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Matching statistic: St000473
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 2
[3,1,1]
=> 1
[2,2,1]
=> 2
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 2
[3,2,1]
=> 2
[3,1,1,1]
=> 0
[2,2,2]
=> 3
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 1
[4,3]
=> 2
[4,2,1]
=> 2
[4,1,1,1]
=> 1
[3,3,1]
=> 2
[3,2,2]
=> 3
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 0
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 1
[5,3]
=> 2
[5,2,1]
=> 2
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000489
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000489: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000489: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? ∊ {0,1}
[1,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1}
[3]
=> []
=> []
=> [] => ? ∊ {0,1}
[2,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[4]
=> []
=> []
=> [] => ? ∊ {0,0}
[3,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0}
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[5]
=> []
=> []
=> [] => ? ∊ {0,1}
[4,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1}
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[6]
=> []
=> []
=> [] => ? ∊ {0,0}
[5,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0}
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[7]
=> []
=> []
=> [] => ? ∊ {1,2}
[6,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,2}
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
[8]
=> []
=> []
=> [] => ? ∊ {0,2,2}
[7,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,2,2}
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,2,2}
[9]
=> []
=> []
=> [] => ? ∊ {1,2,3,3}
[8,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {1,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {1,2,3,3}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,2,2,3,3,3}
[9,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,2,2,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,0,2,2,3,3,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {0,0,2,2,3,3,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {0,0,2,2,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,2,2,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,2,2,3,3,3}
[11]
=> []
=> []
=> [] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[10,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,0,0,1,2,2,3,3,3,3,4,4}
[12]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[11,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,2] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,8,3] => ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4}
Description
The number of cycles of a permutation of length at most 3.
Matching statistic: St000714
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> ?
=> ? = 0
[2]
=> []
=> []
=> ?
=> ? ∊ {0,1}
[1,1]
=> [1]
=> [1]
=> []
=> ? ∊ {0,1}
[3]
=> []
=> []
=> ?
=> ? ∊ {0,1,1}
[2,1]
=> [1]
=> [1]
=> []
=> ? ∊ {0,1,1}
[1,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,1,1}
[4]
=> []
=> []
=> ?
=> ? ∊ {0,0,1,1,2}
[3,1]
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,2}
[2,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,1,1,2}
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,1,1,2}
[5]
=> []
=> []
=> ?
=> ? ∊ {0,0,1,1,1,2,2}
[4,1]
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,2,2}
[3,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2}
[3,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,2,2}
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2}
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1,2,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,1,1,1,2,2}
[6]
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,2,2,2}
[5,1]
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2}
[4,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,2,2}
[4,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2}
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,2,2}
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2}
[2,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,2,2,2}
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2}
[7]
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[6,1]
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[5,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[5,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2,2,2}
[8]
=> []
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[7,1]
=> [1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[6,2]
=> [2]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[6,1,1]
=> [1,1]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[2,2,2,2]
=> [2,2,2]
=> [3,3]
=> [3]
=> 4
[2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 3
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2}
[6,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[5,4]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[5,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[5,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 3
[4,4,1]
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[4,3,2]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 3
[3,3,3]
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1
[3,3,2,1]
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[3,3,1,1,1]
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [3,3]
=> [3]
=> 4
[3,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 3
[2,2,2,2,1]
=> [2,2,2,1]
=> [4,3]
=> [3]
=> 4
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [5,2]
=> [2]
=> 3
[7,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
[6,4]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[6,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
[6,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 3
[5,5]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[5,4,1]
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[5,3,2]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 2
[5,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 3
[4,4,2]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
[4,4,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[4,3,3]
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 1
[4,3,2,1]
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 2
[4,3,1,1,1]
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 1
[4,2,2,2]
=> [2,2,2]
=> [3,3]
=> [3]
=> 4
[4,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 3
[3,3,3,1]
=> [3,3,1]
=> [3,2,2]
=> [2,2]
=> 1
[3,3,2,2]
=> [3,2,2]
=> [3,3,1]
=> [3,1]
=> 3
[3,3,2,1,1]
=> [3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 2
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 1
Description
The number of semistandard Young tableau of given shape, with entries at most 2.
