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Matching statistic: St000807
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(load all 14 compositions to match this statistic)
St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 0
[1,2] => 0
[2,1] => 0
[3] => 0
[1,1,1,1] => 0
[1,1,2] => 0
[1,2,1] => 0
[1,3] => 0
[2,1,1] => 0
[2,2] => 0
[3,1] => 0
[4] => 0
[1,1,1,1,1] => 0
[1,1,1,2] => 0
[1,1,2,1] => 0
[1,1,3] => 0
[1,2,1,1] => 0
[1,2,2] => 0
[1,3,1] => 0
[1,4] => 0
[2,1,1,1] => 0
[2,1,2] => 1
[2,2,1] => 0
[2,3] => 0
[3,1,1] => 0
[3,2] => 0
[4,1] => 0
[5] => 0
[1,1,1,1,1,1] => 0
[1,1,1,1,2] => 0
[1,1,1,2,1] => 0
[1,1,1,3] => 0
[1,1,2,1,1] => 0
[1,1,2,2] => 0
[1,1,3,1] => 0
[1,1,4] => 0
[1,2,1,1,1] => 0
[1,2,1,2] => 1
[1,2,2,1] => 0
[1,2,3] => 0
[1,3,1,1] => 0
[1,3,2] => 0
[1,4,1] => 0
[1,5] => 0
[2,1,1,1,1] => 0
[2,1,1,2] => 1
[2,1,2,1] => 1
Description
The sum of the heights of the valleys of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Matching statistic: St001960
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1] => ? = 0
[1,1] => [2] => [1,1,0,0]
=> [1,2] => 0
[2] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,2] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[3] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 0
[1,3] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[3,1] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[4] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 0
[1,4] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 0
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
[2,3] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
[3,2] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[4,1] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[5] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ? ∊ {0,1}
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 0
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 0
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 0
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 0
[1,5] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 0
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 0
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 0
[2,4] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[6] => [1] => [1,0]
=> [1] => ? ∊ {0,1}
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,1}
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,1}
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,1,6,2,3,4] => ? ∊ {0,0,0,0,0,0,0,1}
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => ? ∊ {0,0,0,0,0,0,0,1}
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => ? ∊ {0,0,0,0,0,0,0,1}
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ? ∊ {0,0,0,0,0,0,0,1}
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? ∊ {0,0,0,0,0,0,0,1}
[7] => [1] => [1,0]
=> [1] => ? ∊ {0,0,0,0,0,0,0,1}
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,1,7,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,6,1,2,7,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,1,6,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,1,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[2,2,1,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[3,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[8] => [1] => [1,0]
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,1,8,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,7,1,2,8,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001371
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 50%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 50%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1,0]
=> 10 => 0
[1,1] => [2] => [1,1,0,0]
=> 1100 => 0
[2] => [1] => [1,0]
=> 10 => 0
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 111000 => 0
[1,2] => [1,1] => [1,0,1,0]
=> 1010 => 0
[2,1] => [1,1] => [1,0,1,0]
=> 1010 => 0
[3] => [1] => [1,0]
=> 10 => 0
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 11110000 => 0
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 0
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,3] => [1,1] => [1,0,1,0]
=> 1010 => 0
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 0
[2,2] => [2] => [1,1,0,0]
=> 1100 => 0
[3,1] => [1,1] => [1,0,1,0]
=> 1010 => 0
[4] => [1] => [1,0]
=> 10 => 0
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => ? = 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 0
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 0
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 0
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,4] => [1,1] => [1,0,1,0]
=> 1010 => 0
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 0
[2,3] => [1,1] => [1,0,1,0]
=> 1010 => 0
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 0
[3,2] => [1,1] => [1,0,1,0]
=> 1010 => 0
[4,1] => [1,1] => [1,0,1,0]
=> 1010 => 0
[5] => [1] => [1,0]
=> 10 => 0
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => ? ∊ {0,1,1,1,1,1}
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,1,1,1,1,1}
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,1,1,1,1,1}
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 0
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? ∊ {0,1,1,1,1,1}
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => 0
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 0
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? ∊ {0,1,1,1,1,1}
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,5] => [1,1] => [1,0,1,0]
=> 1010 => 0
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {0,1,1,1,1,1}
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => 0
[2,2,2] => [3] => [1,1,1,0,0,0]
=> 111000 => 0
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[2,4] => [1,1] => [1,0,1,0]
=> 1010 => 0
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 110010111000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {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] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 1111111100000000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 11111000001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 11110000101100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 111100001100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> 11100010111000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 111000110010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> 11001011110000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 110010110010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 110011001100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 50%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 33% ●values known / values provided: 50%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1,0]
