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Your data matches 389 different statistics following compositions of up to 3 maps.
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Matching statistic: St000183
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 1
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000480
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ 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
([],2)
=> [1,1]
=> 0
([(0,1)],2)
=> [2]
=> 1
([],3)
=> [1,1,1]
=> 0
([(1,2)],3)
=> [2,1]
=> 1
([(0,2),(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> [1,1,1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([],5)
=> [1,1,1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> 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: St000897
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 1
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 1
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St001335
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 0
([],2)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The cardinality of a minimal cycle-isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.
Matching statistic: St001349
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],1)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 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)
=> 1
([],5)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
Description
The number of different graphs obtained from the given graph by removing an edge.
Matching statistic: St001393
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],1)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 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)
=> 1
([],5)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
Description
The induced matching number of a graph.
An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.
Matching statistic: St000259
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Values
([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000298
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
([],3)
=> ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([],2)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 2 = 1 + 1
([],4)
=> ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([],4)
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 2 = 1 + 1
([],5)
=> ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],5)
=> 2 = 1 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> 2 = 1 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000010
Mp00154: Graphs —core⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> 0
([],2)
=> ([],1)
=> []
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> 1
([],3)
=> ([],1)
=> []
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> ([],1)
=> []
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 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)
=> [6]
=> 1
([],5)
=> ([],1)
=> []
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
Description
The length of the partition.
Matching statistic: St000159
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00154: Graphs —core⟶ Graphs
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%
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%
Values
([],1)
=> ([],1)
=> []
=> 0
([],2)
=> ([],1)
=> []
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> [1]
=> 1
([],3)
=> ([],1)
=> []
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([],4)
=> ([],1)
=> []
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 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)
=> [6]
=> 1
([],5)
=> ([],1)
=> []
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 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.
The following 379 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000160The multiplicity of the smallest part of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000260The radius of a connected graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000481The number of upper covers of a partition in dominance order. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000535The rank-width of a graph. St000548The number of different non-empty partial sums of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000920The logarithmic height of a Dyck path. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001271The competition number of a graph. St001280The number of parts of an integer partition that are at least two. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001484The number of singletons of an integer partition. St001512The minimum rank of a graph. St001673The degree of asymmetry of an integer composition. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000378The diagonal inversion number of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation 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. St001352The number of internal nodes in the modular decomposition of a graph. St001642The Prague dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000023The number of inner peaks of a permutation. St000141The maximum drop size of a permutation. St000143The largest repeated part of a partition. St000147The largest part of an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000310The minimal degree of a vertex of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000353The number of inner valleys of a permutation. St000386The number of factors DDU in a Dyck path. St000624The normalized sum of the minimal distances to a greater element. St000651The maximal size of a rise in a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000761The number of ascents in an integer composition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001092The number of distinct even parts of a partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ 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. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001395The number of strictly unfriendly partitions of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001587Half of the largest even part of an integer partition. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000006The dinv of a Dyck path. St000013The height of a Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000286The number of connected components of the complement of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000397The Strahler number of a rooted tree. St000451The length of the longest pattern of the form k 1 2. St000542The number of left-to-right-minima of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000759The smallest missing part in an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000862The number of parts of the shifted shape of a permutation. St000903The number of different parts of an integer composition. St000917The open packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001111The weak 2-dynamic chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001316The domatic number of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. 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. St001530The depth of a Dyck path. St001746The coalition number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001814The number of partitions interlacing the given partition. St001884The number of borders of a binary word. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000671The maximin edge-connectivity for choosing a subgraph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000442The maximal area to the right of an up step of a Dyck path. 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. St000919The number of maximal left branches of a binary tree. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001204Call 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. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000444The length of the maximal rise of a Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001890The maximum magnitude of the Möbius function of a poset. St000003The number of standard Young tableaux of the partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000088The row sums of the character table of the symmetric group. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000284The Plancherel distribution on integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000517The Kreweras number of an integer partition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000531The leading coefficient of the rook polynomial of an integer partition. St000627The exponent of a binary word. St000628The balance of a binary word. St000644The number of graphs with given frequency partition. St000655The length of the minimal rise of a Dyck path. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000706The product of the factorials of the multiplicities of an integer partition. St000717The number of ordinal summands of a poset. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000847The number of standard Young tableaux whose descent set is the binary word. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000906The length of the shortest maximal chain in a poset. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001128The exponens consonantiae of a partition. St001129The product of the squares of the parts of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001780The order of promotion on the set of standard tableaux of given shape. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000306The bounce count of a Dyck path. St000390The number of runs of ones in a binary word. St000418The number of Dyck paths that are weakly below a Dyck path. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St000762The sum of the positions of the weak records of an integer composition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001531Number of partial orders contained in the poset determined by the Dyck path. St001571The Cartan determinant of the integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001959The product of the heights of the peaks of a Dyck path. St000287The number of connected components of a graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001057The Grundy value of the game of creating an independent set in a graph. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001754The number of tolerances of a finite lattice. St001732The number of peaks visible from the left. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000763The sum of the positions of the strong records of an integer composition. St000805The number of peaks of the associated bargraph. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001597The Frobenius rank of a skew partition. St000658The number of rises of length 2 of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000053The number of valleys of the Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000079The number of alternating sign matrices for a given 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. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001363The Euler characteristic of a graph according to Knill. St001463The number of distinct columns in the nullspace of a graph. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000344The number of strongly connected outdegree sequences of a graph. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001256Number of simple reflexive modules that are 2-stable reflexive. St001367The smallest number which does not occur as degree of a vertex in a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. 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 $eA$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000331The number of upper interactions of a Dyck path. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St000456The monochromatic index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001592The maximal number of simple paths between any two different vertices of a graph. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000907The number of maximal antichains of minimal length in a poset. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001520The number of strict 3-descents. St001877Number of indecomposable injective modules with projective dimension 2. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000914The sum of the values of the Möbius function of a poset. St000934The 2-degree of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St001383The BG-rank of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001651The Frankl number of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in 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. St001645The pebbling number of a connected graph.
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