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Your data matches 49 different statistics following compositions of up to 3 maps.
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St000070: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> 4
([(0,1)],2)
=> 3
([(1,2)],3)
=> 6
([(0,1),(0,2)],3)
=> 5
([(0,2),(2,1)],3)
=> 4
([(0,2),(1,2)],3)
=> 5
([(0,2),(0,3),(3,1)],4)
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> 6
([(0,3),(3,1),(3,2)],4)
=> 6
([(0,3),(1,3),(3,2)],4)
=> 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7
Description
The number of antichains in a poset. An antichain in a poset P is a subset of elements of P which are pairwise incomparable. An order ideal is a subset I of P such that aI and bPa implies bI. 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.
Mp00306: Posets rowmotion cycle typeInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> [2,2]
=> 4
([(0,1)],2)
=> [3]
=> 3
([(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
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 6
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 7
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00195: Posets order idealsLattices
St000550: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3
([(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
([(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
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of modular elements of a lattice. A pair (x,y) of elements of a lattice L is a modular pair if for every zy we have that (yx)z=y(xz). An element x is left-modular if (x,y) is a modular pair for every yL, and is modular if both (x,y) and (y,x) are modular pairs for every yL.
Mp00195: Posets order idealsLattices
St000551: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3
([(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
([(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
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of left modular elements of a lattice. A pair (x,y) of elements of a lattice L is a modular pair if for every zy we have that (yx)z=y(xz). An element x is left-modular if (x,y) is a modular pair for every yL.
Matching statistic: St001279
Mp00306: Posets rowmotion cycle typeInteger partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> [2,2]
=> 4
([(0,1)],2)
=> [3]
=> 3
([(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
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 6
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 7
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001527
Mp00306: Posets rowmotion cycle typeInteger partitions
St001527: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> [2,2]
=> 4
([(0,1)],2)
=> [3]
=> 3
([(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
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 6
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 7
Description
The cyclic permutation representation number of an integer partition. This is the size of the largest cyclic group C such that the fake degree is the character of a permutation representation of C.
Mp00195: Posets order idealsLattices
St001616: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3
([(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
([(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
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of neutral elements in a lattice. An element e of the lattice L is neutral if the sublattice generated by e, x and y is distributive for all x,yL.
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000293: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> [2,2]
=> 1100 => 4
([(0,1)],2)
=> [3]
=> 1000 => 3
([(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
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 10000000 => 7
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 100100 => 6
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 100100 => 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 100100 => 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 1000100 => 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1000100 => 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 1000100 => 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1000000 => 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> 1000100 => 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 10000000 => 7
Description
The number of inversions of a binary word.
Mp00198: Posets incomparability graphGraphs
Mp00111: Graphs complementGraphs
St000300: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1)],2)
=> ([],2)
=> 4
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 3
([(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
([(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
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
Description
The number of independent sets of vertices of a graph. An independent set of vertices of a graph G is a subset UV(G) such that no two vertices in U are adjacent. This is also the number of vertex covers of G as the map UV(G)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]]
Mp00306: Posets rowmotion cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 4
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(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
([(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
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 6
([(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
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [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].
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001614The cyclic permutation representation number of a skew partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000081The number of edges of a graph. St000290The major index of a binary word. St000395The sum of the heights of the peaks of a Dyck path. 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. St000012The area of a Dyck path. St000438The position of the last up step in a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. 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. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000189The number of elements in the poset. 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. St001875The number of simple modules with projective dimension at most 1. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St000699The toughness times the least common multiple of 1,.