Your data matches 57 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001534
(load all 2 compositions to match this statistic)
Mapping
St001534: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => 0
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => -1
([(0,1),(0,2)],3)
 => 0
([(0,2),(2,1)],3)
 => 1
([(0,2),(1,2)],3)
 => 0
([],4)
 => 0
([(2,3)],4)
 => 2
([(1,2),(1,3)],4)
 => 0
([(0,1),(0,2),(0,3)],4)
 => 0
([(0,2),(0,3),(3,1)],4)
 => -1
([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(1,2),(2,3)],4)
 => -2
([(0,3),(3,1),(3,2)],4)
 => 0
([(1,3),(2,3)],4)
 => 0
([(0,3),(1,3),(3,2)],4)
 => 0
([(0,3),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2)],4)
 => -2
([(0,3),(1,2),(1,3)],4)
 => -1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
([(0,3),(2,1),(3,2)],4)
 => 1
([(0,3),(1,2),(2,3)],4)
 => -1
([],5)
 => 0
([(3,4)],5)
 => -6
([(2,3),(2,4)],5)
 => 0
([(1,2),(1,3),(1,4)],5)
 => 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(4,2)],5)
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0
([(0,3),(0,4),(3,2),(4,1)],5)
 => -2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => -1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 0
([(2,3),(3,4)],5)
 => 6
([(1,4),(4,2),(4,3)],5)
 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => 0
([(2,4),(3,4)],5)
 => 0
([(1,4),(2,4),(4,3)],5)
 => 0
([(0,4),(1,4),(4,2),(4,3)],5)
 => 0
([(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
Description
The alternating sum of the coefficients of the Poincare polynomial of the poset cone. For a poset $P$ on $\{1,\dots,n\}$, let $\mathcal K_P = \{\vec x\in\mathbb R^n| x_i < x_j \text{ for } i < _P j\}$. Furthermore let $\mathcal L(\mathcal A)$ be the intersection lattice of the braid arrangement $A_{n-1}$ and let $\mathcal L^{int} = \{ X \in \mathcal L(\mathcal A) | X \cap \mathcal K_P \neq \emptyset \}$. Then the Poincare polynomial of the poset cone is $Poin(t) = \sum_{X\in\mathcal L^{int}} |\mu(0, X)| t^{codim X}$. This statistic records its $Poin(-1)$.
Matching statistic: St000205
(load all 3 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 44%●distinct values known / distinct values provided: 6%
Values
([],1)
 => ([],1)
 => [1]
 => []
 => ? = 1
([],2)
 => ([(0,1)],2)
 => [2]
 => []
 => ? = 1
([(0,1)],2)
 => ([],2)
 => [1,1]
 => [1]
 => 0
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,1}
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,1}
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => [1,1]
 => 0
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 0
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [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)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [4,1,1]
 => [1,1]
 => 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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 2 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 44%●distinct values known / distinct values provided: 6%
Values
([],1)
 => ([],1)
 => [1]
 => []
 => ? = 1
([],2)
 => ([(0,1)],2)
 => [2]
 => []
 => ? = 1
([(0,1)],2)
 => ([],2)
 => [1,1]
 => [1]
 => 0
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,1}
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,1}
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => [1,1]
 => 0
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 0
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [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)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [4,1,1]
 => [1,1]
 => 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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St001175
(load all 2 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 44%●distinct values known / distinct values provided: 11%
Values
([],1)
 => ([],1)
 => [1]
 => []
 => ? = 1
([],2)
 => ([(0,1)],2)
 => [2]
 => []
 => ? = 1
([(0,1)],2)
 => ([],2)
 => [1,1]
 => [1]
 => 0
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,1}
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,1}
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => [1,1]
 => 0
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 0
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [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)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [4,1,1]
 => [1,1]
 => 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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
Description
The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001912
(load all 2 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001912: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 44%●distinct values known / distinct values provided: 22%
Values
([],1)
 => ([],1)
 => [1]
 => []
 => ? = 1
([],2)
 => ([(0,1)],2)
 => [2]
 => []
 => ? = 1
([(0,1)],2)
 => ([],2)
 => [1,1]
 => [1]
 => 0
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,1}
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,1}
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
([(0,2),(2,1)],3)
 => ([],3)
 => [1,1,1]
 => [1,1]
 => 0
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,1}
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,1}
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,1}
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,1}
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,1}
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,1}
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,1}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 0
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [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)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(1,2),(1,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,3,3,3,6,6}
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 2
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 2
([],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]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [5,1]
 => [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)
 => [4,1,1]
 => [1,1]
 => 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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(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)
 => [6]
 => []
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
Description
The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.
Matching statistic: St001095
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001095: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 43%●distinct values known / distinct values provided: 6%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-1,0,1}
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-1,0,1}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-1,0,1}
([],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,1,2}
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,1,2}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,1,2}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,1,2}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,1,2}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,1,2}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,1,2}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,1,2}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,1,2}
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(2,3),(2,4),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
([(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
Description
The number of non-isomorphic posets with precisely one further covering relation.
