searching the database
Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001264: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.
Matching statistic: St000617
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> []
=> []
=> 1 = 0 + 1
[1,0,1,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1,0,0]
=> []
=> []
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [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]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [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 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 0 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 0 + 1
Description
The number of global maxima of a Dyck path.
Matching statistic: St000733
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> []
=> []
=> ? = 0 + 1
[1,0,1,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 0 + 1
[1,1,0,0]
=> []
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [[1,2]]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,6],[3,4],[5]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,5],[3],[4]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> [[1,2]]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [[1,2,3,10],[4,5,6],[7,8],[9]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,2,5],[3,4,9],[6,7],[8]]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,2,3,9],[4,5],[6,7],[8]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,2,8],[3,4],[5,6],[7]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3],[2,5],[4,7],[6]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,2,3,9],[4,5,6],[7],[8]]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,2,5],[3,4,8],[6],[7]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,2,3,8],[4,5],[6],[7]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,2,7],[3,4],[5],[6]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3],[2,6],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,2,3,7],[4],[5],[6]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,2,6],[3],[4],[5]]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,2,3,9],[4,5,6],[7,8]]
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,2,5],[3,4,8],[6,7]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,2,3,8],[4,5],[6,7]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,2,7],[3,4],[5,6]]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3],[2,5],[4,6]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,2,3,8],[4,5,6],[7]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,2,5],[3,4,7],[6]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,2,3,7],[4,5],[6]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,2,6],[3,4],[5]]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,2,3,6],[4],[5]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,2,5],[3],[4]]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,2,3,7],[4,5,6]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,2,5],[3,4,6]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,2,3,6],[4,5]]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> [[1],[2]]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [[1,2,3,4,15],[5,6,7,8],[9,10,11],[12,13],[14]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [[1,2,3,7],[4,5,6,14],[8,9,10],[11,12],[13]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [[1,2,3,4,14],[5,6,7],[8,9,10],[11,12],[13]]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [[1,2,3,13],[4,5,6],[7,8,9],[10,11],[12]]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [[1,2,5],[3,4,8],[6,7,12],[9,10],[11]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [[1,2,3,4,14],[5,6,7,8],[9,10],[11,12],[13]]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,7],[4,5,6,13],[8,9],[10,11],[12]]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [[1,2,3,4,13],[5,6,7],[8,9],[10,11],[12]]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [[1,2,3,12],[4,5,6],[7,8],[9,10],[11]]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [[1,2,5],[3,4,11],[6,7],[8,9],[10]]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [[1,2,3,4,12],[5,6],[7,8],[9,10],[11]]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [[1,2,3,11],[4,5],[6,7],[8,9],[10]]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [[1,2,10],[3,4],[5,6],[7,8],[9]]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [[1,2,3,4,14],[5,6,7,8],[9,10,11],[12],[13]]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [[1,2,3,7],[4,5,6,13],[8,9,10],[11],[12]]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [[1,2,3,4,13],[5,6,7],[8,9,10],[11],[12]]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> [[1,2,3,12],[4,5,6],[7,8,9],[10],[11]]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [[1,2,5],[3,4,8],[6,7,11],[9],[10]]
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [[1,2,3,4,13],[5,6,7,8],[9,10],[11],[12]]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> [[1,2,3,7],[4,5,6,12],[8,9],[10],[11]]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [[1,2,3,4,12],[5,6,7],[8,9],[10],[11]]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [[1,2,3,11],[4,5,6],[7,8],[9],[10]]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [[1,2,5],[3,4,10],[6,7],[8],[9]]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [[1,2,3,4,11],[5,6],[7,8],[9],[10]]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [[1,2,3,10],[4,5],[6,7],[8],[9]]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,2,9],[3,4],[5,6],[7],[8]]
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [[1,2,3,4,12],[5,6,7,8],[9],[10],[11]]
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [[1,2,3,7],[4,5,6,11],[8],[9],[10]]
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [[1,2,3,4,11],[5,6,7],[8],[9],[10]]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [[1,2,3,10],[4,5,6],[7],[8],[9]]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,2,5],[3,4,9],[6],[7],[8]]
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> [[1,2,3,4,10],[5,6],[7],[8],[9]]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,2,3,9],[4,5],[6],[7],[8]]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [[1,2,3,4,9],[5],[6],[7],[8]]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> [[1,2,3,4,14],[5,6,7,8],[9,10,11],[12,13]]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> [[1,2,3,7],[4,5,6,13],[8,9,10],[11,12]]
