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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St001389
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 3
[2,2]
=> 3
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 5
[4,1]
=> 4
[3,2]
=> 5
[3,1,1]
=> 3
[2,2,1]
=> 3
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 5
[4,2]
=> 7
[4,1,1]
=> 4
[3,3]
=> 6
[3,2,1]
=> 5
[3,1,1,1]
=> 3
[2,2,2]
=> 4
[2,2,1,1]
=> 3
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 7
[6,1]
=> 6
[5,2]
=> 9
[5,1,1]
=> 5
[4,3]
=> 9
[4,2,1]
=> 7
[4,1,1,1]
=> 4
[3,3,1]
=> 6
[3,2,2]
=> 7
[3,2,1,1]
=> 5
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 4
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 8
[7,1]
=> 7
[6,2]
=> 11
[6,1,1]
=> 6
[5,3]
=> 12
[5,2,1]
=> 9
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000108
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 1
[2]
=> []
=> 1
[1,1]
=> [1]
=> 2
[3]
=> []
=> 1
[2,1]
=> [1]
=> 2
[1,1,1]
=> [1,1]
=> 3
[4]
=> []
=> 1
[3,1]
=> [1]
=> 2
[2,2]
=> [2]
=> 3
[2,1,1]
=> [1,1]
=> 3
[1,1,1,1]
=> [1,1,1]
=> 4
[5]
=> []
=> 1
[4,1]
=> [1]
=> 2
[3,2]
=> [2]
=> 3
[3,1,1]
=> [1,1]
=> 3
[2,2,1]
=> [2,1]
=> 5
[2,1,1,1]
=> [1,1,1]
=> 4
[1,1,1,1,1]
=> [1,1,1,1]
=> 5
[6]
=> []
=> 1
[5,1]
=> [1]
=> 2
[4,2]
=> [2]
=> 3
[4,1,1]
=> [1,1]
=> 3
[3,3]
=> [3]
=> 4
[3,2,1]
=> [2,1]
=> 5
[3,1,1,1]
=> [1,1,1]
=> 4
[2,2,2]
=> [2,2]
=> 6
[2,2,1,1]
=> [2,1,1]
=> 7
[2,1,1,1,1]
=> [1,1,1,1]
=> 5
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[7]
=> []
=> 1
[6,1]
=> [1]
=> 2
[5,2]
=> [2]
=> 3
[5,1,1]
=> [1,1]
=> 3
[4,3]
=> [3]
=> 4
[4,2,1]
=> [2,1]
=> 5
[4,1,1,1]
=> [1,1,1]
=> 4
[3,3,1]
=> [3,1]
=> 7
[3,2,2]
=> [2,2]
=> 6
[3,2,1,1]
=> [2,1,1]
=> 7
[3,1,1,1,1]
=> [1,1,1,1]
=> 5
[2,2,2,1]
=> [2,2,1]
=> 9
[2,2,1,1,1]
=> [2,1,1,1]
=> 9
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 6
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 7
[8]
=> []
=> 1
[7,1]
=> [1]
=> 2
[6,2]
=> [2]
=> 3
[6,1,1]
=> [1,1]
=> 3
[5,3]
=> [3]
=> 4
[5,2,1]
=> [2,1]
=> 5
Description
The number of partitions contained in the given partition.