This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St000777
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 86%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 86%
Values
[1]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2 = 0 + 2
[2]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? = 0 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? = 1 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0} + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,0} + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 1 + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0} + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,0} + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 1 + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2} + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2} + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,2} + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,0,0,2} + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1} + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [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 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1} + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1} + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1} + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 2 + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => ([(6,7)],8)
=> ? ∊ {0,0,0,0,1} + 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 3 + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [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 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [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)
=> 4 = 2 + 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 2 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,8] => ([(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,1,2,4} + 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 3 + 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [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 + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ([(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,3,4} + 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,4,5} + 2
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001525
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 57%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 57%
Values
[1]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> []
=> ? = 0
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? = 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 3
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? ∊ {0,2,2,3}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> 3
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 0
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? ∊ {0,2,2,3}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? ∊ {0,2,2,3}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? ∊ {0,2,2,3}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> ? ∊ {1,2,2,2,2,3,3,3,4}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1]
=> ? ∊ {1,2,2,2,2,3,3,3,4}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1]
=> 2
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,1]
=> ? ∊ {1,2,2,2,2,3,3,3,4}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1]
=> 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,2,1]
=> ? ∊ {1,2,2,2,2,3,3,3,4}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> 0
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> ? ∊ {1,2,2,2,2,3,3,3,4}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> 0
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2,1]
=> ? ∊ {1,2,2,2,2,3,3,3,4}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 0
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? ∊ {1,2,2,2,2,3,3,3,4}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> ? ∊ {1,2,2,2,2,3,3,3,4}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> ? ∊ {1,2,2,2,2,3,3,3,4}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1]
=> 3
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,3,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,4,3,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,6,5,5,4,3,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,4,3,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,7,6,5,4,3,2,1]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,3,3,4,4}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,5,4,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,1,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1,1,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [8,7,6,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,4,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,1,1,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,5,4,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1,1,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
Description
The number of symmetric hooks on the diagonal of a partition.
Matching statistic: St000314
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 71%
Values
[1]
=> [1,0,1,0]
=> [1,2] => [2,1] => 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [3,2,1] => 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,4,3] => 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,2,1] => 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [3,2,1,5,4] => 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,4,3] => 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,3,2] => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,3,1] => 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,2,1] => 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [4,3,2,1,6,5] => 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [2,3,1,5,4] => 3 = 2 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,2,4,1] => 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,4,3,1] => 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [5,3,4,2,1] => 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [6,5,4,3,2,1] => 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [5,4,3,2,1,7,6] => ? ∊ {0,3} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => [4,2,3,1,6,5] => 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [2,1,3,5,4] => 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,3,2,5,4] => 3 = 2 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,1,5,4,3] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [5,3,2,4,1] => 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,5,4,3,2] => 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,3,1] => 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => [6,5,3,4,2,1] => 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ? ∊ {0,3} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [6,5,4,3,2,1,8,7] => ? ∊ {0,1,1,3} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,5,3,4,2,1,7] => [5,3,4,2,1,7,6] => ? ∊ {0,1,1,3} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => [3,2,4,1,6,5] => 3 = 2 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => [2,4,3,1,6,5] => 3 = 2 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,1,4,5,3] => 3 = 2 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => 4 = 3 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,2,5,1] => 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,4,3] => 3 = 2 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,5,3,4,2] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,2,3,4,1] => 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => [6,4,3,5,2,1] => 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,5,4,3,1] => 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => [6,3,5,4,2,1] => 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => [7,6,4,5,3,2,1] => ? ∊ {0,1,1,3} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [8,7,6,5,4,3,2,1] => ? ∊ {0,1,1,3} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [7,6,5,4,3,2,1,9,8] => ? ∊ {0,0,0,0,2,2,3,4} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => [6,5,3,4,2,1,8,7] => ? ∊ {0,0,0,0,2,2,3,4} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => [5,3,2,4,1,7,6] => ? ∊ {0,0,0,0,2,2,3,4} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => [5,2,4,3,1,7,6] => ? ∊ {0,0,0,0,2,2,3,4} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [3,2,1,4,6,5] => 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => [2,3,4,1,6,5] => 4 = 3 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [1,4,3,2,6,5] => 3 = 2 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [3,2,1,6,5,4] => 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,4,5,3] => 4 = 3 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => 3 = 2 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,2,3,5,1] => 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [6,4,3,2,5,1] => 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => 3 = 2 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,5,3,4,1] => 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => [6,3,4,5,2,1] => 1 = 0 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,7,5,4,6,3,2] => [7,6,4,3,5,2,1] => ? ∊ {0,0,0,0,2,2,3,4} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => [7,6,3,5,4,2,1] => ? ∊ {0,0,0,0,2,2,3,4} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,8,7,5,6,4,3,2] => [8,7,6,4,5,3,2,1] => ? ∊ {0,0,0,0,2,2,3,4} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [9,8,7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,2,2,3,4} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [8,7,6,5,4,3,2,1,10,9] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,7,6,4,5,3,2,1,9] => [7,6,4,5,3,2,1,9,8] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,6,4,3,5,2,1,8] => [6,4,3,5,2,1,8,7] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,6,3,5,4,2,1,8] => [6,3,5,4,2,1,8,7] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,4,3,2,5,1,7] => [4,3,2,5,1,7,6] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,3,4,5,2,1,7] => [5,2,3,4,1,7,6] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,2,5,4,3,1,7] => [2,5,4,3,1,7,6] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,7,5,4,3,6,2] => [7,5,4,3,6,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => [7,6,3,4,5,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,8,7,5,4,6,3,2] => [8,7,5,4,6,3,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,7,3,6,5,4,2] => [7,3,6,5,4,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => [8,7,4,6,5,3,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,9,8,7,5,6,4,3,2] => [9,8,7,5,6,4,3,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [10,9,8,7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,3,3,4,4} + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [9,8,7,6,5,4,3,2,1,11,10] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,5,6,4,3,2,1,10] => [8,7,6,4,5,3,2,1,10,9] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,7,5,4,6,3,2,1,9] => [7,6,4,3,5,2,1,9,8] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,7,4,6,5,3,2,1,9] => [7,6,3,5,4,2,1,9,8] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,5,4,3,6,2,1,8] => [6,4,3,2,5,1,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [7,6,3,4,5,2,1,8] => [6,3,4,5,2,1,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [7,3,6,5,4,2,1,8] => [6,2,5,4,3,1,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => [4,3,2,1,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,3,4,2,5,1,7] => [4,2,3,5,1,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,3,2,5,4,1,7] => [3,2,5,4,1,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,4,5,3,1,7] => [2,5,3,4,1,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [1,5,4,3,2,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => [4,3,2,1,7,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [7,5,4,3,2,6,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,7,4,5,3,6,2] => [7,5,3,4,6,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,8,6,5,4,7,3,2] => [8,7,5,4,3,6,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,7,4,3,6,5,2] => [7,4,3,6,5,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => [7,3,6,4,5,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,8,7,4,5,6,3,2] => [8,7,4,5,6,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,9,8,6,5,7,4,3,2] => [9,8,7,5,4,6,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [1,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [7,2,6,5,4,3,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,3,3,4,4,4,5} + 1
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St001232
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 71%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 71%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 1
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3]
=> []
=> []
=> []
=> ? = 1
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4]
=> []
=> []
=> []
=> ? ∊ {1,2}
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,2}
[5]
=> []
=> []
=> []
=> ? ∊ {1,2,2}
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,2,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2}
[6]
=> []
=> []
=> []
=> ? ∊ {0,1,2,2,3}
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,1,2,2,3}
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,1,2,2,3}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,2,2,3}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,2,2,3}
[7]
=> []
=> []
=> []
=> ? ∊ {0,1,1,2,2,2,3,3}
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,1,1,2,2,2,3,3}
