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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St000108
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
[7]
=> 8
[6,1]
=> 13
[5,2]
=> 15
[5,1,1]
=> 16
[4,3]
=> 14
[4,2,1]
=> 19
[4,1,1,1]
=> 17
[3,3,1]
=> 16
[3,2,2]
=> 16
[3,2,1,1]
=> 19
[3,1,1,1,1]
=> 16
[2,2,2,1]
=> 14
[2,2,1,1,1]
=> 15
[2,1,1,1,1,1]
=> 13
[1,1,1,1,1,1,1]
=> 8
[8]
=> 9
[7,1]
=> 15
[6,2]
=> 18
[6,1,1]
=> 19
[5,3]
=> 18
[5,2,1]
=> 24
Description
The number of partitions contained in the given partition.
Matching statistic: St000070
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 8
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 13
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 15
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 16
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 19
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> 17
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 16
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 16
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> 19
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> 16
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 14
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> 15
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> 13
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 8
[8]
=> [[8],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 9
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> 15
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> 18
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> 19
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> 18
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> 24
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 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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
[7]
=> 10000000 => 01111111 => 8
[6,1]
=> 10000010 => 01111101 => 13
[5,2]
=> 1000100 => 0111011 => 15
[5,1,1]
=> 10000110 => 01111001 => 16
[4,3]
=> 101000 => 010111 => 14
[4,2,1]
=> 1001010 => 0110101 => 19
[4,1,1,1]
=> 10001110 => 01110001 => 17
[3,3,1]
=> 110010 => 001101 => 16
[3,2,2]
=> 101100 => 010011 => 16
[3,2,1,1]
=> 1010110 => 0101001 => 19
[3,1,1,1,1]
=> 10011110 => 01100001 => 16
[2,2,2,1]
=> 111010 => 000101 => 14
[2,2,1,1,1]
=> 1101110 => 0010001 => 15
[2,1,1,1,1,1]
=> 10111110 => 01000001 => 13
[1,1,1,1,1,1,1]
=> 11111110 => 00000001 => 8
[8]
=> 100000000 => 011111111 => 9
[7,1]
=> 100000010 => 011111101 => 15
[6,2]
=> 10000100 => 01111011 => 18
[6,1,1]
=> 100000110 => 011111001 => 19
[5,3]
=> 1001000 => 0110111 => 18
[5,2,1]
=> 10001010 => 01110101 => 24
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: St000420
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
St000420: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●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
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 13
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 15
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 16
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 14
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 19
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 17
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 16
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 16
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 19
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 16
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 14
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 15
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 13
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 8
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {9,9,15}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {9,9,15}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 18
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 19
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 18
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 24
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 21
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 15
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 23
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {9,9,15}
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000419
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000419: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
St000419: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●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
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 12 = 13 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 14 = 15 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 15 = 16 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 13 = 14 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 18 = 19 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 16 = 17 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 15 = 16 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 15 = 16 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 18 = 19 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 15 = 16 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 13 = 14 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 14 = 15 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 12 = 13 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7 = 8 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {9,9,15} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {9,9,15} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 17 = 18 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 18 = 19 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 17 = 18 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 23 = 24 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 20 = 21 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 14 = 15 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 22 = 23 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {9,9,15} - 1
Description
The number of Dyck paths that are weakly above the Dyck path, except for the path itself.
Matching statistic: St001664
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001664: Posets ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 70%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001664: Posets ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 70%
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
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
=> 17
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
=> 16
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {8,13,14,14,15,16,16,16,19,19}
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
=> 15
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ([(5,6)],7)
=> 13
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([],7)
=> 8
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => ([(3,4),(4,7),(6,5),(7,6)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ([(4,5),(5,7),(7,6)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ([(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ([],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
Description
The number of non-isomorphic subposets of a poset.
Matching statistic: St000087
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000087: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 52%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000087: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 52%
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
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19} - 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 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
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 35%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 35%
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
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 8
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(0,1),(1,2),(1,3),(2,7),(2,14),(3,6),(3,14),(4,11),(5,12),(6,4),(6,15),(7,5),(7,16),(9,8),(10,8),(11,9),(12,10),(13,9),(13,10),(14,15),(14,16),(15,11),(15,13),(16,12),(16,13)],17)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19}
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 8
[8]
=> [[8],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
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
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 35%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 35%
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
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 6 = 8 - 2
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(0,1),(1,2),(1,3),(2,7),(2,14),(3,6),(3,14),(4,11),(5,12),(6,4),(6,15),(7,5),(7,16),(9,8),(10,8),(11,9),(12,10),(13,9),(13,10),(14,15),(14,16),(15,11),(15,13),(16,12),(16,13)],17)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? ∊ {13,13,14,14,15,15,16,16,16,16,17,19,19} - 2
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 6 = 8 - 2
[8]
=> [[8],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 2
[1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26} - 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
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 26%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 26%
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
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(0,1),(1,2),(1,3),(2,7),(2,14),(3,6),(3,14),(4,11),(5,12),(6,4),(6,15),(7,5),(7,16),(9,8),(10,8),(11,9),(12,10),(13,9),(13,10),(14,15),(14,16),(15,11),(15,13),(16,12),(16,13)],17)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,1),(1,2),(1,3),(2,5),(2,13),(3,7),(3,13),(4,12),(5,11),(6,4),(6,15),(7,6),(7,14),(9,10),(10,8),(11,9),(12,8),(13,11),(13,14),(14,9),(14,15),(15,10),(15,12)],16)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,7),(1,8),(2,9),(3,5),(3,8),(4,6),(4,10),(5,4),(5,12),(6,2),(6,11),(7,1),(7,3),(8,12),(10,11),(11,9),(12,10)],13)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {8,8,13,13,14,14,15,15,16,16,16,16,17,19,19}
[8]
=> [[8],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ?
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,1),(1,2),(1,3),(2,5),(2,15),(3,6),(3,15),(4,14),(5,13),(6,7),(6,16),(7,8),(7,18),(8,4),(8,17),(10,12),(11,10),(12,9),(13,11),(14,9),(15,13),(15,16),(16,11),(16,18),(17,12),(17,14),(18,10),(18,17)],19)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,1),(1,3),(1,4),(2,12),(3,10),(4,6),(4,10),(5,14),(6,7),(6,15),(7,8),(7,17),(8,5),(8,16),(10,2),(10,15),(11,13),(12,11),(13,9),(14,9),(15,12),(15,17),(16,13),(16,14),(17,11),(17,16)],18)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,8),(1,9),(2,10),(3,6),(3,9),(4,5),(4,12),(5,7),(5,11),(6,4),(6,14),(7,2),(7,13),(8,1),(8,3),(9,14),(11,13),(12,11),(13,10),(14,12)],15)
=> ? ∊ {9,9,15,15,15,15,18,18,18,18,19,19,19,21,21,22,22,23,23,24,24,26}
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 8 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. St000071The number of maximal chains in a poset. St001684The reduced word complexity of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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.
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