Your data matches 25 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000007
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1
[1,0,1,0]
 => [1,2] => [2,1] => [2,1] => 2
[1,1,0,0]
 => [2,1] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => [2,3,1] => 2
[1,1,0,0,1,0]
 => [2,1,3] => [3,1,2] => [3,1,2] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [1,3,2] => [1,3,2] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => [1,3,2] => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,3,4,1] => [2,4,3,1] => 3
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,3,4,2] => [1,4,3,2] => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,4,2,3] => [1,4,3,2] => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,4,5,2,1] => [3,5,4,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [2,4,5,3,1] => [2,5,4,3,1] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,5,3,4,1] => [2,5,4,3,1] => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,4,3,5,1] => [2,5,4,3,1] => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,3,4,5,1] => [2,5,4,3,1] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => [5,3,4,1,2] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => [3,5,4,1,2] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,4,5,1,2] => [3,5,4,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => [4,5,1,3,2] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,3,2] => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [5,1,3,4,2] => [5,1,4,3,2] => 4
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,5,3,4,2] => [1,5,4,3,2] => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [1,4,3,5,2] => [1,5,4,3,2] => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,3,4,5,2] => [1,5,4,3,2] => 4
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000393
(load all 2 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 1 => 1 => 1
[1,0,1,0]
 => [1,1] => 11 => 11 => 2
[1,1,0,0]
 => [2] => 10 => 01 => 1
[1,0,1,0,1,0]
 => [1,1,1] => 111 => 111 => 3
[1,0,1,1,0,0]
 => [1,2] => 110 => 011 => 2
[1,1,0,0,1,0]
 => [2,1] => 101 => 101 => 2
[1,1,0,1,0,0]
 => [3] => 100 => 001 => 2
[1,1,1,0,0,0]
 => [3] => 100 => 001 => 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 1111 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 0111 => 3
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 1011 => 3
[1,0,1,1,0,1,0,0]
 => [1,3] => 1100 => 0011 => 3
[1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 0011 => 3
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 1101 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
 => [3,1] => 1001 => 1001 => 3
[1,1,0,1,0,1,0,0]
 => [4] => 1000 => 0001 => 3
[1,1,0,1,1,0,0,0]
 => [4] => 1000 => 0001 => 3
[1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 1001 => 3
[1,1,1,0,0,1,0,0]
 => [4] => 1000 => 0001 => 3
[1,1,1,0,1,0,0,0]
 => [4] => 1000 => 0001 => 3
[1,1,1,1,0,0,0,0]
 => [4] => 1000 => 0001 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 11111 => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 11110 => 01111 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 11101 => 10111 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 11100 => 00111 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 11100 => 00111 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 11011 => 11011 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 11010 => 01011 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 11001 => 10011 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 11001 => 10011 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 00011 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 11101 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 01101 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 10101 => 10101 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 10100 => 00101 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 10100 => 00101 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 10011 => 11001 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => 10010 => 01001 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => 10001 => 10001 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [5] => 10000 => 00001 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [5] => 10000 => 00001 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 10001 => 10001 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [5] => 10000 => 00001 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [5] => 10000 => 00001 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [5] => 10000 => 00001 => 4
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St000553
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => [2,1] => ([],2)
 => ([],2)
 => 2
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => ([],3)
 => 3
[1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([],4)
 => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([],5)
 => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ([(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
Description
The number of blocks of a graph. A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
Matching statistic: St001068
Mapping
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => [[1]]
 => [1,0]
 => 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => [[1,0],[0,1]]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [[0,1],[1,0]]
 => [[0,1],[1,0]]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[0,1,0],[0,0,1],[1,0,0]]
 => [1,1,0,1,0,0]
 => 2
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [1,0,1,1,0,1,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [1,1,0,1,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [1,1,0,1,0,1,0,0]
 => 3
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000053
Mapping
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1]]
 => [[1]]
 => [1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [[1,0],[0,1]]
 => [[1,0],[0,1]]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0]
 => [[0,1],[1,0]]
 => [[0,1],[1,0]]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[0,1,0],[0,0,1],[1,0,0]]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 4 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St001211
(load all 2 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [2] => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001492
(load all 4 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [2] => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St001622
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
Description
The number of join-irreducible elements of a lattice. An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St001615
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,5,6,7,1,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 6 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 6 - 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [3,4,6,7,1,2,5] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 6 - 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [3,5,6,1,7,2,4] => ([(0,3),(0,5),(1,4),(1,6),(2,5),(3,6),(4,2)],7)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 6 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,5,6,7,1,2,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 - 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,6,7,1,2,4,5] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,1,6,7,2,3] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [4,5,6,1,2,7,3] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,5,6,1,7,2,3] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [4,5,7,1,2,3,6] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [4,6,1,7,2,3,5] => ([(0,3),(0,5),(1,4),(1,6),(2,5),(3,6),(4,2)],7)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,6,7,1,2,3,5] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 6 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,1,6,7,2,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,1,2,7,3,4] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,1,7,2,3,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 6 - 1
Description
The number of join prime elements of a lattice. An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001617
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => [2,1] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,5,6,7,1,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 6 - 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 6 - 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [3,4,6,7,1,2,5] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 6 - 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [3,5,6,1,7,2,4] => ([(0,3),(0,5),(1,4),(1,6),(2,5),(3,6),(4,2)],7)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 6 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,5,6,7,1,2,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 - 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,6,7,1,2,4,5] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,1,6,7,2,3] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [4,5,6,1,2,7,3] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,5,6,1,7,2,3] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [4,5,7,1,2,3,6] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [4,6,1,7,2,3,5] => ([(0,3),(0,5),(1,4),(1,6),(2,5),(3,6),(4,2)],7)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 6 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [4,6,7,1,2,3,5] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 6 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,1,6,7,2,3,4] => ([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,1,2,7,3,4] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,1,7,2,3,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 6 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 6 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 6 - 1
Description
The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying $$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001875The number of simple modules with projective dimension at most 1. St000015The number of peaks of a Dyck path. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000021The number of descents of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001626The number of maximal proper sublattices of a lattice.