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,1,1,2,2,2,3,3}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,2,2,2,3,3}
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,1,1,2,2,2,3,3}
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,2,2,2,3,3}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,2,2,2,3,3}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,2,2,2,3,3}
[8]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,1,2,2,2,2,3,3,4}
[9]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[8,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[7,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[9,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[8,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[8,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[7,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[6,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000454
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 57%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 57%
Values
[1]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,2} + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2} + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2} + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,2,2,3} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,3} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,2,2,2,3,3} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,2,2,2,3,3} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,3,3} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,3,3} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,3} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,3} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,2,3,3} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => ([(6,7)],8)
=> ? ∊ {1,1,1,2,2,2,3,3} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> 1 = 0 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7] => ([(6,7)],8)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,8] => ([(7,8)],9)
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,3,3,3,4} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6] => ([(5,6)],7)
=> 1 = 0 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4} + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> 1 = 0 + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> 1 = 0 + 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6] => ([(5,6)],7)
=> 1 = 0 + 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [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 = 3 + 1
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,6] => ([(5,6)],7)
=> 1 = 0 + 1
[3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1 = 0 + 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000352
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 0
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 2
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => 0
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 2
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,4,2,5,6] => 0
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,4,5,6,3] => 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [4,3,2,1,5,6] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,3,5,6,4,2] => 0
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,5,2,1,6,4] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [5,4,3,2,1,6] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 3
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,3,4,2,5,6,7] => 0
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,4,5,6,3,7] => 0
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => ? ∊ {1,1,1,2,2}
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 2
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,1,5,3,6,7,4] => ? ∊ {1,1,1,2,2}
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,3,5,6,4,2,7] => 0
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [3,5,2,1,6,4,7] => ? ∊ {1,1,1,2,2}
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 2
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [2,4,6,1,7,5,3] => ? ∊ {1,1,1,2,2}
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [4,6,3,2,1,7,5] => ? ∊ {1,1,1,2,2}
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 3
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 3
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,3,4,2,5,6,7,8] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,4,5,6,3,7,8] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,4,1,5,3,6,7,8] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 2
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,8,4] => 0
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,1,5,3,6,7,4,8] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,3,5,6,4,2,7,8] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [3,5,2,1,6,4,7,8] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 2
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,3,4,6,7,2,8,5] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [3,2,6,1,4,7,8,5] => 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [2,4,6,1,7,5,3,8] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [4,6,3,2,1,7,5,8] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 3
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,3,5,7,8,6,4,2] => 0
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [3,5,7,2,1,8,6,4] => ? ∊ {0,0,0,0,1,1,2,2,2,2}
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [5,7,4,3,2,1,8,6] => 3
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 3
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,3,4,2,5,6,7,8,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,4,5,6,3,7,8,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [2,4,1,5,3,6,7,8,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,5,6,7,8,4,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,1,5,3,6,7,4,8,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,3,5,6,4,2,7,8,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [3,5,2,1,6,4,7,8,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,1,3,6,4,7,8,9,5] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,3,4,6,7,2,8,5,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [3,2,6,1,4,7,8,5,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [2,4,6,1,7,5,3,8,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [4,6,3,2,1,7,5,8,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,2,4,5,7,8,9,6,3] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [2,4,1,7,5,8,3,9,6] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [4,3,7,2,1,5,8,9,6] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,3,5,7,8,6,4,2,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [3,5,7,2,1,8,6,4,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [5,7,4,3,2,1,8,6,9] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [2,4,6,8,1,9,7,5,3] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [4,6,8,3,2,1,9,7,5] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,9,1,7] => [6,8,5,4,3,2,1,9,7] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3}