=> 10 => 0
[1,1] => [2] => [1,1,0,0]
=> 1100 => 0
[2] => [1] => [1,0]
=> 10 => 0
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 111000 => 0
[1,2] => [1,1] => [1,0,1,0]
=> 1010 => 0
[2,1] => [1,1] => [1,0,1,0]
=> 1010 => 0
[3] => [1] => [1,0]
=> 10 => 0
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 11110000 => 0
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 0
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,3] => [1,1] => [1,0,1,0]
=> 1010 => 0
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 0
[2,2] => [2] => [1,1,0,0]
=> 1100 => 0
[3,1] => [1,1] => [1,0,1,0]
=> 1010 => 0
[4] => [1] => [1,0]
=> 10 => 0
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => ? = 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 0
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 0
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 0
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,4] => [1,1] => [1,0,1,0]
=> 1010 => 0
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 0
[2,3] => [1,1] => [1,0,1,0]
=> 1010 => 0
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 0
[3,2] => [1,1] => [1,0,1,0]
=> 1010 => 0
[4,1] => [1,1] => [1,0,1,0]
=> 1010 => 0
[5] => [1] => [1,0]
=> 10 => 0
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 111111000000 => ? ∊ {0,1,1,1,1,1}
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,1,1,1,1,1}
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,1,1,1,1,1}
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => 0
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? ∊ {0,1,1,1,1,1}
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => 0
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 0
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? ∊ {0,1,1,1,1,1}
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,5] => [1,1] => [1,0,1,0]
=> 1010 => 0
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {0,1,1,1,1,1}
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => 0
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => 0
[2,2,2] => [3] => [1,1,1,0,0,0]
=> 111000 => 0
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[2,4] => [1,1] => [1,0,1,0]
=> 1010 => 0
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 0
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 11111110000000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 110010111000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {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] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 1111111100000000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 11111000001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 11110000101100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 111100001100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> 11100010111000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 111000101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 111000110010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> 11001011110000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 110010110010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 110010101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 110011001100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001208
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 50%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 50%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1,0]
=> [2,1] => 1 = 0 + 1
[1,1] => [2] => [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[2] => [1] => [1,0]
=> [2,1] => 1 = 0 + 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[1,2] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[2,1] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[3] => [1] => [1,0]
=> [2,1] => 1 = 0 + 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 0 + 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,3] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[2,2] => [2] => [1,1,0,0]
=> [2,3,1] => 1 = 0 + 1
[3,1] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[4] => [1] => [1,0]
=> [2,1] => 1 = 0 + 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1 + 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 0 + 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 0 + 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,4] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 0 + 1
[2,3] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 0 + 1
[3,2] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[4,1] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[5] => [1] => [1,0]
=> [2,1] => 1 = 0 + 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,1,1,1,1,1} + 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? ∊ {0,1,1,1,1,1} + 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? ∊ {0,1,1,1,1,1} + 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1 = 0 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? ∊ {0,1,1,1,1,1} + 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 0 + 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? ∊ {0,1,1,1,1,1} + 1
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,5] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {0,1,1,1,1,1} + 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 0 + 1
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[2,4] => [1,1] => [1,0,1,0]
=> [3,1,2] => 1 = 0 + 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,1,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} + 1
[3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {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] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,2,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,1,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,2,1,3] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2} + 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St000225
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 0
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 0
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St001586
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 0
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 0
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 0
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
Description
The number of odd parts smaller than the largest even part in an integer partition.
