => [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,1,1,2,2] => [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,3,1] => [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,1,4] => [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,1,2,1,2] => [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,2,2,1] => [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,2,3] => [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,1,3,1,1] => [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,1,3,2] => [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,1,4,1] => [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,5] => [5,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
[1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,2,1,1,2] => [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,1,2,1] => [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,1,3] => [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,2,2,1,1] => [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,2,2,2] => [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,3,1] => [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,2,4] => [4,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
[1,3,1,1,1] => [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,3,1,2] => [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,3,2,1] => [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 2
[1,3,3] => [3,3,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
[1,4,1,1] => [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2
[1,4,2] => [4,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
[1,5,1] => [5,1,1]
=> [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
[1,6] => [6,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[2,1,4] => [4,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
[2,4,1] => [4,2,1]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
[2,5] => [5,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
[3,1,3] => [3,3,1]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3}
Description
The smallest positive integer that does not appear twice in the partition.
The following 36 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001330The hat guessing number of a graph. St001096The size of the overlap set of a permutation. St001642The Prague dimension of a graph. St000822The Hadwiger number of the graph. St001933The largest multiplicity of a part in an integer partition. St000035The number of left outer peaks of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000782The indicator function of whether a given perfect matching is an L & P matching. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001845The number of join irreducibles minus the rank of a lattice. St000993The multiplicity of the largest part of an integer partition. St000031The number of cycles in the cycle decomposition of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001621The number of atoms of a lattice. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001737The number of descents of type 2 in a permutation. St000764The number of strong records in an integer composition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!