Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000765
(load all 2 compositions to match this statistic)
Mapping
St000765: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 2
[2,1] => 1
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 3
[1,2,1] => 2
[1,3] => 2
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 4
[1,1,2,1] => 3
[1,1,3] => 3
[1,2,1,1] => 2
[1,2,2] => 3
[1,3,1] => 2
[1,4] => 2
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 2
[3,1,1] => 1
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 5
[1,1,1,2,1] => 4
[1,1,1,3] => 4
[1,1,2,1,1] => 3
[1,1,2,2] => 4
[1,1,3,1] => 3
[1,1,4] => 3
[1,2,1,1,1] => 2
[1,2,1,2] => 3
[1,2,2,1] => 3
[1,2,3] => 3
[1,3,1,1] => 2
[1,3,2] => 2
[1,4,1] => 2
[1,5] => 2
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 2
Description
The number of weak records in an integer composition. A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$.
Matching statistic: St001733
(load all 5 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 89%
Values
[1] => [1,0]
 => 1
[1,1] => [1,0,1,0]
 => 2
[2] => [1,1,0,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => 3
[1,2] => [1,0,1,1,0,0]
 => 2
[2,1] => [1,1,0,0,1,0]
 => 1
[3] => [1,1,1,0,0,0]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 2
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 1
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 1
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 1
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 1
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 9
[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 8
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 7
[1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 7
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 6
[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 7
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 6
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 6
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5
[1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 6
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 6
[1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 6
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 5
[1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5
[1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 5
Description
The number of weak left to right maxima of a Dyck path. A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its left.
Matching statistic: St000542
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
 => [1] => [1] => 1
[1,1] => [1,0,1,0]
 => [2,1] => [2,1] => 2
[2] => [1,1,0,0]
 => [1,2] => [1,2] => 1
[1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 3
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => 2
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 1
[3] => [1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,3,1] => 3
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,5,4,2] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,4,2,5,3] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [5,6,4,3,2,1] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [4,6,5,3,2,1] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [4,5,6,3,2,1] => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [5,3,6,4,2,1] => 4
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [3,4,6,5,2,1] => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [3,4,5,6,2,1] => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [2,6,5,4,3,1] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [5,2,6,4,3,1] => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [4,2,6,5,3,1] => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [4,5,2,6,3,1] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [2,3,6,5,4,1] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => [2,5,3,6,4,1] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [2,3,4,6,5,1] => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [1,6,5,4,3,2] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => [5,1,6,4,3,2] => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => [4,1,6,5,3,2] => 2
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 7
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 6
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 5
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 5
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 4
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [6,4,7,5,3,2,1] => ? = 5
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,2,1] => [4,5,7,6,3,2,1] => ? = 4
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [4,5,6,7,3,2,1] => ? = 4
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [3,7,6,5,4,2,1] => ? = 3
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [6,3,7,5,4,2,1] => ? = 4
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,2,1] => [5,3,7,6,4,2,1] => ? = 4
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [5,6,3,7,4,2,1] => ? = 4
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [3,4,7,6,5,2,1] => ? = 3
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [3,6,4,7,5,2,1] => ? = 3
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,2,1] => [3,4,5,7,6,2,1] => ? = 3
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [3,4,5,6,7,2,1] => ? = 3
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,3,1] => [2,7,6,5,4,3,1] => ? = 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [6,2,7,5,4,3,1] => ? = 3
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [5,2,7,6,4,3,1] => ? = 3
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [5,6,2,7,4,3,1] => ? = 3
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [4,2,7,6,5,3,1] => ? = 3
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [6,4,2,7,5,3,1] => ? = 4
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => [4,5,2,7,6,3,1] => ? = 3
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [4,5,6,2,7,3,1] => ? = 3
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,3,4,1] => [2,3,7,6,5,4,1] => ? = 2
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => [2,6,3,7,5,4,1] => ? = 2
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => [2,5,3,7,6,4,1] => ? = 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => [5,2,6,3,7,4,1] => ? = 3
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,4,5,1] => [2,3,4,7,6,5,1] => ? = 2
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,7,2,3,4,5,1] => [2,3,6,4,7,5,1] => ? = 2
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => [2,3,4,5,7,6,1] => ? = 2
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? = 2
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => ? = 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,1,2] => [6,1,7,5,4,3,2] => ? = 2
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,1,2] => [5,1,7,6,4,3,2] => ? = 2
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,1,2] => [5,6,1,7,4,3,2] => ? = 2
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,1,2] => [4,1,7,6,5,3,2] => ? = 2
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,1,2] => [6,4,1,7,5,3,2] => ? = 3
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,1,2] => [4,5,1,7,6,3,2] => ? = 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,1,2] => [4,5,6,1,7,3,2] => ? = 2
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,1,2] => [3,1,7,6,5,4,2] => ? = 2
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,1,2] => [6,3,1,7,5,4,2] => ? = 3
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,1,2] => [5,3,1,7,6,4,2] => ? = 3
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,1,2] => [5,6,3,1,7,4,2] => ? = 3
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,1,2] => [3,4,1,7,6,5,2] => ? = 2
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,1,2] => [3,6,4,1,7,5,2] => ? = 2
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,1,2] => [3,4,5,1,7,6,2] => ? = 2
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => [3,4,5,6,1,7,2] => ? = 2
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,1,2,3] => [1,6,2,7,5,4,3] => ? = 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,1,2,3] => [1,5,2,7,6,4,3] => ? = 1
Description
The number of left-to-right-minima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000541
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
 => [1] => [1] => ? = 1 - 1
[1,1] => [1,0,1,0]
 => [2,1] => [2,1] => 1 = 2 - 1
[2] => [1,1,0,0]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,2] => [1,0,1,1,0,0]
 => [2,3,1] => [2,3,1] => 1 = 2 - 1
[2,1] => [1,1,0,0,1,0]
 => [3,1,2] => [1,3,2] => 0 = 1 - 1
[3] => [1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,4,2,1] => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,4,3,2] => 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => 4 = 5 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,5,3,2,1] => 3 = 4 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [3,5,4,2,1] => 2 = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [3,4,5,2,1] => 2 = 3 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [4,2,5,3,1] => 2 = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,5,4,1] => 1 = 2 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => 1 = 2 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,5,4,3,2] => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => 1 = 2 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [3,1,5,4,2] => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [3,4,1,5,2] => 1 = 2 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,4,2,5,3] => 0 = 1 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5 = 6 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [5,6,4,3,2,1] => 4 = 5 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [4,6,5,3,2,1] => 3 = 4 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [4,5,6,3,2,1] => 3 = 4 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 2 = 3 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [5,3,6,4,2,1] => 3 = 4 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [3,4,6,5,2,1] => 2 = 3 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [3,4,5,6,2,1] => 2 = 3 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [2,6,5,4,3,1] => 1 = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [5,2,6,4,3,1] => 2 = 3 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [4,2,6,5,3,1] => 2 = 3 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [4,5,2,6,3,1] => 2 = 3 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [2,3,6,5,4,1] => 1 = 2 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,4,1] => [2,5,3,6,4,1] => 1 = 2 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,5,1] => [2,3,4,6,5,1] => 1 = 2 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 1 = 2 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,2] => [1,6,5,4,3,2] => 0 = 1 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,1,2] => [5,1,6,4,3,2] => 1 = 2 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => [4,1,6,5,3,2] => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => [4,5,1,6,3,2] => 1 = 2 - 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 7 - 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 6 - 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 5 - 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 5 - 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 4 - 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [6,4,7,5,3,2,1] => ? = 5 - 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,2,1] => [4,5,7,6,3,2,1] => ? = 4 - 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [4,5,6,7,3,2,1] => ? = 4 - 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,2,1] => [3,7,6,5,4,2,1] => ? = 3 - 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [6,3,7,5,4,2,1] => ? = 4 - 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,2,1] => [5,3,7,6,4,2,1] => ? = 4 - 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [5,6,3,7,4,2,1] => ? = 4 - 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,2,1] => [3,4,7,6,5,2,1] => ? = 3 - 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [3,6,4,7,5,2,1] => ? = 3 - 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,2,1] => [3,4,5,7,6,2,1] => ? = 3 - 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [3,4,5,6,7,2,1] => ? = 3 - 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,2,3,1] => [2,7,6,5,4,3,1] => ? = 2 - 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [6,2,7,5,4,3,1] => ? = 3 - 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => [5,2,7,6,4,3,1] => ? = 3 - 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [5,6,2,7,4,3,1] => ? = 3 - 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => [4,2,7,6,5,3,1] => ? = 3 - 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [6,4,2,7,5,3,1] => ? = 4 - 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => [4,5,2,7,6,3,1] => ? = 3 - 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [4,5,6,2,7,3,1] => ? = 3 - 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,3,4,1] => [2,3,7,6,5,4,1] => ? = 2 - 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => [2,6,3,7,5,4,1] => ? = 2 - 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => [2,5,3,7,6,4,1] => ? = 2 - 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => [5,2,6,3,7,4,1] => ? = 3 - 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [7,6,2,3,4,5,1] => [2,3,4,7,6,5,1] => ? = 2 - 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [6,7,2,3,4,5,1] => [2,3,6,4,7,5,1] => ? = 2 - 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [7,2,3,4,5,6,1] => [2,3,4,5,7,6,1] => ? = 2 - 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? = 2 - 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => ? = 1 - 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,1,2] => [6,1,7,5,4,3,2] => ? = 2 - 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,3,1,2] => [5,1,7,6,4,3,2] => ? = 2 - 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,1,2] => [5,6,1,7,4,3,2] => ? = 2 - 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,3,1,2] => [4,1,7,6,5,3,2] => ? = 2 - 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,1,2] => [6,4,1,7,5,3,2] => ? = 3 - 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,3,1,2] => [4,5,1,7,6,3,2] => ? = 2 - 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,1,2] => [4,5,6,1,7,3,2] => ? = 2 - 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,4,1,2] => [3,1,7,6,5,4,2] => ? = 2 - 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,1,2] => [6,3,1,7,5,4,2] => ? = 3 - 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [7,5,6,3,4,1,2] => [5,3,1,7,6,4,2] => ? = 3 - 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,1,2] => [5,6,3,1,7,4,2] => ? = 3 - 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [7,6,3,4,5,1,2] => [3,4,1,7,6,5,2] => ? = 2 - 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,1,2] => [3,6,4,1,7,5,2] => ? = 2 - 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [7,3,4,5,6,1,2] => [3,4,5,1,7,6,2] => ? = 2 - 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => [3,4,5,6,1,7,2] => ? = 2 - 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,1,2,3] => [1,6,2,7,5,4,3] => ? = 1 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 78%
