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Matching statistic: St000481
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 1
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 1
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 1
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 1
[3,3]
=> 1
[3,2,1]
=> 2
[3,1,1,1]
=> 1
[2,2,2]
=> 1
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 1
[4,3]
=> 1
[4,2,1]
=> 2
[4,1,1,1]
=> 1
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 1
[5,3]
=> 1
[5,2,1]
=> 2
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000480
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 0
[2]
=> [1,1]
=> 0
[1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> 0
[2,1]
=> [2,1]
=> 1
[1,1,1]
=> [3]
=> 1
[4]
=> [1,1,1,1]
=> 0
[3,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> 1
[2,1,1]
=> [3,1]
=> 1
[1,1,1,1]
=> [4]
=> 1
[5]
=> [1,1,1,1,1]
=> 0
[4,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> 1
[2,2,1]
=> [3,2]
=> 1
[2,1,1,1]
=> [4,1]
=> 1
[1,1,1,1,1]
=> [5]
=> 1
[6]
=> [1,1,1,1,1,1]
=> 0
[5,1]
=> [2,1,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> 1
[3,3]
=> [2,2,2]
=> 1
[3,2,1]
=> [3,2,1]
=> 2
[3,1,1,1]
=> [4,1,1]
=> 1
[2,2,2]
=> [3,3]
=> 1
[2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> 1
[1,1,1,1,1,1]
=> [6]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> 1
[4,3]
=> [2,2,2,1]
=> 1
[4,2,1]
=> [3,2,1,1]
=> 2
[4,1,1,1]
=> [4,1,1,1]
=> 1
[3,3,1]
=> [3,2,2]
=> 1
[3,2,2]
=> [3,3,1]
=> 1
[3,2,1,1]
=> [4,2,1]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> 1
[2,2,2,1]
=> [4,3]
=> 1
[2,2,1,1,1]
=> [5,2]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 1
[5,3]
=> [2,2,2,1,1]
=> 1
[5,2,1]
=> [3,2,1,1,1]
=> 2
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for n≠2) can cover is
12(√1+8n−3)
and an element which covers this number of elements is given by (c+i,c,c−1,…,3,2,1), where 1≤i≤c+2.
Matching statistic: St000711
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 75%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 75%
Values
[1]
=> [1,0]
=> [1,0]
=> [2,1] => 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,2] => ? = 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,2] => ? = 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? = 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,2] => ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => ? = 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,2] => ? = 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,2] => ? = 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,8,4,5,6,7,1,3] => ? = 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ? = 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,8,7,5,6,1,3,4] => ? = 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ? = 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [2,8,4,6,1,7,3,5] => ? = 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => ? = 2
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ? = 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,1,8,7,3,6] => ? = 2
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => ? = 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ? = 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [4,3,1,5,6,7,8,9,2] => ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? = 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 0
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [10,3,4,5,6,7,8,9,1,2] => ? = 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2,9,4,5,6,7,8,1,3] => ? = 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,10,2] => ? = 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,7,4,5,6,1,8,3] => ? = 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [2,9,8,5,6,7,1,3,4] => ? = 2
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,10,2] => ? = 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ? = 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [2,8,6,5,1,7,3,4] => ? = 2
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => ? = 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [2,9,4,7,6,1,8,3,5] => ? = 2
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,10,2] => ? = 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ? = 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 1
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation π is an index i such that π(i)−i>1.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St001195
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1]
=> [1]
=> [1,0]
=> ? = 0
[2]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> ? = 0
[1,1]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> ? = 1
[3]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[2,1]
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [3]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[4]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,1]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2]
=> [2,2]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,1]
=> [3,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [4]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[5]
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[4,1]
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[3,2]
=> [2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,1]
=> [3,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[2,2,1]
=> [3,2]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[2,1,1,1]
=> [4,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [5]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[6]
=> [1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,1]
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[4,2]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[4,1,1]
=> [3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[3,3]
=> [2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,2,1]
=> [3,2,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,1,1,1]
=> [4,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,2,2]
=> [3,3]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[2,2,1,1]
=> [4,2]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1]
=> [5,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [6]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[5,2]
=> [2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[5,1,1]
=> [3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[4,3]
=> [2,2,2,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[4,2,1]
=> [3,2,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[4,1,1,1]
=> [4,1,1,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[3,3,1]
=> [3,2,2]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,2,2]
=> [3,3,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,2,1,1]
=> [4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,1,1,1,1]
=> [5,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[2,2,2,1]
=> [4,3]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[2,2,1,1,1]
=> [5,2]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1]
=> [6,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[8]
=> [1,1,1,1,1,1,1,1]
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[6,2]
=> [2,2,1,1,1,1]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[5,3]
=> [2,2,2,1,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[5,2,1]
=> [3,2,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[5,1,1,1]
=> [4,1,1,1,1]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[4,4]
=> [2,2,2,2]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[4,3,1]
=> [3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 2
[4,2,2]
=> [3,3,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[4,2,1,1]
=> [4,2,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[4,1,1,1,1]
=> [5,1,1,1]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[3,3,2]
=> [3,3,2]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,3,1,1]
=> [4,2,2]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,2,2,1]
=> [4,3,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,2,1,1,1]
=> [5,2,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[3,1,1,1,1,1]
=> [6,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,2,2,2]
=> [4,4]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[2,2,2,1,1]
=> [5,3]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[2,2,1,1,1,1]
=> [6,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1,1]
=> [7,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [8]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,1]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[7,2]
=> [2,2,1,1,1,1,1]
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[7,1,1]
=> [3,1,1,1,1,1,1]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[6,3]
=> [2,2,2,1,1,1]
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[6,2,1]
=> [3,2,1,1,1,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[6,1,1,1]
=> [4,1,1,1,1,1]
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[5,4]
=> [2,2,2,2,1]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[5,3,1]
=> [3,2,2,1,1]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2
[5,2,2]
=> [3,3,1,1,1]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[5,2,1,1]
=> [4,2,1,1,1]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[5,1,1,1,1]
=> [5,1,1,1,1]
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
Description
The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> ? = 0 - 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 0 - 1
[1,1]
=> [[1],[2]]
=> [2] => ([],2)
=> ? = 1 - 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 0 - 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3] => ([],3)
=> ? = 1 - 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 0 - 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4] => ([],4)
=> ? = 1 - 1
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 0 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> ? = 1 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 0 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6] => ([],6)
=> ? = 1 - 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ? = 0 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,5] => ([(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7] => ([],7)
=> ? = 1 - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => ([],8)
=> ? = 0 - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [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)
=> ? = 2 - 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [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)
=> ? = 2 - 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1 - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ([(0,7),(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)
=> ? = 1 - 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 1 - 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8] => ([],8)
=> ? = 1 - 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [9] => ([],9)
=> ? = 0 - 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 - 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 - 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 - 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
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