Matching statistic: St000070
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 96%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000070: Posets ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 96%
Values
[1]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[2]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[1,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[3]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[2,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[1,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[4]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[3,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[2,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[2,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[5]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[4,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[3,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[3,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[2,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[6]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[5,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[4,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[4,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[3,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 4
[3,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[2,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6
[7]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[6,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[5,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[5,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[4,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 4
[4,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[4,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[3,3,1]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[3,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[3,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 9
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7
[8]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[7,1]
=> [1]
=> [[1],[]]
=> ([],1)
=> 2
[6,2]
=> [2]
=> [[2],[]]
=> ([(0,1)],2)
=> 3
[6,1,1]
=> [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 3
[5,3]
=> [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 4
[5,2,1]
=> [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 5
[5,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 4
[4,4]
=> [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[4,3,1]
=> [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[4,2,2]
=> [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
[4,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 7
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 5
[3,3,2]
=> [3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 9
[3,3,1,1]
=> [3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 10
[9]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[10]
=> []
=> [[],[]]
=> ([],0)
=> ? = 1
[11]
=> []
=> [[],[]]
=> ([],0)
=> ? ∊ {1,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1,1,1],[]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? ∊ {1,11}
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 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 96%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 96%
Values
[1]
=> []
=> => => ? = 1
[2]
=> []
=> => => ? = 1
[1,1]
=> [1]
=> 10 => 01 => 2
[3]
=> []
=> => => ? = 1
[2,1]
=> [1]
=> 10 => 01 => 2
[1,1,1]
=> [1,1]
=> 110 => 001 => 3
[4]
=> []
=> => => ? = 1
[3,1]
=> [1]
=> 10 => 01 => 2
[2,2]
=> [2]
=> 100 => 011 => 3
[2,1,1]
=> [1,1]
=> 110 => 001 => 3
[1,1,1,1]
=> [1,1,1]
=> 1110 => 0001 => 4
[5]
=> []
=> => => ? = 1
[4,1]
=> [1]
=> 10 => 01 => 2
[3,2]
=> [2]
=> 100 => 011 => 3
[3,1,1]
=> [1,1]
=> 110 => 001 => 3
[2,2,1]
=> [2,1]
=> 1010 => 0101 => 5
[2,1,1,1]
=> [1,1,1]
=> 1110 => 0001 => 4
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 00001 => 5
[6]
=> []
=> => => ? = 1
[5,1]
=> [1]
=> 10 => 01 => 2
[4,2]
=> [2]
=> 100 => 011 => 3
[4,1,1]
=> [1,1]
=> 110 => 001 => 3
[3,3]
=> [3]
=> 1000 => 0111 => 4
[3,2,1]
=> [2,1]
=> 1010 => 0101 => 5
[3,1,1,1]
=> [1,1,1]
=> 1110 => 0001 => 4
[2,2,2]
=> [2,2]
=> 1100 => 0011 => 6
[2,2,1,1]
=> [2,1,1]
=> 10110 => 01001 => 7
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 00001 => 5
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 000001 => 6
[7]
=> []
=> => => ? = 1
[6,1]
=> [1]
=> 10 => 01 => 2
[5,2]
=> [2]
=> 100 => 011 => 3
[5,1,1]
=> [1,1]
=> 110 => 001 => 3
[4,3]
=> [3]
=> 1000 => 0111 => 4
[4,2,1]
=> [2,1]
=> 1010 => 0101 => 5
[4,1,1,1]
=> [1,1,1]
=> 1110 => 0001 => 4
[3,3,1]
=> [3,1]
=> 10010 => 01101 => 7
[3,2,2]
=> [2,2]
=> 1100 => 0011 => 6
[3,2,1,1]
=> [2,1,1]
=> 10110 => 01001 => 7
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 00001 => 5
[2,2,2,1]
=> [2,2,1]
=> 11010 => 00101 => 9
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 010001 => 9
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 000001 => 6
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0000001 => 7
[8]
=> []
=> => => ? = 1
[7,1]
=> [1]
=> 10 => 01 => 2
[6,2]
=> [2]
=> 100 => 011 => 3
[6,1,1]
=> [1,1]
=> 110 => 001 => 3
[5,3]
=> [3]
=> 1000 => 0111 => 4
[5,2,1]
=> [2,1]
=> 1010 => 0101 => 5
[5,1,1,1]
=> [1,1,1]
=> 1110 => 0001 => 4
[4,4]
=> [4]
=> 10000 => 01111 => 5
[4,3,1]
=> [3,1]
=> 10010 => 01101 => 7
[4,2,2]
=> [2,2]
=> 1100 => 0011 => 6
[4,2,1,1]
=> [2,1,1]
=> 10110 => 01001 => 7
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 00001 => 5
[3,3,2]
=> [3,2]
=> 10100 => 01011 => 9
[3,3,1,1]
=> [3,1,1]
=> 100110 => 011001 => 10
[9]
=> []
=> => => ? = 1
[10]
=> []
=> => => ? ∊ {1,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => 0000000001 => ? ∊ {1,10}
[11]
=> []
=> => => ? ∊ {1,10,11,17}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 0100000001 => ? ∊ {1,10,11,17}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => 0000000001 => ? ∊ {1,10,11,17}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 00000000001 => ? ∊ {1,10,11,17}
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)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 96%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 96%
Values
[1]
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> ? = 1
[1,1]
=> [1]
=> [1,0,1,0]
=> 2
[3]
=> []
=> []
=> ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[4]
=> []
=> []
=> ? = 1
[3,1]
=> [1]
=> [1,0,1,0]
=> 2
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[5]
=> []
=> []
=> ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> 2
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[6]
=> []
=> []
=> ? = 1
[5,1]
=> [1]
=> [1,0,1,0]
=> 2
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[7]
=> []
=> []
=> ? = 1
[6,1]
=> [1]
=> [1,0,1,0]
=> 2
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 7
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 9
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 9
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 7
[8]
=> []
=> []
=> ? = 1
[7,1]
=> [1]
=> [1,0,1,0]
=> 2
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 7
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 9
