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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000070
Values
([],1)
=> 2
([],2)
=> 4
([(0,1)],2)
=> 3
([],3)
=> 8
([(1,2)],3)
=> 6
([(0,1),(0,2)],3)
=> 5
([(0,2),(2,1)],3)
=> 4
([(0,2),(1,2)],3)
=> 5
([],4)
=> 16
([(2,3)],4)
=> 12
([(1,2),(1,3)],4)
=> 10
([(0,1),(0,2),(0,3)],4)
=> 9
([(0,2),(0,3),(3,1)],4)
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
([(1,2),(2,3)],4)
=> 8
([(0,3),(3,1),(3,2)],4)
=> 6
([(1,3),(2,3)],4)
=> 10
([(0,3),(1,3),(3,2)],4)
=> 6
([(0,3),(1,3),(2,3)],4)
=> 9
([(0,3),(1,2)],4)
=> 9
([(0,3),(1,2),(1,3)],4)
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> 7
([(0,3),(2,1),(3,2)],4)
=> 5
([(0,3),(1,2),(2,3)],4)
=> 7
Description
The number of antichains in a poset.
An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable.
An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Matching statistic: St001279
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 2
([],2)
=> [2,2]
=> 4
([(0,1)],2)
=> [3]
=> 3
([],3)
=> [2,2,2,2]
=> 8
([(1,2)],3)
=> [6]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> 16
([(2,3)],4)
=> [6,6]
=> 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 6
([(1,2),(2,3)],4)
=> [4,4]
=> 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 7
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St000300
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 2
([],2)
=> ([(0,1)],2)
=> ([],2)
=> 4
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 3
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 8
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 6
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 5
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 4
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 5
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 16
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 12
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 10
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 9
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 8
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 10
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 9
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 9
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 7
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 7
Description
The number of independent sets of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is [[St000093]]
Matching statistic: St001034
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 8
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> 16
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000293
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%
Values
([],1)
=> [2]
=> 100 => 2
([],2)
=> [2,2]
=> 1100 => 4
([(0,1)],2)
=> [3]
=> 1000 => 3
([],3)
=> [2,2,2,2]
=> 111100 => 8
([(1,2)],3)
=> [6]
=> 1000000 => 6
([(0,1),(0,2)],3)
=> [3,2]
=> 10100 => 5
([(0,2),(2,1)],3)
=> [4]
=> 10000 => 4
([(0,2),(1,2)],3)
=> [3,2]
=> 10100 => 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? = 16
([(2,3)],4)
=> [6,6]
=> 11000000 => 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> 100001100 => 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1011100 => 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 10000000 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 100100 => 6
([(1,2),(2,3)],4)
=> [4,4]
=> 110000 => 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 100100 => 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> 100001100 => 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 100100 => 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1011100 => 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111000 => 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 1001000 => 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 101100 => 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 100000 => 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 10000000 => 7
Description
The number of inversions of a binary word.
Matching statistic: St000290
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 96%●distinct values known / distinct values provided: 91%
Values
([],1)
=> [2]
=> 100 => 010 => 2
([],2)
=> [2,2]
=> 1100 => 1010 => 4
([(0,1)],2)
=> [3]
=> 1000 => 0010 => 3
([],3)
=> [2,2,2,2]
=> 111100 => 111010 => 8
([(1,2)],3)
=> [6]
=> 1000000 => 0000010 => 6
([(0,1),(0,2)],3)
=> [3,2]
=> 10100 => 10010 => 5
([(0,2),(2,1)],3)
=> [4]
=> 10000 => 00010 => 4
([(0,2),(1,2)],3)
=> [3,2]
=> 10100 => 10010 => 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> 1111111100 => ? => ? = 16
([(2,3)],4)
=> [6,6]
=> 11000000 => 00001010 => 12
([(1,2),(1,3)],4)
=> [6,2,2]
=> 100001100 => 110000010 => 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1011100 => 1110010 => 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 10000000 => 00000010 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 100100 => 100010 => 6
([(1,2),(2,3)],4)
=> [4,4]
=> 110000 => 001010 => 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 100100 => 100010 => 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> 100001100 => 110000010 => 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 100100 => 100010 => 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1011100 => 1110010 => 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111000 => 101010 => 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 1001000 => 0100010 => 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 101100 => 110010 => 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 100000 => 000010 => 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 10000000 => 00000010 => 7
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000228
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 92%●distinct values known / distinct values provided: 82%
St000228: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 92%●distinct values known / distinct values provided: 82%
Values
([],1)
=> [2]
=> 2
([],2)
=> [2,2]
=> 4
([(0,1)],2)
=> [3]
=> 3
([],3)
=> [2,2,2,2]
=> 8
([(1,2)],3)
=> [6]
=> 6
([(0,1),(0,2)],3)
=> [3,2]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> ? ∊ {12,16}
([(2,3)],4)
=> [6,6]
=> ? ∊ {12,16}
([(1,2),(1,3)],4)
=> [6,2,2]
=> 10
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 9
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 6
([(1,2),(2,3)],4)
=> [4,4]
=> 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 6
([(1,3),(2,3)],4)
=> [6,2,2]
=> 10
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> 9
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 7
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000395
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 88%●distinct values known / distinct values provided: 82%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 88%●distinct values known / distinct values provided: 82%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 8
([(1,2)],3)
=> [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
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {10,10,16}
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 12
([(1,2),(1,3)],4)
=> [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]
=> ? ∊ {10,10,16}
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 9
([(0,2),(0,3),(3,1)],4)
=> [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
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 6
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 8
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 6
([(1,3),(2,3)],4)
=> [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]
=> ? ∊ {10,10,16}
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 6
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 9
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 9
([(0,3),(1,2),(1,3)],4)
=> [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]
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 7
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,3),(1,2),(2,3)],4)
=> [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
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001616
Values
([],1)
=> ([(0,1)],2)
=> 2
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3
([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 8
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {10,10,12,16}
([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {10,10,12,16}
([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {10,10,12,16}
([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> 9
([(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
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 8
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {10,10,12,16}
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> 9
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 9
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 8
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
Description
The number of neutral elements in a lattice.
An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St000550
Values
([],1)
=> ([(0,1)],2)
=> 2
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3
([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 8
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {8,8,9,9,9,10,10,12,16}
([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {8,8,9,9,9,10,10,12,16}
([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {8,8,9,9,9,10,10,12,16}
([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? ∊ {8,8,9,9,9,10,10,12,16}
([(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
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,9,9,9,10,10,12,16}
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {8,8,9,9,9,10,10,12,16}
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? ∊ {8,8,9,9,9,10,10,12,16}
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,9,9,9,10,10,12,16}
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {8,8,9,9,9,10,10,12,16}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
Description
The number of modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
The following 19 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. St001614The cyclic permutation representation number of a skew partition. St000479The Ramsey number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001622The number of join-irreducible elements of a lattice. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000189The number of elements in the poset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001875The number of simple modules with projective dimension at most 1. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral.
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