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Your data matches 114 different statistics following compositions of up to 3 maps.
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Matching statistic: St000717
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
Description
The number of ordinal summands of a poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the length of the longest ordinal decomposition of a poset.
Matching statistic: St001637
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 75% ●values known / values provided: 92%●distinct values known / distinct values provided: 75%
Values
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 4 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 5 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 5 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 5 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3,2],[2,1]]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,1,1],[2,2]]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ? = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,2,1],[2,2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[5,5,4,1],[4,3]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[6,4,1],[3]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,2,2],[2,2,1,1,1]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[4,3,2,2,2],[2,1,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[4,4,4,2,2],[3,3,1,1]]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[4,3,3,3,2],[2,2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? = 4 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[6,6,3],[5,2]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[6,5,4],[4,3]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 5 - 1
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001624
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Values
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1,1],[1,1]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(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)
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(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)
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(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)
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3,1],[2]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,2],[2,2,1,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(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)
=> ? = 5 - 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(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)
=> ? = 5 - 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(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)
=> ? = 5 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[5,5,3],[4,2]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [[5,5,3],[3,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[5,4,3],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [[5,5,3],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [[5,4,2],[2]]
=> ([(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)
=> ? = 4 - 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3,2],[2,1]]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(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)
=> ? = 5 - 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3,2],[1,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4,2],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3,2],[2]]
=> ([(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)
=> ? = 4 - 1
[1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [[5,4,3],[2]]
=> ([(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)
=> ? = 4 - 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [[4,4,3,3],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1,1,1],[1,1,1]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1,1,1],[1,1]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(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)
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,1,1],[1,1,1]]
=> ([(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)
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,1,1],[2,2]]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3,1,1],[2]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,2,1],[2,2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(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)
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[5,5,4,1],[4,3]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[6,4,1],[3]]
=> ([(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)
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2,2,2],[2,2,1,1,1]]
=> ([(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)
=> ? = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[4,3,2,2,2],[2,1,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,2,2],[2,2,2,1,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[4,4,4,2,2],[3,3,1,1]]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[4,3,3,3,2],[2,2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[4,4,4,3,3],[3,3,2,2]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[5,5,5,3],[4,4,2]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[6,6,3],[5,2]]
=> ([(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)
=> ? = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[6,5,4],[4,3]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(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)
=> ? = 5 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[6,6,4],[5,3]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[7,4],[3]]
=> ([(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)
=> ? = 4 - 1
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001630
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Values
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(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)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [[5,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [[5,4],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(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)
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[6,4],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001877
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 34%●distinct values known / distinct values provided: 25%
Values
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(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)
=> ? = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(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)
=> ? = 4 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(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)
=> ? = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(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)
=> ? = 4 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [[5,3],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [[5,4],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(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)
=> ? = 4 - 2
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2 - 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 3 - 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[6,4],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St000261
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000261: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 75%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000261: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 75%
Values
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,3),(0,6),(1,2),(1,6),(2,4),(3,5),(6,4),(6,5)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(2,6),(4,2),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(3,5),(4,2),(4,6),(7,5),(7,6)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(2,6),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(3,6),(4,2),(4,7),(6,5),(7,5),(8,6),(8,7)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(2,7),(3,5),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [[5,3],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(6,4),(7,4),(7,5)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> ([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(2,6),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [[5,4],[1]]
=> ([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(4,6),(7,6)],8)
=> ([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2],[3]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(4,6),(7,6)],8)
=> ([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(6,4),(7,4),(7,5)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(3,5),(4,2),(4,6),(7,5),(7,6)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1]]
=> ([(0,3),(0,4),(1,2),(1,8),(2,6),(3,7),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,6),(0,8),(1,4),(1,8),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(2,5),(3,6),(4,2),(4,7),(7,5),(8,6),(8,7)],9)
=> ([(0,6),(0,8),(1,4),(1,8),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(2,6),(4,2),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [[3,3,2,2],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(2,5),(3,6),(4,2),(4,7),(7,5),(8,6),(8,7)],9)
=> ([(0,6),(0,8),(1,4),(1,8),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2],[1]]
=> ([(0,3),(0,9),(1,4),(1,9),(2,5),(3,2),(3,8),(4,7),(7,6),(8,5),(8,6),(9,7),(9,8)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,6),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,9),(5,7),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,7),(1,3),(1,4),(2,6),(2,7),(3,5),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
Description
The edge connectivity of a graph.