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10
[9]
=> []
=> []
=> ? ∊ {1,9}
[1,1,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {1,9}
[10]
=> []
=> []
=> ? ∊ {1,9,10}
[2,1,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {1,9,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,9,10}
[11]
=> []
=> []
=> ? ∊ {1,9,10,11,17}
[3,1,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {1,9,10,11,17}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,9,10,11,17}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,9,10,11,17}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,9,10,11,17}
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000110
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 79%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 79%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 5
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 5
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 7
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 4
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 6
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => 6
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => 9
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => 5
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => 9
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,5,3,6] => 7
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 6
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 7
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => 5
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7] => 3
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => 2
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => 7
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,7,5] => 11
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8] => 6
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => 12
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,1,2,3,6,4,7] => 9
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,2,5,3,6,7] => ? ∊ {3,4,5,7}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8] => ? ∊ {3,4,5,7}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => ? ∊ {3,4,5,7}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8] => ? ∊ {3,4,5,7}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,8,6] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,1,2,3,4,5,6,8,9] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,1,2,3,4,7,5,8] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8,9] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [5,1,2,3,6,4,7,8] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8,9] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3,6,7] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [4,1,2,5,6,3,7] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,2,5,3,6,7,8] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8,9] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7,8] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ? ∊ {1,2,3,3,4,5,5,6,7,7,7,8,9,9,9,10,11,13}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,1,2,3,4,5,6,7,8,9] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8,10] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7,9,10] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [7,1,2,3,4,8,5,6] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [7,1,2,3,4,5,8,6,9] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6,8,9,10] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [6,1,2,3,7,4,5,8] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [6,1,2,3,4,7,8,5] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [6,1,2,3,4,7,5,8,9] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8,9,10] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,1,2,6,3,7,4] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [5,1,2,6,3,4,7,8] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [5,1,2,3,6,7,4,8] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [5,1,2,3,6,4,7,8,9] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,1,2,3,4,6,7,8,9,10] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,1,2,3,6,7] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3,7] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,5,2,3,6,7,8] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,2,5,6,3,7,8] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,5,3,6,7,8,9] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7,8,9,10] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,4,1,5,2,6,7] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,1,4,5,6,2,7] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,2,6,7,8] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7,8,9] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7,8,9,10] => ? ∊ {1,2,3,3,4,4,5,5,6,7,7,7,8,9,9,9,9,9,10,10,10,11,12,13,13,13,14,15,15,16,18,19}
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St001464
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 58%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 3
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 3
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 3
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 5
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 4
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 3
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => 4
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 6
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 7
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => 5
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,3,6,4,5] => 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => 3
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 4
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => 5
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,4] => 4
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 7
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 6
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,2,5,6,3] => 7
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,3] => 5
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 9
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => 9
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => 6
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ? = 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ? ∊ {1,2,3,4,5,6,7,8,11}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => ? ∊ {1,2,3,4,5,6,7,8,11}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => 3
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,7,8,6] => ? ∊ {1,2,3,4,5,6,7,8,11}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,6,3,4,5] => 4
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5] => 5
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,5] => ? ∊ {1,2,3,4,5,6,7,8,11}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 5
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => 7
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => 6
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,2,5,3,6,7,4] => 7