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,3,4,2,5,6,7,8,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [1,2,4,5,6,3,7,8,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[7,2,1]
=> [[1,3,6,7,8,9,10],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9,10] => [2,4,1,5,3,6,7,8,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,2,3,5,6,7,8,4,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [2,1,5,3,6,7,4,8,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9,10] => [1,3,5,6,4,2,7,8,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[6,2,1,1]
=> [[1,4,7,8,9,10],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9,10] => [3,5,2,1,6,4,7,8,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,6,7,8,9,10,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [2,1,3,6,4,7,8,9,5,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9,10] => [1,3,4,6,7,2,8,5,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[5,3,1,1]
=> [[1,4,5,9,10],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9,10] => [3,2,6,1,4,7,8,5,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9,10] => [2,4,6,1,7,5,3,8,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9,10] => [4,6,3,2,1,7,5,8,9,10] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
[4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6] => [1,3,4,2,7,8,5,9,10,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St000259
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 43%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 43%
Values
[1]
=> 1 => [1] => ([],1)
=> 0
[2]
=> 0 => [1] => ([],1)
=> 0
[1,1]
=> 11 => [2] => ([],2)
=> ? = 1
[3]
=> 1 => [1] => ([],1)
=> 0
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[1,1,1]
=> 111 => [3] => ([],3)
=> ? = 1
[4]
=> 0 => [1] => ([],1)
=> 0
[3,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,1,1,2}
[2,2]
=> 00 => [2] => ([],2)
=> ? ∊ {0,1,1,2}
[2,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,1,1,2}
[1,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,1,1,2}
[5]
=> 1 => [1] => ([],1)
=> 0
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[3,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,1,2}
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,1,2}
[1,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {0,1,2}
[6]
=> 0 => [1] => ([],1)
=> 0
[5,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,0,1,1,2,2,2,3}
[4,2]
=> 00 => [2] => ([],2)
=> ? ∊ {0,0,0,1,1,2,2,2,3}
[4,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,2,2,2,3}
[3,3]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,0,1,1,2,2,2,3}
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,0,0,1,1,2,2,2,3}
[2,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {0,0,0,1,1,2,2,2,3}
[2,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,2,3}
[2,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,2,2,2,3}
[1,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? ∊ {0,0,0,1,1,2,2,2,3}
[7]
=> 1 => [1] => ([],1)
=> 0
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[5,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
[3,3,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
[3,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
[3,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
[3,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,2,1,1,1]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
[2,1,1,1,1,1]
=> 011111 => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
[1,1,1,1,1,1,1]
=> 1111111 => [7] => ([],7)
=> ? ∊ {0,0,0,1,1,2,2,3,3}
[8]
=> 0 => [1] => ([],1)
=> 0
[7,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[6,2]
=> 00 => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[6,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[5,3]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[4,4]
=> 00 => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[4,3,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[4,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[4,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[4,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,3,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[3,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[2,2,2,2]
=> 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[2,2,2,1,1]
=> 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[2,2,1,1,1,1]
=> 001111 => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[2,1,1,1,1,1,1]
=> 0111111 => [1,6] => ([(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[1,1,1,1,1,1,1,1]
=> 11111111 => [8] => ([],8)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[9]
=> 1 => [1] => ([],1)
=> 0
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[7,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4}
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[6,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4}
[5,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[5,3,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4}
[5,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4}
[5,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,3,3,3,3,3,4,4}
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[10]
=> 0 => [1] => ([],1)
=> 0
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[11]
=> 1 => [1] => ([],1)
=> 0
[10,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[9,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[8,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[7,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[6,5]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[6,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001060The distinguishing index of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000741The Colin de Verdière graph invariant. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001557The number of inversions of the second entry of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001964The interval resolution global dimension of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000663The number of right floats of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000366The number of double descents of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001488The number of corners of a skew partition. St000764The number of strong records in an integer composition. 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 $eA$.
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