This is the minimum number of edges that has to be removed to make the graph disconnected.
Matching statistic: St000262
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000262: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 75%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000262: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 75%
Values
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,3),(0,6),(1,2),(1,6),(2,4),(3,5),(6,4),(6,5)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(2,6),(4,2),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(3,5),(4,2),(4,6),(7,5),(7,6)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(2,6),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(3,6),(4,2),(4,7),(6,5),(7,5),(8,6),(8,7)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(2,7),(3,5),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [[5,3],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(6,4),(7,4),(7,5)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> ([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(2,6),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [[5,4],[1]]
=> ([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(4,6),(7,6)],8)
=> ([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2],[3]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(4,6),(7,6)],8)
=> ([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(6,4),(7,4),(7,5)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(3,5),(4,2),(4,6),(7,5),(7,6)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1]]
=> ([(0,3),(0,4),(1,2),(1,8),(2,6),(3,7),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,6),(0,8),(1,4),(1,8),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(2,5),(3,6),(4,2),(4,7),(7,5),(8,6),(8,7)],9)
=> ([(0,6),(0,8),(1,4),(1,8),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(2,6),(4,2),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [[3,3,2,2],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(2,5),(3,6),(4,2),(4,7),(7,5),(8,6),(8,7)],9)
=> ([(0,6),(0,8),(1,4),(1,8),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2],[1]]
=> ([(0,3),(0,9),(1,4),(1,9),(2,5),(3,2),(3,8),(4,7),(7,6),(8,5),(8,6),(9,7),(9,8)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,6),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,9),(5,7),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,7),(1,3),(1,4),(2,6),(2,7),(3,5),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
Description
The vertex connectivity of a graph.
For non-complete graphs, this is the minimum number of vertices that has to be removed to make the graph disconnected.
Matching statistic: St000310
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000310: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 75%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000310: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 75%
Values
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,3),(0,6),(1,2),(1,6),(2,4),(3,5),(6,4),(6,5)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(2,6),(4,2),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(3,5),(4,2),(4,6),(7,5),(7,6)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> ([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(2,6),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(3,6),(4,2),(4,7),(6,5),(7,5),(8,6),(8,7)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[4,2,2],[1,1]]
=> ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2],[1]]
=> ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,6),(1,3),(1,4),(3,5),(3,6),(4,2),(4,5)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(2,7),(3,5),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[4,3,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [[5,3],[1]]
=> ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [[4,4,3],[2,1]]
=> ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(6,4),(7,4),(7,5)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> ([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(2,6),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [[5,4],[1]]
=> ([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1]]
=> ([(0,4),(0,6),(1,2),(1,3),(3,6),(4,5),(6,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5),(3,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2],[1]]
=> ([(0,3),(0,6),(1,4),(1,6),(3,5),(4,2),(6,5)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(4,6),(7,6)],8)
=> ([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,4),(0,6),(1,2),(1,3),(2,5),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2],[3]]
=> ([(0,6),(1,3),(1,4),(2,6),(3,5),(4,2),(4,5)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [[4,4,2],[2]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(4,6),(7,6)],8)
=> ([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,6),(1,3),(1,7),(2,6),(2,7),(3,5),(6,4),(7,4),(7,5)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(3,5),(4,2),(4,6),(7,5),(7,6)],8)
=> ([(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,6),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1]]
=> ([(0,3),(0,4),(1,2),(1,8),(2,6),(3,7),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,6),(0,8),(1,4),(1,8),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(2,5),(3,6),(4,2),(4,7),(7,5),(8,6),(8,7)],9)
=> ([(0,6),(0,8),(1,4),(1,8),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2],[1]]
=> ([(0,3),(0,7),(1,4),(1,7),(2,6),(4,2),(4,5),(5,6),(7,5)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [[3,3,2,2],[2]]
=> ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [[3,3,2,2],[1]]
=> ([(0,3),(0,8),(1,4),(1,8),(2,5),(3,6),(4,2),(4,7),(7,5),(8,6),(8,7)],9)
=> ([(0,6),(0,8),(1,4),(1,8),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,8),(4,7),(4,8),(7,5),(8,5),(8,6)],9)
=> ([(0,8),(1,5),(1,7),(2,3),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2],[1]]
=> ([(0,3),(0,9),(1,4),(1,9),(2,5),(3,2),(3,8),(4,7),(7,6),(8,5),(8,6),(9,7),(9,8)],10)
=> ([(0,5),(0,9),(1,4),(1,8),(2,6),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,9),(5,7),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
=> ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,7),(1,3),(1,4),(2,6),(2,7),(3,5),(4,2),(4,5),(5,6)],8)
=> ([(0,7),(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 2 - 1
Description
The minimal degree of a vertex of a graph.
