Your data matches 69 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000147
Mp00198: Posets incomparability graphGraphs
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> 0
([],2)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1)],2)
=> ([],2)
=> []
=> 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [3]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St001814
Mp00198: Posets incomparability graphGraphs
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
St001814: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> 1 = 0 + 1
([],2)
=> ([(0,1)],2)
=> [1]
=> 2 = 1 + 1
([(0,1)],2)
=> ([],2)
=> []
=> 1 = 0 + 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 3 = 2 + 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 2 = 1 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 2 = 1 + 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 4 = 3 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 3 = 2 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 3 = 2 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 3 = 2 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> 1 = 0 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 5 = 4 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> 2 = 1 + 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> 7 = 6 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [3]
=> 4 = 3 + 1
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 5 = 4 + 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> 1 = 0 + 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 4 = 3 + 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 4 = 3 + 1
Description
The number of partitions interlacing the given partition.
Matching statistic: St000010
Mp00198: Posets incomparability graphGraphs
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> []
=> 0
([],2)
=> ([(0,1)],2)
=> [1]
=> [1]
=> 1
([(0,1)],2)
=> ([],2)
=> []
=> []
=> 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> []
=> 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> []
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> []
=> 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> [1,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000160
Mp00198: Posets incomparability graphGraphs
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> []
=> 0
([],2)
=> ([(0,1)],2)
=> [1]
=> [1]
=> 1
([(0,1)],2)
=> ([],2)
=> []
=> []
=> 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> []
=> 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> []
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> []
=> 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> [1,1,1]
=> 3
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000271
Mp00195: Posets order idealsLattices
Mp00193: Lattices to posetPosets
Mp00198: Posets incomparability graphGraphs
St000271: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> 3
Description
The chromatic index of a graph. This is the minimal number of colours needed such that no two adjacent edges have the same colour.
Matching statistic: St000548
Mp00198: Posets incomparability graphGraphs
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> []
=> 0
([],2)
=> ([(0,1)],2)
=> [1]
=> [1]
=> 1
([(0,1)],2)
=> ([],2)
=> []
=> []
=> 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> []
=> 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> []
=> 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> [1]
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> []
=> 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> [1,1,1]
=> 3
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St000667
Mp00198: Posets incomparability graphGraphs
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
St000667: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 82%distinct values known / distinct values provided: 83%
Values
([],1)
=> ([],1)
=> []
=> ? = 0
([],2)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1)],2)
=> ([],2)
=> []
=> ? = 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> ? = 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [3]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> ? = 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 3
Description
The greatest common divisor of the parts of the partition.
Mp00198: Posets incomparability graphGraphs
Mp00156: Graphs line graphGraphs
St001110: Graphs ⟶ ℤResult quality: 82% values known / values provided: 82%distinct values known / distinct values provided: 83%
Values
([],1)
=> ([],1)
=> ([],0)
=> ? = 0
([],2)
=> ([(0,1)],2)
=> ([],1)
=> 1
([(0,1)],2)
=> ([],2)
=> ([],0)
=> ? = 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([],0)
=> ? = 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([],1)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([],1)
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],0)
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],0)
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([],1)
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],0)
=> ? = 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 3
Description
The 3-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $3$-dynamic chromatic number of a graph.
Matching statistic: St001389
Mp00198: Posets incomparability graphGraphs
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
St001389: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 82%distinct values known / distinct values provided: 83%
Values
([],1)
=> ([],1)
=> []
=> ? = 0
([],2)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1)],2)
=> ([],2)
=> []
=> ? = 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> ? = 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [3]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> ? = 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 3
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001527
Mp00198: Posets incomparability graphGraphs
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
St001527: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 82%distinct values known / distinct values provided: 83%
Values
([],1)
=> ([],1)
=> []
=> ? = 0
([],2)
=> ([(0,1)],2)
=> [1]
=> 1
([(0,1)],2)
=> ([],2)
=> []
=> ? = 0
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> ? = 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> ? = 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [2]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> ? = 0
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> 1
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> 6
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [3]
=> 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> ? = 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> [3]
=> 3
Description
The cyclic permutation representation number of an integer partition. This is the size of the largest cyclic group $C$ such that the fake degree is the character of a permutation representation of $C$.
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001571The Cartan determinant of the integer partition. St001725The harmonious chromatic number of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001933The largest multiplicity of a part in an integer partition. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001581The achromatic number of a graph. St000171The degree of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St000172The Grundy number of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000537The cutwidth of a graph. St001108The 2-dynamic chromatic number of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001963The tree-depth of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000993The multiplicity of the largest part of an integer partition. St001644The dimension of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001118The acyclic chromatic index of a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000450The number of edges minus the number of vertices plus 2 of a 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. St000455The second largest eigenvalue of a graph if it is integral. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. 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 $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000259The diameter of a connected graph. St000260The radius 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree.