Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000619
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0]
 => [3,1,2] => [3,2,1] => [3,2,1] => 2
[1,1,0,0]
 => [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,3,1,2] => [1,4,3,2] => 3
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,3,2,1,4] => [3,5,2,1,4] => 3
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,4,2,1,3] => [2,5,4,1,3] => 3
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [5,3,4,2,1] => [3,5,4,2,1] => 3
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,2,1,3,4] => [2,1,5,3,4] => 3
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,4,3,1,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,3,1,2,4] => [1,3,5,2,4] => 2
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [5,4,2,3,1] => [2,5,4,3,1] => 3
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,2,5,3,1] => [4,5,2,3,1] => 2
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,3,1,4] => [2,3,5,1,4] => 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,3,4,1,2] => [1,3,5,4,2] => 3
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,4,1] => [2,3,5,4,1] => 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,4,3,2,1,5] => [4,6,3,2,1,5] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,5,3,2,1,4] => [3,6,5,2,1,4] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [6,4,5,3,2,1] => [4,6,5,3,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,3,2,1,4,5] => [3,2,6,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,5,4,2,1,3] => [2,6,5,4,1,3] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,4,2,1,3,5] => [2,4,6,1,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 4
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [5,3,6,4,2,1] => [5,6,3,4,2,1] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [6,3,4,2,1,5] => [3,4,6,2,1,5] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,5,2,1,3,4] => [2,1,6,5,3,4] => 4
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [6,4,5,2,1,3] => [2,4,6,5,1,3] => 3
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [6,3,4,5,2,1] => [3,4,6,5,2,1] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,2,1,3,4,5] => [2,1,3,6,4,5] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => 5
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6,4,3,1,2,5] => [1,4,6,3,2,5] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6,5,3,1,2,4] => [1,3,6,5,2,4] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [6,4,5,3,1,2] => [1,4,6,5,3,2] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,3,1,2,4,5] => [1,3,2,6,4,5] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [6,5,4,2,3,1] => [2,6,5,4,3,1] => 4
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [6,4,2,3,1,5] => [2,4,6,3,1,5] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2,6,5,3,1] => [6,4,5,2,3,1] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => 2
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2,6,3,1,5] => [4,2,6,3,1,5] => 4
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [6,5,2,3,1,4] => [2,3,6,5,1,4] => 3
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [6,4,5,2,3,1] => [2,4,6,5,3,1] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,2,4,6,3,1] => [5,2,4,6,3,1] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,3,1,4,5] => [2,3,1,6,4,5] => 3
Description
The number of cyclic descents of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000871
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [3,2,1,4] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2 = 3 - 1
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [5,2,1,4,6,3] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [3,2,6,4,1,5] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [3,2,1,5,4,6] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [4,3,2,1,5,6] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [7,2,1,4,6,3,8,5] => ? = 3 - 1
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,5,4,3,6,8,7] => ? = 3 - 1
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [5,2,1,4,3,7,8,6] => ? = 3 - 1
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => [6,5,2,1,4,7,8,3] => ? = 3 - 1
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [3,2,6,4,8,5,1,7] => ? = 4 - 1
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => [3,2,1,4,7,6,5,8] => ? = 2 - 1
[1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [3,2,5,4,1,8,6,7] => ? = 3 - 1
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [3,2,1,5,4,7,6,8] => ? = 2 - 1
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [6,3,2,1,5,7,4,8] => ? = 2 - 1
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [4,3,8,5,2,1,6,7] => ? = 3 - 1
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [4,3,2,7,5,1,6,8] => ? = 3 - 1
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [4,3,2,1,6,5,7,8] => ? = 2 - 1
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [5,4,3,2,1,6,7,8] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => [9,2,1,4,6,3,8,5,10,7] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,7,8,6,5,10,9] => [2,1,7,4,3,6,8,5,10,9] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => [7,2,1,4,3,6,9,5,10,8] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => [8,7,2,1,4,6,9,3,10,5] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,5,6,4,3,8,7,10,9] => [2,1,5,4,8,6,3,7,10,9] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,5,6,4,3,9,10,8,7] => [5,2,1,4,9,6,3,8,10,7] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,5,7,4,8,6,3,10,9] => [2,1,5,4,3,7,6,8,10,9] => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => [5,2,1,4,3,7,6,9,10,8] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,5,8,4,9,10,7,6,3] => [8,5,2,1,4,7,3,9,10,6] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [2,1,6,7,8,5,4,3,10,9] => [2,1,6,5,4,3,7,8,10,9] => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => [6,2,1,5,4,3,7,9,10,8] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => [6,5,2,1,4,3,8,9,10,7] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => [7,6,5,2,1,4,8,9,10,3] => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [3,2,6,4,8,5,10,7,1,9] => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [3,4,2,1,6,5,9,10,8,7] => [3,2,1,4,6,5,9,8,7,10] => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [3,4,2,1,7,8,6,5,10,9] => [3,2,7,4,1,6,10,8,5,9] => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => [3,4,2,1,7,9,6,10,8,5] => [3,2,1,7,4,6,9,8,5,10] => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [3,4,2,1,8,9,10,7,6,5] => [8,3,2,1,4,7,9,6,5,10] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => [3,2,5,4,8,6,1,10,7,9] => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => [3,5,2,6,4,1,9,10,8,7] => [3,2,1,5,9,6,4,8,7,10] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [3,5,2,7,4,8,6,1,10,9] => [3,2,5,4,1,7,6,10,8,9] => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => [3,2,1,5,4,7,6,9,8,10] => ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [3,5,2,8,4,9,10,7,6,1] => [8,3,2,1,5,7,4,9,6,10] => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [3,6,2,7,8,5,4,1,10,9] => [3,2,6,5,4,1,7,10,8,9] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [3,6,2,7,9,5,4,10,8,1] => [6,3,2,5,4,1,7,9,8,10] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [3,6,2,8,9,5,10,7,4,1] => [6,3,2,1,5,4,8,9,7,10] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [3,7,2,8,9,10,6,5,4,1] => [7,6,3,2,1,5,8,9,4,10] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [4,5,6,3,2,1,8,7,10,9] => [4,3,8,5,10,6,2,1,7,9] => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [4,5,6,3,2,1,9,10,8,7] => [4,3,2,5,9,8,6,1,7,10] => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [4,5,7,3,2,8,6,1,10,9] => [4,3,7,5,2,10,6,1,8,9] => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => [4,3,2,7,5,9,6,1,8,10] => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [4,5,8,3,2,9,10,7,6,1] => [4,3,2,1,5,8,7,6,9,10] => ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [4,6,7,3,8,5,2,1,10,9] => [4,3,6,5,2,1,10,7,8,9] => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => [4,3,2,6,5,1,9,7,8,10] => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => [4,3,2,1,6,5,8,7,9,10] => ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => [7,4,3,2,1,6,8,5,9,10] => ? = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [5,6,7,8,4,3,2,1,10,9] => [5,4,10,6,3,2,1,7,8,9] => ? = 3 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => [5,4,3,9,6,2,1,7,8,10] => ? = 3 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [6,5,4,3,2,1,7,8,9,10] => 1 = 2 - 1
Description
The number of very big ascents of a permutation. A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$. For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
Matching statistic: St001491
(load all 2 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 20%
Values
[1,0]
 => [1] => [1] => => ? = 1 - 1
[1,0,1,0]
 => [1,2] => [2,1] => 0 => ? = 2 - 1
[1,1,0,0]
 => [2,1] => [1,2] => 1 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => 00 => ? = 3 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => 00 => ? = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,3,1] => 00 => ? = 3 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1,3] => 01 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,3,2] => 10 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => 000 => ? = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,3,1,2] => 000 => ? = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => 000 => ? = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [4,2,1,3] => 000 => ? = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [4,1,3,2] => 000 => ? = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [3,4,2,1] => 000 => ? = 4 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => 000 => ? = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,4,1] => 000 => ? = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,2,1,4] => 001 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => 000 => ? = 2 - 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,4,3,1] => 000 => ? = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 000 => ? = 3 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,4,3] => 010 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,4,3,2] => 100 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => 0000 => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,4,3,1,2] => 0000 => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,4,2,3,1] => 0000 => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [5,4,2,1,3] => 0000 => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [5,4,1,3,2] => 0000 => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,3,4,2,1] => 0000 => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [5,3,4,1,2] => 0000 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,3,2,4,1] => 0000 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [5,3,2,1,4] => 0000 => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [5,3,1,4,2] => 0000 => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [5,2,4,3,1] => 0000 => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [5,2,4,1,3] => 0000 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [5,2,1,4,3] => 0000 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [5,1,4,3,2] => 0000 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [4,5,3,2,1] => 0000 => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0000 => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [4,5,2,3,1] => 0000 => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [4,5,2,1,3] => 0000 => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [4,5,1,3,2] => 0000 => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [4,3,5,2,1] => 0000 => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,3,5,1,2] => 0000 => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,3,2,5,1] => 0000 => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [4,3,2,1,5] => 0001 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,3,1,5,2] => 0000 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [4,2,5,3,1] => 