Matching statistic: St000455
(load all 7 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 39%●distinct values known / distinct values provided: 6%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
([],2)
 => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => ? ∊ {0,1}
([(0,1)],2)
 => ([],2)
 => ([],2)
 => ([],2)
 => ? ∊ {0,1}
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ? ∊ {-1,0,0,1}
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => ? ∊ {-1,0,0,1}
([(0,2),(2,1)],3)
 => ([],3)
 => ([],3)
 => ([],3)
 => ? ∊ {-1,0,0,1}
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([],2)
 => ([],2)
 => ? ∊ {-1,0,0,1}
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([],3)
 => ([],3)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ([],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([],2)
 => ([],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([],4)
 => ([],4)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,1,2}
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([],2)
 => ([],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ([],3)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([],4)
 => ([],4)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(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)
 => ([],2)
 => ([],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(1,4),(2,3)],5)
 => ([],3)
 => ([],3)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ([],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(1,4),(3,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(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)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 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)
 => ([(1,3),(2,3)],4)
 => ([(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)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 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)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 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)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 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)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 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)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 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)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(1,4),(1,5),(5,2),(5,3)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(1,4),(1,5),(4,3),(5,2)],6)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
([(1,5),(5,2),(5,3),(5,4)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(1,4),(4,5),(5,2),(5,3)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(1,5),(2,5),(5,3),(5,4)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St001570
(load all 3 compositions to match this statistic)
Mapping
Mp00074: Posets —to graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 36%●distinct values known / distinct values provided: 6%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 1
([],2)
 => ([],2)
 => ([],2)
 => ([],1)
 => ? ∊ {0,1}
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,1}
([],3)
 => ([],3)
 => ([],3)
 => ([],1)
 => ? ∊ {-1,0,0,0,1}
([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {-1,0,0,0,1}
([(0,1),(0,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {-1,0,0,0,1}
([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {-1,0,0,0,1}
([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {-1,0,0,0,1}
([],4)
 => ([],4)
 => ([],4)
 => ([],1)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(2,3)],4)
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(1,2),(1,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,1),(0,2),(0,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,2),(0,3),(3,1)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,0,0,0,0,1,2}
([],5)
 => ([],5)
 => ([],5)
 => ([],1)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,3),(2,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(1,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,3),(1,4),(4,2)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],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),(2,4),(3,4),(4,1)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],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
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(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),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(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)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],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)
 => ([(0,1),(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
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],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)
 => ([(0,1),(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
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(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)
 => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(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
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(2,3),(2,4),(3,5),(4,5)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],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
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(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)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],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)
 => ([(0,1),(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
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
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: St000175
(load all 2 compositions to match this statistic)
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 33%●distinct values known / distinct values provided: 17%
Values
([],1)
 => [1]
 => []
 => ? = 1
([],2)
 => [2]
 => []
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => [3,3]
 => [3]
 => 0
([(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,0,0,1}
([(0,1),(0,2)],3)
 => [2]
 => []
 => ? ∊ {-1,0,0,1}
([(0,2),(2,1)],3)
 => [1]
 => []
 => ? ∊ {-1,0,0,1}
([(0,2),(1,2)],3)
 => [2]
 => []
 => ? ∊ {-1,0,0,1}
([],4)
 => [4,4,4,4,4,4]
 => [4,4,4,4,4]
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(2,3)],4)
 => [4,4,4]
 => [4,4]
 => 0
([(1,2),(1,3)],4)
 => [8]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [3]
 => 0
([(0,2),(0,3),(3,1)],4)
 => [3]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,3),(3,1),(3,2)],4)
 => [2]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(1,3),(2,3)],4)
 => [8]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,3),(1,3),(3,2)],4)
 => [2]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [3]
 => 0
([(0,3),(1,2)],4)
 => [4,2]
 => [2]
 => 0
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2]
 => 0
([(0,3),(2,1),(3,2)],4)
 => [1]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [5,5,5,5,5,5,5,5,5,5,5]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [10,10,10]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [15]
 => 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [4,4,4,4,4]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [4,4]
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 0
([(1,3),(1,4),(4,2)],5)
 => [15]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [5]
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [2]
 => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [2]
 => 0
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [5,5,5]
 => 0
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [5]
 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [3]
 => 0
([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [10,10,10]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [5]
 => 0
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [2]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [15]
 => 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [3]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [4,4,4,4,4]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [10]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [4]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [6]
 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [4,4]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [4,4]
 => 0
([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [5,5,5,5,5]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [5,5]
 => 0
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [4]
 => 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [5,5,5]
 => 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [2]
 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
 => [7]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [10]
 => 0
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [4,4]
 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [4,3]
 => 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [4]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [6]
 => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [3]
 => 0
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [4]
 => 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(3,2),(4,3)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(3,4)],5)
 => [15]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [5]
 => 0
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [2]
 => 0
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,2,2,2,2,2,2,3,3,3,6,6}
([],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ?
