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Your data matches 129 different statistics following compositions of up to 3 maps.
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Matching statistic: St000318
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 1
([],2)
=> []
=> 1
([(0,1)],2)
=> [1]
=> 2
([],3)
=> []
=> 1
([(1,2)],3)
=> [1]
=> 2
([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 2
([],4)
=> []
=> 1
([(2,3)],4)
=> [1]
=> 2
([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 2
([],5)
=> []
=> 1
([(3,4)],5)
=> [1]
=> 2
([(2,4),(3,4)],5)
=> [2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 2
([(1,4),(2,3)],5)
=> [1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000159
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0 = 1 - 1
([],2)
=> []
=> 0 = 1 - 1
([(0,1)],2)
=> [1]
=> 1 = 2 - 1
([],3)
=> []
=> 0 = 1 - 1
([(1,2)],3)
=> [1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([],4)
=> []
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1 = 2 - 1
([],5)
=> []
=> 0 = 1 - 1
([(3,4)],5)
=> [1]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1 = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 1 = 2 - 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000480
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 0 = 1 - 1
([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,1)],2)
=> [2]
=> 1 = 2 - 1
([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(1,2)],3)
=> [2,1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> 0 = 1 - 1
([(2,3)],4)
=> [2,1,1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
([],5)
=> [1,1,1,1,1]
=> 0 = 1 - 1
([(3,4)],5)
=> [2,1,1,1]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> 1 = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> 1 = 2 - 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000533
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000533: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0 = 1 - 1
([],2)
=> []
=> 0 = 1 - 1
([(0,1)],2)
=> [1]
=> 1 = 2 - 1
([],3)
=> []
=> 0 = 1 - 1
([(1,2)],3)
=> [1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([],4)
=> []
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1 = 2 - 1
([],5)
=> []
=> 0 = 1 - 1
([(3,4)],5)
=> [1]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1 = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 1 = 2 - 1
Description
The minimum of the number of parts and the size of the first part of an integer partition.
This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.
Matching statistic: St000783
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000783: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0 = 1 - 1
([],2)
=> []
=> 0 = 1 - 1
([(0,1)],2)
=> [1]
=> 1 = 2 - 1
([],3)
=> []
=> 0 = 1 - 1
([(1,2)],3)
=> [1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([],4)
=> []
=> 0 = 1 - 1
([(2,3)],4)
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1 = 2 - 1
([],5)
=> []
=> 0 = 1 - 1
([(3,4)],5)
=> [1]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 2 = 3 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1 = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 1 = 2 - 1
Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Matching statistic: St000453
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([],2)
=> ([],2)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([],3)
=> ([],3)
=> ([],3)
=> 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],4)
=> ([],4)
=> ([],4)
=> 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> ([],5)
=> ([],5)
=> 1
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000758
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00259: Graphs —vertex addition⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],2)
=> [2] => 1
([],2)
=> ([],3)
=> [3] => 1
([(0,1)],2)
=> ([(1,2)],3)
=> [1,2] => 2
([],3)
=> ([],4)
=> [4] => 1
([(1,2)],3)
=> ([(2,3)],4)
=> [1,3] => 2
([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> [1,1,2] => 2
([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
([],4)
=> ([],5)
=> [5] => 1
([(2,3)],4)
=> ([(3,4)],5)
=> [1,4] => 2
([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> [1,1,3] => 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> [1,2,2] => 2
([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> [2,3] => 2
([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> [1,1,1,2] => 2
([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> [2,3] => 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,2] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [1,2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2] => 2
([],5)
=> ([],6)
=> [6] => 1
([(3,4)],5)
=> ([(4,5)],6)
=> [1,5] => 2
([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> [1,1,4] => 2
([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,3,2] => 2
([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> [2,4] => 2
([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => 2
([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> [2,4] => 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,2] => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,2,2] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,2,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1,2] => 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [1,2,1,2] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,2,1,2] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,1,2] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,1,1,1,2] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => 2
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St001432
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00259: Graphs —vertex addition⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],2)
=> [1,1]
=> 1
([],2)
=> ([],3)
=> [1,1,1]
=> 1
([(0,1)],2)
=> ([(1,2)],3)
=> [2,1]
=> 2
([],3)
=> ([],4)
=> [1,1,1,1]
=> 1
([(1,2)],3)
=> ([(2,3)],4)
=> [2,1,1]
=> 2
([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
([],4)
=> ([],5)
=> [1,1,1,1,1]
=> 1
([(2,3)],4)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([],5)
=> ([],6)
=> [1,1,1,1,1,1]
=> 1
([(3,4)],5)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> [2,2,1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 2
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St000481
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1]
=> 0 = 1 - 1
([],2)
=> [1,1]
=> [2]
=> 0 = 1 - 1
([(0,1)],2)
=> [2]
=> [1,1]
=> 1 = 2 - 1
([],3)
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2,2]
=> [2,2]
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [3,2]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [3,3]
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [2,2,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [4,4]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [2,2,2]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([],5)
=> [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [4,2]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [4,3]
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [3,3]
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [3,2,1]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [5,4]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [5,5]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [3,2,2]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,2,2]
=> [6,6]
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [4,4]
=> 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [3,3,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [5,5]
=> 1 = 2 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [4,4,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [3,3,3]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [3,3,2]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [2,2,2,1]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [4,4,2]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [4,4,4]
=> 1 = 2 - 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St000535
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([],2)
=> ([],2)
=> ([],2)
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],3)
=> ([],3)
=> ([],3)
=> 0 = 1 - 1
([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
([],4)
=> ([],4)
=> ([],4)
=> 0 = 1 - 1
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([],5)
=> ([],5)
=> ([],5)
=> 0 = 1 - 1
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 3 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
Description
The rank-width of a graph.
The following 119 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000897The number of different multiplicities of parts of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001349The number of different graphs obtained from the given graph by removing an edge. St000258The burning number of a graph. St000443The number of long tunnels of a Dyck path. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000784The maximum of the length and the largest part of the integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000291The number of descents of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000628The balance of a binary word. St000638The number of up-down runs of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000920The logarithmic height of a Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. 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. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000630The length of the shortest palindromic decomposition of a binary word. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001616The number of neutral elements in a lattice. St001754The number of tolerances of a finite lattice. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St000918The 2-limited packing number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000260The radius of a connected graph. St000097The order of the largest clique of the graph. St000668The least common multiple of the parts of the partition. St000444The length of the maximal rise of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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:
St001471The magnitude of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000675The number of centered multitunnels of a Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St001261The Castelnuovo-Mumford regularity of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001333The cardinality of a minimal edge-isolating set of a graph. St001393The induced matching number of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000015The number of peaks of a Dyck path. St000335The difference of lower and upper interactions. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001545The second Elser number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000464The Schultz index of a connected graph. 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. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000907The number of maximal antichains of minimal length in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001520The number of strict 3-descents. St001118The acyclic chromatic index of a graph. St001060The distinguishing index of a graph. St001668The number of points of the poset minus the width of the poset. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001651The Frankl number of a lattice. St001645The pebbling number of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph.
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