Matching statistic: St000793
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 3
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13}}
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13}}
=> ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12}}
=> ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11}}
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13}}
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12}}
=> ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12}}
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11}}
=> ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> {{1,2,3},{4,5,6},{7,8},{9,10}}
=> ? = 4
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11}}
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> {{1,2,3,4},{5,6},{7,8},{9,10}}
=> ? = 3
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12}}
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12}}
=> ? = 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11}}
=> ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> {{1,2,3},{4,5,6},{7,8,9},{10}}
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11}}
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> {{1,2,3,4},{5,6,7,8},{9},{10}}
=> ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 2
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 2
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> {{1,2,3,4,5,6},{7,8}}
=> 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition.
To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$.
Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row.
Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$.
This statistic is the length of the longest partition on the diagonal of the diagram.
Matching statistic: St000232
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 2 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 3 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 2 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 2 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 2 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 3 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 3 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 2 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 2 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 3 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 4 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 3 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 4 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 2 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 3 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 2 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 2 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 3 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 3 - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 2 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 3 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 3 - 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 3 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 2 - 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 2 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13}}
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13}}
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12}}
=> ? = 2 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12}}
=> ? = 4 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12}}
=> ? = 3 - 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11}}
=> ? = 2 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> {{1,2,3},{4,5,6},{7,8},{9,10}}
=> ? = 4 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11}}
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> {{1,2,3,4},{5,6},{7,8},{9,10}}
=> ? = 3 - 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> ? = 2 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 0 = 2 - 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12}}
=> ? = 2 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12}}
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11}}
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> {{1,2,3},{4,5,6},{7,8,9},{10}}
=> ? = 2 - 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11}}
=> ? = 2 - 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 2 - 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10]]
=> {{1,2,3,4},{5,6,7,8},{9},{10}}
=> ? = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 2 - 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 2 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 2 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> 0 = 2 - 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 2 - 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 0 = 2 - 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 2 - 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 2 - 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> 0 = 2 - 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 2 - 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 2 - 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> {{1,2,3,4,5,6},{7,8}}
=> 0 = 2 - 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 2 - 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 2 - 2
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
The following 104 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000730The maximal arc length of a set partition. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000058The order of a permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000091The descent variation of a composition. St000234The number of global ascents of a permutation. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000632The jump number of the poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001330The hat guessing number of a graph. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000732The number of double deficiencies of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000670The reversal length of a permutation. St000485The length of the longest cycle of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001741The largest integer such that all patterns of this size are contained in the permutation. St000056The decomposition (or block) number of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000570The Edelman-Greene number of a permutation. St000652The maximal difference between successive positions of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001729The number of visible descents of a permutation. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000709The number of occurrences of 14-2-3 or 14-3-2. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001381The fertility of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001536The number of cyclic misalignments of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset.
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