Your data matches 78 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000108
Mapping
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 2
[2]
 => 3
[1,1]
 => 3
[3]
 => 4
[2,1]
 => 5
[1,1,1]
 => 4
[4]
 => 5
[3,1]
 => 7
[2,2]
 => 6
[2,1,1]
 => 7
[1,1,1,1]
 => 5
[5]
 => 6
[4,1]
 => 9
[3,2]
 => 9
[3,1,1]
 => 10
[2,2,1]
 => 9
[2,1,1,1]
 => 9
[1,1,1,1,1]
 => 6
[6]
 => 7
[5,1]
 => 11
[4,2]
 => 12
[4,1,1]
 => 13
[3,3]
 => 10
[3,2,1]
 => 14
[3,1,1,1]
 => 13
[2,2,2]
 => 10
[2,2,1,1]
 => 12
[2,1,1,1,1]
 => 11
[1,1,1,1,1,1]
 => 7
Description
The number of partitions contained in the given partition.
Matching statistic: St000420
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 2
[2]
 => [1,1,0,0,1,0]
 => 3
[1,1]
 => [1,0,1,1,0,0]
 => 3
[3]
 => [1,1,1,0,0,0,1,0]
 => 4
[2,1]
 => [1,0,1,0,1,0]
 => 5
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 4
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 7
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 6
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 7
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 5
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 9
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 9
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 9
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 9
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 6
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 7
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 11
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 12
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 13
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 10
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 14
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 13
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 10
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 12
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 11
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 7
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 1 = 2 - 1
[2]
 => [1,1,0,0,1,0]
 => 2 = 3 - 1
[1,1]
 => [1,0,1,1,0,0]
 => 2 = 3 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 3 = 4 - 1
[2,1]
 => [1,0,1,0,1,0]
 => 4 = 5 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 5 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 6 = 7 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 5 = 6 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 6 = 7 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 5 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5 = 6 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 8 = 9 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 8 = 9 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 9 = 10 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 8 = 9 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 8 = 9 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5 = 6 - 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 6 = 7 - 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 10 = 11 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 11 = 12 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 12 = 13 - 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 9 = 10 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 13 = 14 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 12 = 13 - 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 9 = 10 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 11 = 12 - 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 10 = 11 - 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 6 = 7 - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St000070
(load all 2 compositions to match this statistic)
Mapping
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1],[]]
 => ([],1)
 => 2
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => 3
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => 3
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 4
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 5
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 4
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 6
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 7
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 5
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 9
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 9
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 10
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 9
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 9
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 11
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => 12
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 13
[3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 10
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => 14
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 13
[2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 10
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => 12
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 11
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
Description
The number of antichains in a poset. An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable. An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St001313
(load all 3 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 01 => 2
[2]
 => 100 => 011 => 3
[1,1]
 => 110 => 001 => 3
[3]
 => 1000 => 0111 => 4
[2,1]
 => 1010 => 0101 => 5
[1,1,1]
 => 1110 => 0001 => 4
[4]
 => 10000 => 01111 => 5
[3,1]
 => 10010 => 01101 => 7
[2,2]
 => 1100 => 0011 => 6
[2,1,1]
 => 10110 => 01001 => 7
[1,1,1,1]
 => 11110 => 00001 => 5
[5]
 => 100000 => 011111 => 6
[4,1]
 => 100010 => 011101 => 9
[3,2]
 => 10100 => 01011 => 9
[3,1,1]
 => 100110 => 011001 => 10
[2,2,1]
 => 11010 => 00101 => 9
[2,1,1,1]
 => 101110 => 010001 => 9
[1,1,1,1,1]
 => 111110 => 000001 => 6
[6]
 => 1000000 => 0111111 => 7
[5,1]
 => 1000010 => 0111101 => 11
[4,2]
 => 100100 => 011011 => 12
[4,1,1]
 => 1000110 => 0111001 => 13
[3,3]
 => 11000 => 00111 => 10
[3,2,1]
 => 101010 => 010101 => 14
[3,1,1,1]
 => 1001110 => 0110001 => 13
[2,2,2]
 => 11100 => 00011 => 10
[2,2,1,1]
 => 110110 => 001001 => 12
[2,1,1,1,1]
 => 1011110 => 0100001 => 11
[1,1,1,1,1,1]
 => 1111110 => 0000001 => 7
Description
The number of Dyck paths above the lattice path given by a binary word. One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$. See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St001664
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001664: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => ([],1)
 => 2
[2]
 => [[1,2]]
 => [1,2] => ([(0,1)],2)
 => 3
[1,1]
 => [[1],[2]]
 => [2,1] => ([],2)
 => 3
[3]
 => [[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 4
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => 5
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([],3)
 => 4
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 5
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 7
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 6
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => 7
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => 5
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 9
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 9
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 10
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 9
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 9
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => 6
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => 11
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => 12
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => 13
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => 10
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => 14
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => 13
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => 10
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
 => 12
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(4,5)],6)
 => 11
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([],6)
 => 7
Description
The number of non-isomorphic subposets of a poset.
Matching statistic: St000087
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000087: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => ([],1)
 => 1 = 2 - 1
[2]
 => [[1,2]]
 => [1,2] => ([],2)
 => 2 = 3 - 1
[1,1]
 => [[1],[2]]
 => [2,1] => ([(0,1)],2)
 => 2 = 3 - 1
[3]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => 3 = 4 - 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 4 = 5 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 4 - 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([],4)
 => 4 = 5 - 1
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 6 = 7 - 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 5 = 6 - 1
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6 = 7 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([],5)
 => 5 = 6 - 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 8 = 9 - 1
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 8 = 9 - 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 9 = 10 - 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 8 = 9 - 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8 = 9 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 6 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([],6)
 => 6 = 7 - 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10 = 11 - 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 11 = 12 - 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 12 = 13 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 9 = 10 - 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 13 = 14 - 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [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)
 => 12 = 13 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 9 = 10 - 1
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 11 = 12 - 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 10 = 11 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 7 - 1
Description
The number of induced subgraphs. A subgraph $H \subseteq G$ is induced if $E(H)$ consists of all edges in $E(G)$ that connect the vertices of $H$.
Matching statistic: St001616
Mapping
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 58% ●values known / values provided: 66%●distinct values known / distinct values provided: 58%
Values
[1]
 => [[1],[]]
 => ([],1)
 => ([(0,1)],2)
 => 2
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 7
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 6
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 7
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => 9
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => 9
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(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)
 => ? = 10
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => 9
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => 9
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
Description
The number of neutral elements in a lattice. An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St001846
Mapping
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 58% ●values known / values provided: 66%●distinct values known / distinct values provided: 58%
Values
[1]
 => [[1],[]]
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 3 - 2
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 3 = 5 - 2
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 4 - 2
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 3 = 5 - 2
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 5 = 7 - 2
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 4 = 6 - 2
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 5 = 7 - 2
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 3 = 5 - 2
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 4 = 6 - 2
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => 7 = 9 - 2
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => 7 = 9 - 2
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(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)
 => ? = 10 - 2
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => 7 = 9 - 2
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => 7 = 9 - 2
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 4 = 6 - 2
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 5 = 7 - 2
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? ∊ {10,10,11,11,12,12,13,13,14} - 2
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? ∊ {10,10,11,11,12,12,13,13,14} - 2
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {10,10,11,11,12,12,13,13,14} - 2
[3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? ∊ {10,10,11,11,12,12,13,13,14} - 2
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ? ∊ {10,10,11,11,12,12,13,13,14} - 2
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {10,10,11,11,12,12,13,13,14} - 2
[2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? ∊ {10,10,11,11,12,12,13,13,14} - 2
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? ∊ {10,10,11,11,12,12,13,13,14} - 2
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? ∊ {10,10,11,11,12,12,13,13,14} - 2
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 5 = 7 - 2
Description
The number of elements which do not have a complement in the lattice. A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
Matching statistic: St000550
(load all 2 compositions to match this statistic)
Mapping
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Values
[1]
 => [[1],[]]
 => ([],1)
 => ([(0,1)],2)
 => 2
[2]
 => [[2],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
[1,1]
 => [[1,1],[]]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3
[3]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,1]
 => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
[1,1,1]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[4]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[3,1]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 7
[2,2]
 => [[2,2],[]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 6
[2,1,1]
 => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 7
[1,1,1,1]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[5]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,1]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? ∊ {9,9,9,9,10}
[3,2]
 => [[3,2],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {9,9,9,9,10}
[3,1,1]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(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)
 => ? ∊ {9,9,9,9,10}
[2,2,1]
 => [[2,2,1],[]]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? ∊ {9,9,9,9,10}
[2,1,1,1]
 => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? ∊ {9,9,9,9,10}
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[5,1]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[4,2]
 => [[4,2],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[4,1,1]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[3,3]
 => [[3,3],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[3,2,1]
 => [[3,2,1],[]]
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[3,1,1,1]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[2,2,2]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[2,2,1,1]
 => [[2,2,1,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => ? ∊ {10,10,11,11,12,12,13,13,14}
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
Description
The number of modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
The following 68 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000551The number of left modular elements of a lattice. St001684The reduced word complexity of a permutation. St000071The number of maximal chains in a poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000909The number of maximal chains of maximal size in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St000080The rank of the poset. St000570The Edelman-Greene number of a permutation. St000572The dimension exponent of a set partition. St001298The number of repeated entries in the Lehmer code of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001760The number of prefix or suffix reversals needed to sort a permutation. St000222The number of alignments in the permutation. St000516The number of stretching pairs of a permutation. St000519The largest length of a factor maximising the subword complexity. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000906The length of the shortest maximal chain in a poset. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001375The pancake length of a permutation. St001535The number of cyclic alignments of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001821The sorting index of a signed permutation. St001841The number of inversions of a set partition. St001911A descent variant minus the number of inversions. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001565The number of arithmetic progressions of length 2 in a permutation. St001782The order of rowmotion on the set of order ideals of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001902The number of potential covers of a poset. St000528The height of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001926Sparre Andersen's position of the maximum of a signed permutation. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000101The cocharge of a semistandard tableau. St000454The largest eigenvalue of a graph if it is integral. St001556The number of inversions of the third entry of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one.