Matching statistic: St000805
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000805: Integer compositions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000805: Integer compositions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> [1] => 1 = 0 + 1
[1,1] => ([(0,1)],2)
=> [1,1] => 1 = 0 + 1
[2] => ([],2)
=> [2] => 1 = 0 + 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 1 = 0 + 1
[1,2] => ([(1,2)],3)
=> [1,2] => 1 = 0 + 1
[2,1] => ([(0,2),(1,2)],3)
=> [1,1,1] => 1 = 0 + 1
[3] => ([],3)
=> [3] => 1 = 0 + 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1 = 0 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 1 = 0 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1 = 0 + 1
[1,3] => ([(2,3)],4)
=> [1,3] => 1 = 0 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1 = 0 + 1
[2,2] => ([(1,3),(2,3)],4)
=> [1,1,2] => 1 = 0 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1 = 0 + 1
[4] => ([],4)
=> [4] => 1 = 0 + 1
[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)
=> [4,1] => 1 = 0 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 1 = 0 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1 = 0 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [2,3] => 1 = 0 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1 = 0 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 1 = 0 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1 = 0 + 1
[1,4] => ([(3,4)],5)
=> [1,4] => 1 = 0 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 1 = 0 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2 = 1 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1 = 0 + 1
[2,3] => ([(2,4),(3,4)],5)
=> [1,1,3] => 1 = 0 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 1 = 0 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 1 = 0 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,3,1] => 1 = 0 + 1
[5] => ([],5)
=> [5] => 1 = 0 + 1
[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)
=> [5,1] => 1 = 0 + 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)
=> [4,2] => 1 = 0 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,3,1,1] => 1 = 0 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => 1 = 0 + 1
[1,1,2,1,1] => ([(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)
=> [2,2,1,1] => 1 = 0 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => 2 = 1 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,2,1] => 1 = 0 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [2,4] => 1 = 0 + 1
[1,2,1,1,1] => ([(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)
=> [3,1,1,1] => 1 = 0 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,2] => 2 = 1 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,1,1] => 1 = 0 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => 1 = 0 + 1
[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)
=> [2,1,2,1] => 2 = 1 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => 1 = 0 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,3,1] => 1 = 0 + 1
[1,5] => ([(4,5)],6)
=> [1,5] => 1 = 0 + 1
[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)
=> [4,1,1] => 1 = 0 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => 2 = 1 + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,1,1] => 1 = 0 + 1
[1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,1,2,1,3] => ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,1,4,2] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,2,1,1,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,7] => ([(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[2,1,1,1,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[2,1,1,4] => ([(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[2,6] => ([(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
[1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,1,2,1] => ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,2,1,1] => ([(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,2,2] => ([(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,3,1] => ([(0,8),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,2,1,1,1] => ([(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,2,1,2] => ([(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,2,2,1] => ([(0,8),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,2,3] => ([(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,3,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,3,2] => ([(1,8),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,4,1] => ([(0,8),(1,8),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,1,5] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,1,1,2] => ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,1,2,1] => ([(0,8),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,1,3] => ([(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,2,1,1] => ([(0,7),(0,8),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,2,2] => ([(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,2,4] => ([(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,3,1,1,1] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,3,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
[1,1,1,3,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2} + 1
Description
The number of peaks of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.