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,4] => ? ∊ {1,2,3,4,5,6,7,8,11}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,3] => ? ∊ {1,2,3,4,5,6,7,8,11}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,2,3,4,5,6,7,8,11}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {1,2,3,4,5,6,7,8,11}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {1,2,3,4,5,6,7,8,11}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,8,6,7] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,6,8,9,7] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,2,3,4,7,5,8,6] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,5,7,8,9,6] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,2,3,6,4,7,8,5] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,9,5] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,2,3,6,7,4] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,2,5,3,6,7,8,4] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,9,4] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,3] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,4,2,5,6,7,8,3] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,8,9,3] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,2] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,9,2] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {1,2,3,3,4,5,5,6,7,7,8,9,9,10,11,13,13,15}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9,10] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,2,3,4,5,6,9,7,8] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,5,6,7,9,10,8] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,2,3,4,8,5,6,7] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,2,3,4,5,8,6,9,7] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,8,9,10,7] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,2,3,7,4,5,8,6] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,2,3,4,7,8,5,6] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [1,2,3,4,7,5,8,9,6] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,7,8,9,10,6] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,2,6,3,4,7,8,5] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,2,3,6,7,4,8,5] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,2,3,6,4,7,8,9,5] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,6,7,8,9,10,5] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,2,3,6,7,4] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,5,2,6,3,7,4] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [1,5,2,3,6,7,8,4] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,2,5,6,3,7,8,4] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,5,3,6,7,8,9,4] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,8,9,10,4] => ? ∊ {1,2,3,3,4,4,5,5,6,6,7,7,7,8,9,9,9,9,10,10,11,12,13,13,13,14,15,15,16,18,18,19}
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St000883
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 42%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 42%
Values
[1]
=> [[1]]
=> [[1]]
=> [1] => 1
[2]
=> [[1,2]]
=> [[1],[2]]
=> [2,1] => 2
[1,1]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 1
[3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 3
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 5
[4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4
[3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 5
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 3
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 5
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => 7
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => 6
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => 5
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 4
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 3
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 7
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => 6
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[1,6],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6] => ? ∊ {4,5,6,7,7,9,9}
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => 5
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => ? ∊ {4,5,6,7,7,9,9}
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ? ∊ {4,5,6,7,7,9,9}
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 4
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ? ∊ {4,5,6,7,7,9,9}
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => ? ∊ {4,5,6,7,7,9,9}
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => ? ∊ {4,5,6,7,7,9,9}
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => 3
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => ? ∊ {4,5,6,7,7,9,9}
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => 8
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => 7
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => 6
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [[1,6],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[1,6,8],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => 5
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,5,8],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,5,7],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,5,7,8],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => 4
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,4,7],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,4,6,8],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [[1,4,6,7,8],[2,5],[3]]
=> [3,2,5,1,4,6,7,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => 3
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => 5
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => ? ∊ {3,4,5,6,7,7,9,9,9,10,10,11,12}
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => 2
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => 9
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => 8
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [[1,8],[2,9],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,9,1,8] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => 7
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [[1,7],[2,8],[3,9],[4],[5],[6]]
=> [6,5,4,3,9,2,8,1,7] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [[1,7,9],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => 6
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [[1,6],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [[1,6,9],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [[1,6,8],[2,7,9],[3],[4],[5]]
=> [5,4,3,2,7,9,1,6,8] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [[1,6,8,9],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> [4,3,7,2,6,9,1,5,8] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,5,8,9],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,5,7,9],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,5,7,8,9],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> [3,6,9,2,5,8,1,4,7] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> [3,2,5,7,9,1,4,6,8] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> [3,2,5,1,4,6,7,8,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> [2,4,6,1,3,5,7,8,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,3,5,6,7,8,9],[2,4]]
=> [2,4,1,3,5,6,7,8,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,13,13,14,14,15}
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [[1,9],[2,10],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,10,1,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,12,13,13,13,14,14,15,15,15,16,16,16,18,18,19}