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> [[1,2,3,4,13],[5,6,7],[8,9,10],[11,12]]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> [[1,2,3,12],[4,5,6],[7,8,9],[10,11]]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> [[1,2,3,4,13],[5,6,7,8],[9,10],[11,12]]
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> [[1,2,3,7],[4,5,6,12],[8,9],[10,11]]
=> ? = 1 + 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001038
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [1,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 2 + 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000993
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> []
=> ? = 0 + 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2 = 0 + 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? = 0 + 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 0 + 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3 = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2 = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4 = 2 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2 = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3 = 1 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 2 = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> 3 = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 2 = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 3 = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 2 = 0 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4 = 2 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 2 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 2 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> ? = 3 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> ? = 3 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,1,1]
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> ? = 2 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,1,1]
=> ? = 1 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1,1]
=> ? = 1 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 1 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1,1]
=> ? = 2 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,1,1]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1,1]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1,1]
=> ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,5,3,1,1,1]
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,4,3,1,1,1]
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,5,2,1,1,1]
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,4,2,1,1,1]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1,1]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> ? = 1 + 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ? = 2 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> ? = 2 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2]
=> ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> ? = 0 + 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001803
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1]
=> [[1]]
=> 0
[1,1,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,3],[2]]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> 0
[1,1,0,1,0,0]
=> [1]
=> [[1]]
=> 0
[1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,3,4],[2]]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,3],[2]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 0
[1,1,1,0,1,0,0,0]
=> [1]
=> [[1]]
=> 0
[1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,3,6,10],[2,5,9],[4,8],[7]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,3,4],[2]]
=> 0
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,3],[2]]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> ? = 2
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 32%●distinct values known / distinct values provided: 20%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 32%●distinct values known / distinct values provided: 20%
Values
[1,0]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[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] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[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] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[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] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(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)
=> ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[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] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[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] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[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] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 2 + 2
[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] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[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] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ([(0,3),(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 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ([(0,3),(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)
=> ? = 0 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[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] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001875
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 30%●distinct values known / distinct values provided: 20%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 30%●distinct values known / distinct values provided: 20%
Values
[1,0]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
[1,0,1,0]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,0]
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 2 + 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 2 + 3
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 2 + 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 3 + 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 2 + 3
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 2 + 3
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,12),(4,13),(4,14),(5,11),(5,15),(6,12),(6,15),(8,11),(9,7),(10,7),(11,9),(12,10),(13,8),(14,8),(15,9),(15,10)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,9),(4,12),(5,12),(6,10),(6,11),(8,7),(9,7),(10,13),(11,13),(12,8),(13,8),(13,9)],14)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ?
=> ?
