- FindStat

Your data matches 27 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000936

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000936: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
([],4)
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
([(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [2,2]
 => [2]
 => 0
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([(2,3),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(3,2),(4,3)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([(0,3),(3,4),(4,1),(4,2)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([(1,4),(2,3),(3,4)],5)
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([(0,4),(3,2),(4,1),(4,3)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [2,2,1]
 => [2,1]
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [3,2]
 => [2]
 => 0
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(3,4),(3,5)],6)
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(2,3),(2,4),(2,5)],6)
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(1,2),(1,3),(1,4),(1,5)],6)
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0

Description

The number of even values of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugace class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $4$. It is shown in [1] that the sum of the values of the statistic over all partitions of a given size is even.

Matching statistic: St001570
(load all 2 compositions to match this statistic)

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 10%●distinct values known / distinct values provided: 4%

Values

([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 0
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,2),(1,3),(1,4)],5)
 => ([(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)
 => 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(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)
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0
([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(3,4),(3,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(2,3),(2,4),(2,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => ? = 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1)],2)
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => ? = 1
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 0
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 2

Description

The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.

Matching statistic: St000264
(load all 2 compositions to match this statistic)

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 10%●distinct values known / distinct values provided: 4%

Values

([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => ? = 0 + 3
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 3
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(1,2),(1,3),(1,4)],5)
 => ([(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)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(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)
 => 3 = 0 + 3
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => 3 = 0 + 3
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => 3 = 0 + 3
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => ? = 2 + 3
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 3
([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
([(3,4),(3,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
([(2,3),(2,4),(2,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => ? = 1 + 3
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2 + 3
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2 + 3
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2 + 3
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2 + 3
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 2 + 3
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2 + 3
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1)],2)
 => ? = 2 + 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 1 + 3
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 1 + 3
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => ? = 1 + 3
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 2 + 3
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 0 + 3
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 2 + 3

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

Matching statistic: St001629

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 4% ●values known / values provided: 10%●distinct values known / distinct values provided: 4%

Values

([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => [1] => ? = 0
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => 0
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(1,2),(1,3),(1,4)],5)
 => ([(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)
 => [3,1] => 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(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)
 => [3,1] => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => [3,1] => 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => [3,1] => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => [1] => ? = 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1] => 0
([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => 0
([(3,4),(3,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => 0
([(2,3),(2,4),(2,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => 0
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [2,1] => 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 2
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 2
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 2
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 2
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 1
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 2
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 0
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1)],2)
 => [1,1] => ? = 2

Description

The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.

Matching statistic: St001603

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 10%●distinct values known / distinct values provided: 4%

Values

([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => [1]
 => ? = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(2,3),(3,4)],5)
 => ([(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 + 1
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => [1]
 => ? = 2 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(3,4),(3,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(2,3),(2,4),(2,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 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)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1

Description

The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St001604

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 10%●distinct values known / distinct values provided: 4%

Values

([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => [1]
 => ? = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(2,3),(3,4)],5)
 => ([(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 + 1
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => [1]
 => ? = 2 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(3,4),(3,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(2,3),(2,4),(2,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 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)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1

Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St001605

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 10%●distinct values known / distinct values provided: 4%

Values

([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],1)
 => [1]
 => ? = 0 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(2,3),(3,4)],5)
 => ([(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 + 1
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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)
 => [4]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([],1)
 => [1]
 => ? = 2 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => 1 = 0 + 1
([(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(3,4),(3,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(2,3),(2,4),(2,5)],6)
 => ([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1 = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 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)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 1 + 1
([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 0 + 1
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1)],2)
 => [2]
 => ? = 2 + 1

Description

The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St001001

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 6%●distinct values known / distinct values provided: 4%

Values

([],3)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
([],4)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 0
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
([(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 0
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 0
([],5)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(1,4),(3,2),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([(0,3),(3,4),(4,1),(4,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([(1,4),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([(0,4),(3,2),(4,1),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0
([(3,4),(3,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0
([(2,3),(2,4),(2,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0
([(1,2),(1,3),(1,4),(1,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(1,3),(1,4),(1,5),(5,2)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(2,3),(2,4),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(1,4),(1,5),(5,2),(5,3)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(2,3),(2,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
([(1,4),(1,5),(4,3),(5,2)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 0
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(3,4),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(2,3),(3,4),(3,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(1,5),(5,2),(5,3),(5,4)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 0
([(1,5),(2,5),(5,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
([(0,5),(1,5),(2,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0

Description

The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001371

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 6%●distinct values known / distinct values provided: 4%

Values

([],3)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
([],4)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? = 0
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([],5)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,4),(3,2),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,3),(3,4),(4,1),(4,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(1,4),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(3,2),(4,1),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => ? = 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 10111111000000 => ? = 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(3,4),(3,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(2,3),(2,4),(2,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(1,2),(1,3),(1,4),(1,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,3),(1,4),(1,5),(5,2)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(2,3),(2,4),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,4),(1,5),(5,2),(5,3)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(2,3),(2,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
([(1,4),(1,5),(4,3),(5,2)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(3,4),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(2,3),(3,4),(3,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,5),(5,2),(5,3),(5,4)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,5),(2,5),(5,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,5),(2,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0

Description

The length of the longest Yamanouchi prefix of a binary word. This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

Matching statistic: St001730

Mapping

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 6%●distinct values known / distinct values provided: 4%

Values

([],3)
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0
([],4)
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0
([(0,2),(0,3),(3,1)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(0,3),(3,1),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => ? = 0
([(0,3),(1,2),(2,3)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 0
([],5)
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => ? = 0
([(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(2,3),(2,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(1,2),(1,3),(1,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,3),(1,4),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,3),(0,4),(4,1),(4,2)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,4),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(1,3),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(1,4),(3,2),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,3),(3,4),(4,1),(4,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(1,4),(2,3),(3,4)],5)
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0
([(0,4),(1,2),(2,4),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(3,2),(4,1),(4,3)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 111110000010 => ? = 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => ? = 0
([],6)
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 10111111000000 => ? = 0
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(3,4),(3,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(2,3),(2,4),(2,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(1,2),(1,3),(1,4),(1,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 101111010000 => ? = 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,3),(1,4),(1,5),(5,2)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(2,3),(2,4),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,4),(1,5),(5,2),(5,3)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(2,3),(2,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
([(1,4),(1,5),(4,3),(5,2)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => ? = 0
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(3,4),(4,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(2,3),(3,4),(3,5)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,5),(5,2),(5,3),(5,4)],6)
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0
([(1,5),(2,5),(5,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0
([(0,5),(1,5),(2,3),(5,4)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0

Description

The number of times the path corresponding to a binary word crosses the base line. Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.

The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001556The number of inversions of the third entry of a permutation.