Values
[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 1 - 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? = 2 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 3 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 4 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 3 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 3 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 3 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 3 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 2 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 2 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => ? = 2 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 1 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2 - 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 3 - 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 2 - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 1 - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1 - 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 1 - 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 1 - 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1 - 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 1 - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,1,1] => [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]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 7 - 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 5 - 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 4 - 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 5 - 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 4 - 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 - 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 4 - 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 4 - 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 4 - 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0 = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000454
Mapping
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[1] => 1 => [1,1] => ([(0,1)],2)
 => 1
[1,1] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2] => 10 => [1,2] => ([(1,2)],3)
 => 1
[1,1,1] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,1] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[3] => 100 => [1,3] => ([(2,3)],4)
 => 1
[1,1,1,1] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,2] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,2,1] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[2,1,1] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,2] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[3,1] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[4] => 1000 => [1,4] => ([(3,4)],5)
 => 1
[1,1,1,1,1] => 11111 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,1,1,2] => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,2,1] => 11101 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,1,3] => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,2,1,1] => 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,2,2] => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,3,1] => 11001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[1,4] => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 2
[2,1,1,1] => 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[2,1,2] => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[2,2,1] => 10101 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[2,3] => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[3,1,1] => 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[3,2] => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[4,1] => 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[5] => 10000 => [1,5] => ([(4,5)],6)
 => 1
[1,1,1,1,1,1] => 111111 => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6
[1,1,1,1,2] => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[1,1,1,2,1] => 111101 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,1,1,3] => 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
[1,1,2,1,1] => 111011 => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[1,1,2,2] => 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4
[1,1,3,1] => 111001 => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[1,1,4] => 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[1,2,1,1,1] => 110111 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,2,1,2] => 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[1,2,2,1] => 110101 => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[1,2,3] => 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[1,3,1,1] => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,3,2] => 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,4,1] => 110001 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,5] => 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 2
[2,1,1,1,1] => 101111 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,1,1,2] => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,1,2,1] => 101101 => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,1,3] => 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,2,1,1] => 101011 => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,2,2] => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,1] => 101001 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,4] => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[3,1,1,1] => 100111 => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,1,2] => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,2,1] => 100101 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[3,3] => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[4,1,1] => 100011 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[4,2] => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[5,1] => 100001 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[6] => 100000 => [1,6] => ([(5,6)],7)
 => 1
[1,1,1,1,1,1,1] => 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7
[1,1,1,1,1,2] => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[1,1,1,1,2,1] => 1111101 => [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,1,1,1,3] => 1111100 => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,1,1,2,1,1] => 1111011 => [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[1,1,1,2,2] => 1111010 => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,1,1,3,1] => 1111001 => [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[1,1,1,4] => 1111000 => [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
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.