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(4,5)],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ?
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(3,4),(3,5)],6)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ?
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [15,15]
 => [15]
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [3]
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [10,10]
 => [10]
 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [12,4]
 => [4]
 => 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6]
 => [6]
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [10,4,4]
 => [4,4]
 => 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [5]
 => 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [15,5,5]
 => [5,5]
 => 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,5,5,5]
 => [5,5,5]
 => 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [15,15]
 => [15]
 => 0
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial $$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$ The statistic of the degree of this polynomial. For example, the partition $(3, 2, 1, 1, 1)$ gives $$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$ which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$. This is the same as the number of unordered pairs of different parts, which follows from: $$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St001283
Mapping
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 33%●distinct values known / distinct values provided: 6%
Values
([],1)
 => [1]
 => []
 => ? = 1
([],2)
 => [2]
 => []
 => ? ∊ {0,1}
([(0,1)],2)
 => [1]
 => []
 => ? ∊ {0,1}
([],3)
 => [3,3]
 => [3]
 => 0
([(1,2)],3)
 => [3]
 => []
 => ? ∊ {-1,0,0,1}
([(0,1),(0,2)],3)
 => [2]
 => []
 => ? ∊ {-1,0,0,1}
([(0,2),(2,1)],3)
 => [1]
 => []
 => ? ∊ {-1,0,0,1}
([(0,2),(1,2)],3)
 => [2]
 => []
 => ? ∊ {-1,0,0,1}
([],4)
 => [4,4,4,4,4,4]
 => [4,4,4,4,4]
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(2,3)],4)
 => [4,4,4]
 => [4,4]
 => 0
([(1,2),(1,3)],4)
 => [8]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [3]
 => 0
([(0,2),(0,3),(3,1)],4)
 => [3]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,3),(3,1),(3,2)],4)
 => [2]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(1,3),(2,3)],4)
 => [8]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,3),(1,3),(3,2)],4)
 => [2]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [3]
 => 0
([(0,3),(1,2)],4)
 => [4,2]
 => [2]
 => 0
([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2]
 => 0
([(0,3),(2,1),(3,2)],4)
 => [1]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => ? ∊ {-2,-2,-1,-1,-1,0,0,0,1,2}
([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => [5,5,5,5,5,5,5,5,5,5,5]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [10,10,10]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [15]
 => 0
([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [4,4,4,4,4]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [4,4]
 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [3]
 => 0
([(1,3),(1,4),(4,2)],5)
 => [15]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [5]
 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [2]
 => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [2]
 => 0
([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [5,5,5]
 => 0
([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [5]
 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [3]
 => 0
([(2,4),(3,4)],5)
 => [10,10,10,10]
 => [10,10,10]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [5]
 => 0
([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [2]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => [15]
 => 0
([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => [3]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => [4,4,4,4,4]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => [10]
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => [4]
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [6]
 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => [4,4]
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [4,4]
 => 0
([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => [5,5,5,5,5]
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => [5,5]
 => 0
([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [4]
 => 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [5,5,5]
 => 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [2]
 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
 => [7]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [10]
 => 0
([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => [4,4]
 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [4,3]
 => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [8]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => [4]
 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [6]
 => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [3]
 => 0
([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [4]
 => 0
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(3,2),(4,3)],5)
 => [5]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(1,4),(2,3),(3,4)],5)
 => [15]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [5]
 => 0
([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,2),(2,4),(3,4)],5)
 => [4,2]
 => [2]
 => 0
([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => []
 => ? ∊ {-6,-3,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,2,2,2,2,2,2,3,3,3,6,6}
([],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ?
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(4,5)],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ?
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(3,4),(3,5)],6)
 => [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12]
 => ?
 => ? ∊ {-24,-24,-12,-12,-12,-8,-8,-8,-8,-8,-8,-6,-6,-6,-6,-6,-6,-4,-4,-4,-4,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,8,8,8,8,8,12,24}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => [15,15]
 => [15]
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => [3]
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => [10,10]
 => [10]
 => 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
 => [12,4]
 => [4]
 => 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6]
 => [6]
 => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
 => [10,4,4]
 => [4,4]
 => 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
 => [5,5]
 => [5]
 => 0
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => [15,5,5]
 => [5,5]
 => 0
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [5,5,5,5]
 => [5,5,5]
 => 0
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
 => [15,15]
 => [15]
 => 0
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers. A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers. The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001284The number of finite groups that are realised by the given partition over the complex numbers. St001525The number of symmetric hooks on the diagonal of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000225Difference between largest and smallest parts in a partition. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000478Another weight of a partition according to Alladi. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000117The number of centered tunnels of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001651The Frankl number of a lattice. St000477The weight of a partition according to Alladi. St000928The sum of the coefficients of the character polynomial of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000268The number of strongly connected orientations of a graph. St000344The number of strongly connected outdegree sequences of a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001363The Euler characteristic of a graph according to Knill. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.