=> ? = 0 + 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,14),(3,14),(4,9),(5,8),(6,8),(6,10),(7,9),(7,10),(8,11),(9,12),(10,11),(10,12),(11,13),(12,13),(13,14)],15)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,11),(4,11),(5,9),(6,8),(7,8),(7,9),(8,10),(9,10),(10,11)],12)
=> ([(0,1)],2)
=> ? = 2 + 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,15),(3,15),(4,13),(5,8),(6,14),(6,16),(7,12),(7,16),(9,11),(10,11),(11,8),(12,10),(13,12),(14,9),(15,13),(16,9),(16,10)],17)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,12),(3,9),(4,8),(5,10),(6,10),(7,8),(7,9),(8,11),(9,11),(10,12),(11,12)],13)
=> ([(0,1)],2)
=> ? = 2 + 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(8,10),(9,10)],11)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 3 + 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,13),(2,16),(3,14),(4,14),(5,12),(6,8),(7,13),(7,15),(9,11),(10,11),(11,8),(12,10),(13,9),(14,16),(15,9),(15,10),(16,12),(16,15)],17)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,15),(2,14),(3,13),(4,12),(5,8),(6,14),(6,15),(7,11),(7,13),(9,12),(10,8),(11,10),(12,11),(13,10),(14,9),(15,9)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,9),(4,9),(5,9),(6,8),(7,8),(8,9)],10)
=> ([(0,1)],2)
=> ? = 3 + 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(8,10),(9,10)],11)
=> ([(0,1)],2)
=> ? = 3 + 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(7,12),(8,14),(9,13),(10,13),(12,14),(13,12),(14,11)],15)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,12),(3,9),(4,8),(5,10),(6,10),(7,8),(7,9),(8,11),(9,11),(10,12),(11,12)],13)
=> ([(0,1)],2)
=> ? = 3 + 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,11),(4,11),(5,9),(6,8),(7,8),(7,9),(8,10),(9,10),(10,11)],12)
=> ([(0,1)],2)
=> ? = 0 + 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1 + 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,13),(2,16),(3,14),(4,14),(5,12),(6,8),(7,13),(7,15),(9,11),(10,11),(11,8),(12,10),(13,9),(14,16),(15,9),(15,10),(16,12),(16,15)],17)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,11),(3,14),(4,13),(5,15),(6,8),(7,13),(7,14),(9,15),(10,8),(11,10),(12,10),(13,9),(14,9),(15,11),(15,12)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(10,8)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,9),(3,8),(4,8),(5,11),(6,10),(7,12),(8,14),(9,14),(10,13),(11,13),(13,12),(14,10),(14,11)],15)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,13),(3,15),(4,12),(5,8),(6,13),(6,14),(7,11),(7,15),(9,11),(10,12),(11,10),(12,8),(13,9),(14,9),(15,10)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,12),(2,11),(3,10),(4,9),(5,8),(6,8),(7,9),(7,10),(8,14),(9,13),(10,13),(11,12),(13,14),(14,11)],15)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,14),(2,15),(3,15),(4,13),(5,8),(6,14),(6,16),(7,12),(7,16),(9,11),(10,11),(11,8),(12,10),(13,12),(14,9),(15,13),(16,9),(16,10)],17)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,9),(4,9),(5,12),(6,11),(7,10),(8,11),(9,12),(11,13),(12,13),(13,10)],14)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,15),(2,14),(3,13),(4,12),(5,8),(6,14),(6,15),(7,11),(7,13),(9,12),(10,8),(11,10),(12,11),(13,10),(14,9),(15,9)],16)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(2,10),(3,10),(4,10),(5,10),(6,10),(7,8),(8,9),(10,8)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,10),(3,10),(4,10),(5,8),(6,8),(7,9),(8,10),(10,9)],11)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,11),(3,9),(4,9),(5,8),(6,8),(7,10),(8,11),(9,11),(11,10)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(2,10),(3,10),(4,9),(5,9),(6,8),(7,8),(7,12),(8,14),(9,13),(10,13),(12,14),(13,12),(14,11)],15)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(2,11),(3,11),(4,11),(5,8),(6,8),(7,9),(8,11),(9,10),(11,9)],12)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001549
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 80%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,2,6,4,3,5] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,2,5,6,3,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,6,4,2,3,5] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,6,3,4,2,5] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,5,6,2,3,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,2,3,6,4,5] => 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,2,6,3,4,5] => 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,6,2,3,4,5] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [1,7,5,3,2,6,4] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [1,5,3,2,6,7,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [1,5,6,3,2,7,4] => ? = 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [1,5,6,7,3,2,4] => ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [1,7,5,2,3,4,6] => ? = 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => [1,7,5,2,4,6,3] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [1,6,3,2,7,4,5] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => ? = 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [1,7,3,2,4,5,6] => ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [1,2,6,4,3,7,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [1,2,6,7,4,3,5] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => [1,2,7,5,6,3,4] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [1,2,7,4,3,5,6] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,2,5,6,7,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [1,6,4,2,3,7,5] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [1,6,7,4,2,3,5] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [1,6,3,4,2,7,5] => ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => [1,6,3,5,2,7,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [1,6,7,3,4,2,5] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => [1,7,5,6,2,3,4] => ? = 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [1,7,4,2,3,5,6] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [1,7,3,4,2,5,6] => ? = 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => [1,5,6,7,2,3,4] => ? = 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [1,2,3,7,5,4,6] => 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [1,2,3,6,7,4,5] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [1,2,7,5,3,4,6] => 0
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [1,2,7,4,5,3,6] => 0
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [1,2,6,7,3,4,5] => 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [1,7,5,2,3,4,6] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [1,7,4,5,2,3,6] => ? = 0
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [1,7,3,4,5,2,6] => ? = 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => [1,6,7,2,3,4,5] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [1,2,3,4,7,5,6] => 0
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => [1,2,3,7,4,5,6] => 0
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => [1,2,7,3,4,5,6] => 0
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => [1,7,2,3,4,5,6] => ? = 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => [1,8,6,4,2,7,5,3] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,6,1,2,3,4,8,5] => [1,7,8,5,3,2,6,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => [1,5,3,2,8,6,7,4] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,6,1,2,3,7,4,5] => [1,8,5,3,2,6,7,4] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => [1,5,3,2,6,7,8,4] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [8,4,1,2,7,3,5,6] => [1,8,6,3,2,4,5,7] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [7,4,1,2,6,3,8,5] => [1,7,8,5,6,3,2,4] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,8,1,2,7,3,4,6] => [1,5,7,4,2,8,6,3] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [8,7,1,2,6,3,4,5] => [1,8,5,6,3,2,7,4] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,7,1,2,6,3,8,4] => [1,5,6,3,2,7,8,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => [1,5,8,6,7,3,2,4] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [8,4,1,2,6,7,3,5] => [1,8,5,6,7,3,2,4] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,8,1,2,6,7,3,4] => [1,5,6,7,3,2,8,4] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [5,4,1,2,6,7,8,3] => [1,5,6,7,8,3,2,4] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [7,3,1,8,2,4,5,6] => [1,7,5,2,3,4,8,6] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [7,3,1,6,2,4,8,5] => [1,7,8,5,2,3,4,6] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [8,3,1,5,2,7,4,6] => [1,8,6,7,4,5,2,3] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [8,3,1,6,2,7,4,5] => [1,8,5,2,3,4,6,7] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [6,3,1,5,2,7,8,4] => [1,6,7,8,4,5,2,3] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [7,4,1,8,2,3,5,6] => [1,7,5,2,4,8,6,3] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [7,4,1,6,2,3,8,5] => [1,7,8,5,2,4,6,3] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [8,7,1,5,2,3,4,6] => [1,8,6,3,2,7,4,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [8,7,1,6,2,3,4,5] => [1,8,5,2,7,4,6,3] => ? = 1
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St001194
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001194: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 60%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001194: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,7,4,1,6,3,5] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,7,6,1,4,5] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,6,1,2,3,4,8,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,6,1,2,3,7,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [8,4,1,2,7,3,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [7,4,1,2,6,3,8,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,8,1,2,7,3,4,6] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
Description
The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module
The following 1 statistic also match your data. Click on any of them to see the details.
St000264The girth of a graph, which is not a tree.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!