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [[1,8],[2,9],[3,10],[4],[5],[6],[7]]
=> [7,6,5,4,3,10,2,9,1,8] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,12,13,13,13,14,14,15,15,15,16,16,16,18,18,19}
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [[1,8,10],[2,9],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,9,1,8,10] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,12,13,13,13,14,14,15,15,15,16,16,16,18,18,19}
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [[1,7],[2,8],[3,9],[4,10],[5],[6]]
=> [6,5,4,10,3,9,2,8,1,7] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,12,13,13,13,14,14,15,15,15,16,16,16,18,18,19}
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [[1,7,10],[2,8],[3,9],[4],[5],[6]]
=> [6,5,4,3,9,2,8,1,7,10] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,12,13,13,13,14,14,15,15,15,16,16,16,18,18,19}
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [[1,7,9],[2,8,10],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,10,1,7,9] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,12,13,13,13,14,14,15,15,15,16,16,16,18,18,19}
[6,2,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10]]
=> [[1,7,9,10],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7,9,10] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,12,13,13,13,14,14,15,15,15,16,16,16,18,18,19}
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> [5,10,4,9,3,8,2,7,1,6] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,12,13,13,13,14,14,15,15,15,16,16,16,18,18,19}
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [[1,6,10],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6,10] => ? ∊ {3,4,5,5,6,7,7,9,9,9,9,10,10,10,11,12,12,13,13,13,14,14,15,15,15,16,16,16,18,18,19}
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001232
(load all 147 compositions to match this statistic)
(load all 147 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 46%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 46%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {4,5,6} - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ? ∊ {4,5,6} - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {4,5,6} - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? ∊ {3,4,7,7,9} - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5 = 6 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {3,4,7,7,9} - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 5 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {3,4,7,7,9} - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ? ∊ {3,4,7,7,9} - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {3,4,7,7,9} - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 8 = 9 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 5 = 6 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 7 = 8 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 10 = 11 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,2,3,4,4,5,5,5,6,7,7,9,9,9,10,10,12} - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ?
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ?
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> 5 = 6 - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 6 = 7 - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> 7 = 8 - 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4 = 5 - 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> 9 = 10 - 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 8 = 9 - 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {1,2,3,3,4,4,5,5,6,7,7,9,9,9,9,10,10,11,12,13,13,14,14,15} - 1
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> 6 = 7 - 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> 8 = 9 - 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 6 = 7 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000456
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 46%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 46%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> ? = 1
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> ? = 2
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? ∊ {2,3}
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {2,3}
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {2,3,4}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {2,3,4}
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,3,4}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ? ∊ {3,3,4,5,5}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {3,3,4,5,5}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {3,3,4,5,5}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {3,3,4,5,5}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {3,3,4,5,5}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? ∊ {3,3,4,5,5,6,7}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {3,3,4,5,5,6,7}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,4,5,5,6,7}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {3,3,4,5,5,6,7}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,3,4,5,5,6,7}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,5,5,6,7}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {3,3,4,5,5,6,7}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {3,4,5,5,6,6,7,7,7,9,9}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,3,4,5,5,6,7,7,7,9,9,9,10,10,11,12}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? ∊ {1,2,3,5,5,6,7,7,7,8,9,9,9,9,9,10,10,11,12,13,13,14,14,15}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? ∊ {1,2,3,5,5,6,7,7,7,8,9,9,9,9,9,10,10,11,12,13,13,14,14,15}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,3,5,5,6,7,7,7,8,9,9,9,9,9,10,10,11,12,13,13,14,14,15}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => ([(6,8),(7,8)],9)
=> ? ∊ {1,2,3,5,5,6,7,7,7,8,9,9,9,9,9,10,10,11,12,13,13,14,14,15}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [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)
=> 10
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => ([(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)
=> 10
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,3,4,2,6,7,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [6,2,3,5,4,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 11
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,2,4,5,3,7,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => ([(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
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000369The dinv deficit of a Dyck path. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001644The dimension of a graph. St000454The largest eigenvalue of a graph if it is integral. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001684The reduced word complexity of a permutation. St000004The major index of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000797The stat`` of a permutation. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000133The "bounce" of a permutation. St001727The number of invisible inversions of a permutation. St001726The number of visible inversions of a permutation. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000259The diameter of a connected graph. St000422The energy of a graph, if it is integral.
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