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Your data matches 154 different statistics following compositions of up to 3 maps.
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Matching statistic: St000528
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1 = 2 - 1
[[]]
=> ([(0,1)],2)
=> 2 = 3 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 3 = 4 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4 = 5 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4 = 5 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3 = 4 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4 = 5 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4 = 5 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4 = 5 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St001343
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St001343: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001343: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1 = 2 - 1
[[]]
=> ([(0,1)],2)
=> 2 = 3 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 3 = 4 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4 = 5 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4 = 5 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3 = 4 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4 = 5 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4 = 5 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4 = 5 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
Description
The dimension of the reduced incidence algebra of a poset.
The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets.
Thus, this statistic returns the number of non-isomorphic intervals of the poset.
Matching statistic: St001717
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 1 = 2 - 1
[[]]
=> ([(0,1)],2)
=> 2 = 3 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 4 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 3 = 4 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4 = 5 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3 = 4 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4 = 5 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3 = 4 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4 = 5 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4 = 5 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4 = 5 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
Description
The largest size of an interval in a poset.
Matching statistic: St000013
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> 0 = 2 - 2
[[]]
=> [1,0]
=> 1 = 3 - 2
[[],[]]
=> [1,0,1,0]
=> 1 = 3 - 2
[[[]]]
=> [1,1,0,0]
=> 2 = 4 - 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> 1 = 3 - 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> 2 = 4 - 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> 2 = 4 - 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> 3 = 5 - 2
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 4 - 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 3 = 5 - 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 2 = 4 - 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 3 = 5 - 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 3 = 5 - 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 5 - 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000080
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000080: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 0 = 2 - 2
[[]]
=> ([(0,1)],2)
=> 1 = 3 - 2
[[],[]]
=> ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 4 - 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 4 - 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 4 - 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 4 - 2
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 4 - 2
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2 = 4 - 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3 = 5 - 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 4 - 2
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 4 - 2
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2 = 4 - 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 3 = 5 - 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 2 = 4 - 2
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3 = 5 - 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3 = 5 - 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 3 = 5 - 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
Description
The rank of the poset.
Matching statistic: St000093
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
[[]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 3 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 4 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 3 = 4 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 5 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 4 = 5 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 6 - 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> [1]
=> 1 = 2 - 1
[[]]
=> ([(0,1)],2)
=> [2]
=> 2 = 3 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 2 = 3 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 4 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 2 = 3 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3 = 4 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3 = 4 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 3 = 4 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 5 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 2 = 3 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4 = 5 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 3 = 4 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4 = 5 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 3 = 4 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4 = 5 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4 = 5 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 4 = 5 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 6 - 1
Description
The largest part of an integer partition.
Matching statistic: St000786
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
[[]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 3 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 4 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 3 = 4 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 5 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 4 = 5 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 6 - 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St001337
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
[[]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 3 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 4 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 3 = 4 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 5 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 4 = 5 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 6 - 1
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
[[]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 3 - 1
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 4 - 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 3 = 4 - 1
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 5 - 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 4 - 1
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 4 = 5 - 1
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 6 - 1
Description
The upper irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The upper irredundance number is the largest size of a maximal irredundant set.
The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
The following 144 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001720The minimal length of a chain of small intervals in a lattice. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000094The depth of an ordered tree. St000299The number of nonisomorphic vertex-induced subtrees. St000521The number of distinct subtrees of an ordered tree. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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. St001674The number of vertices of the largest induced star graph in the graph. St000010The length of the partition. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000166The depth minus 1 of an ordered tree. St000172The Grundy number of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000442The maximal area to the right of an up step of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000676The number of odd rises of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000734The last entry in the first row of a standard tableau. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000907The number of maximal antichains of minimal length in a poset. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001494The Alon-Tarsi number of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001589The nesting number of a perfect matching. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000011The number of touch points (or returns) of a Dyck path. St000141The maximum drop size of a permutation. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000536The pathwidth of a graph. St000662The staircase size of the code of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001277The degeneracy of a graph. St001323The independence gap of a graph. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000308The height of the tree associated to a permutation. St000527The width of the poset. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000982The length of the longest constant subword. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000015The number of peaks of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000877The depth of the binary word interpreted as a path. St000983The length of the longest alternating subword. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001590The crossing number of a perfect matching. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St000245The number of ascents of a permutation. St000392The length of the longest run of ones in a binary word. St000864The number of circled entries of the shifted recording tableau of a permutation. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001651The Frankl number of a lattice. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000702The number of weak deficiencies of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000730The maximal arc length of a set partition. St001330The hat guessing number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001820The size of the image of the pop stack sorting operator. St001621The number of atoms of a lattice. St000454The largest eigenvalue of a graph if it is integral. St000932The number of occurrences of the pattern UDU in a Dyck path. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001864The number of excedances of a signed permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000254The nesting number of a set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001728The number of invisible descents of a permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000075The orbit size of a standard tableau under promotion. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000736The last entry in the first row of a semistandard tableau. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation.
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