Matching statistic: St000257
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> 0
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,0}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> 0
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> 0
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> 0
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 0
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 0
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> 0
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 0
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> 0
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 0
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> 0
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> 0
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 0
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 0
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> 0
[6] => [[6],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1]
=> 0
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0}
[1,6] => [[6,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0}
[7] => [[7],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,6] => [[6,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St000292
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 67%
Values
[1] => [[1],[]]
=> []
=> ? => ? = 0
[1,1] => [[1,1],[]]
=> []
=> ? => ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ? => ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0}
[1,2] => [[2,1],[]]
=> []
=> ? => ? ∊ {0,0,0}
[2,1] => [[2,2],[1]]
=> [1]
=> 1 => 0
[3] => [[3],[]]
=> []
=> ? => ? ∊ {0,0,0}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0}
[1,1,2] => [[2,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1 => 0
[1,3] => [[3,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 11 => 0
[2,2] => [[3,2],[1]]
=> [1]
=> 1 => 0
[3,1] => [[3,3],[2]]
=> [2]
=> 0 => 0
[4] => [[4],[]]
=> []
=> ? => ? ∊ {0,0,0,0}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1 => 0
[1,1,3] => [[3,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 11 => 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1 => 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0 => 0
[1,4] => [[4,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 111 => 0
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 11 => 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 01 => 1
[2,3] => [[4,2],[1]]
=> [1]
=> 1 => 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 00 => 0
[3,2] => [[4,3],[2]]
=> [2]
=> 0 => 0
[4,1] => [[4,4],[3]]
=> [3]
=> 1 => 0
[5] => [[5],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,1}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,1}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1 => 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,1}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 11 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0 => 0
[1,1,4] => [[4,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,1}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 111 => 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 11 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 01 => 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 00 => 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0 => 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1 => 0
[1,5] => [[5,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,1}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1111 => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 111 => 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 011 => 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 11 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 001 => 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 01 => 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 11 => 0
[2,4] => [[5,2],[1]]
=> [1]
=> 1 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 000 => 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 00 => 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 10 => 0
[3,3] => [[5,3],[2]]
=> [2]
=> 0 => 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 11 => 0
[4,2] => [[5,4],[3]]
=> [3]
=> 1 => 0
[5,1] => [[5,5],[4]]
=> [4]
=> 0 => 0
[6] => [[6],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,1}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,1,1,1,1}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,1,1,1,1}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> 1 => 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,1,1,1,1}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> 11 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> 1 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> 0 => 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,1,1,1,1}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 111 => 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> 11 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> 01 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> 1 => 0
[1,1,5] => [[5,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,1,1,1,1}
[1,6] => [[6,1],[]]
=> []
=> ? => ? ∊ {0,0,0,1,1,1,1}
[7] => [[7],[]]
=> []
=> ? => ? ∊ {0,0,0,1,1,1,1}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,1,6] => [[6,1,1],[]]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
Description
The number of ascents of a binary word.
The following 213 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001092The number of distinct even parts of a partition. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001423The number of distinct cubes in a binary word. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000929The constant term of the character polynomial of an integer partition. St000478Another weight of a partition according to Alladi. 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. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000661The number of rises of length 3 of a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001490The number of connected components of a skew partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000649The number of 3-excedences of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St000664The number of right ropes of a permutation. St001715The number of non-records in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St000455The second largest eigenvalue of a graph if it is integral. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000405The number of occurrences of the pattern 1324 in a permutation. St000699The toughness times the least common multiple of 1,. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001344The neighbouring number of a permutation. St000570The Edelman-Greene number of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000370The genus of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001793The difference between the clique number and the chromatic number of a graph. St000323The minimal crossing number of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001736The total number of cycles in a graph. St001797The number of overfull subgraphs of a graph. St001307The number of induced stars on four vertices in a graph. St000322The skewness of a graph. St000379The number of Hamiltonian cycles in a graph. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000366The number of double descents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000914The sum of the values of the Möbius function of a poset. St001847The number of occurrences of the pattern 1432 in a permutation. St001964The interval resolution global dimension of a poset. St001665The number of pure excedances of a permutation. St000516The number of stretching pairs of a permutation. St000260The radius of a connected graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001890The maximum magnitude of the Möbius function of a poset. St001520The number of strict 3-descents. St000889The number of alternating sign matrices with the same antidiagonal sums. 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 $eA$. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001498The normalised height of a Nakayama algebra with magnitude 1. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000068The number of minimal elements in a poset.
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