0000 => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => 0000 => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [4,2,1,5,3] => 0000 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [4,1,5,3,2] => 0000 => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [3,5,4,2,1] => 0000 => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [3,5,4,1,2] => 0000 => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [3,5,2,4,1] => 0000 => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [3,5,2,1,4] => 0000 => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [3,5,1,4,2] => 0000 => ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [3,2,5,4,1] => 0000 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [3,2,5,1,4] => 0000 => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [3,2,1,5,4] => 0010 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [2,1,5,4,3] => 0100 => 1 = 2 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [1,5,4,3,2] => 1000 => 1 = 2 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000237
Mapping
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 60%
Values
[1,0]
 => [[1],[2]]
 => [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => [2,4,1,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [1,3,2,4] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => [1,2,4,3,6,5] => 2 = 3 - 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => [1,2,5,3,6,4] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => [1,3,6,5,2,4] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => [3,5,6,1,2,4] => [1,3,6,4,2,5] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [1,4,2,5,3,6] => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [2,4,6,8,1,3,5,7] => [1,2,4,8,7,5,3,6] => ? = 4 - 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [2,4,7,8,1,3,5,6] => [1,2,4,8,6,3,7,5] => ? = 3 - 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [2,5,6,8,1,3,4,7] => [1,2,5,3,6,4,8,7] => ? = 3 - 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => [2,5,7,8,1,3,4,6] => [1,2,5,3,7,4,8,6] => ? = 3 - 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => [2,6,7,8,1,3,4,5] => [1,2,6,3,7,4,8,5] => ? = 3 - 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [3,4,6,8,1,2,5,7] => [1,3,6,2,4,8,7,5] => ? = 4 - 1
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => [3,4,7,8,1,2,5,6] => [1,3,7,5,2,4,8,6] => ? = 2 - 1
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => [3,5,6,8,1,2,4,7] => [1,3,6,2,5,4,8,7] => ? = 3 - 1
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => [3,5,7,8,1,2,4,6] => [1,3,7,4,8,6,2,5] => ? = 2 - 1
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => [3,6,7,8,1,2,4,5] => [1,3,7,4,8,5,2,6] => ? = 2 - 1
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => [4,5,6,8,1,2,3,7] => [1,4,8,7,3,6,2,5] => ? = 3 - 1
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => [4,5,7,8,1,2,3,6] => [1,4,8,6,2,5,3,7] => ? = 3 - 1
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => [4,6,7,8,1,2,3,5] => [1,4,8,5,2,6,3,7] => ? = 2 - 1
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [2,4,6,8,10,1,3,5,7,9] => [1,2,4,8,5,10,9,7,3,6] => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => [2,4,6,9,10,1,3,5,7,8] => [1,2,4,9,7,3,6,5,10,8] => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => [2,4,7,8,10,1,3,5,6,9] => [1,2,4,8,5,10,9,6,3,7] => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => [2,4,7,9,10,1,3,5,6,8] => [1,2,4,9,6,3,7,5,10,8] => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => [2,4,8,9,10,1,3,5,6,7] => [1,2,4,9,6,3,8,5,10,7] => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => [2,5,6,8,10,1,3,4,7,9] => [1,2,5,10,9,7,3,6,4,8] => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => [2,5,6,9,10,1,3,4,7,8] => [1,2,5,10,8,4,9,7,3,6] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => [2,5,7,8,10,1,3,4,6,9] => [1,2,5,10,9,6,3,7,4,8] => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => [2,5,7,9,10,1,3,4,6,8] => [1,2,5,10,8,4,9,6,3,7] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => [2,5,8,9,10,1,3,4,6,7] => [1,2,5,10,7,3,8,4,9,6] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => [2,6,7,8,10,1,3,4,5,9] => [1,2,6,3,7,4,8,5,10,9] => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => [2,6,7,9,10,1,3,4,5,8] => [1,2,6,3,7,4,9,5,10,8] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => [2,6,8,9,10,1,3,4,5,7] => [1,2,6,3,8,4,9,5,10,7] => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => [2,7,8,9,10,1,3,4,5,6] => [1,2,7,3,8,4,9,5,10,6] => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => [3,4,6,8,10,1,2,5,7,9] => [1,3,6,2,4,8,5,10,9,7] => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => [3,4,6,9,10,1,2,5,7,8] => [1,3,6,2,4,9,7,5,10,8] => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => [3,4,7,8,10,1,2,5,6,9] => [1,3,7,2,4,8,5,10,9,6] => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => [3,4,7,9,10,1,2,5,6,8] => [1,3,7,2,4,9,6,5,10,8] => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [3,4,8,9,10,1,2,5,6,7] => [1,3,8,5,10,7,2,4,9,6] => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => [3,5,6,8,10,1,2,4,7,9] => [1,3,6,2,5,10,9,7,4,8] => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => [3,5,6,9,10,1,2,4,7,8] => [1,3,6,2,5,10,8,4,9,7] => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => [3,5,7,8,10,1,2,4,6,9] => [1,3,7,2,5,10,9,6,4,8] => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => [3,5,7,9,10,1,2,4,6,8] => [1,3,7,2,5,10,8,4,9,6] => ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => [3,5,8,9,10,1,2,4,6,7] => [1,3,8,4,9,6,2,5,10,7] => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => [3,6,7,8,10,1,2,4,5,9] => [1,3,7,2,6,4,8,5,10,9] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => [3,6,7,9,10,1,2,4,5,8] => [1,3,7,2,6,4,9,5,10,8] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => [3,6,8,9,10,1,2,4,5,7] => [1,3,8,4,9,5,10,7,2,6] => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => [3,7,8,9,10,1,2,4,5,6] => [1,3,8,4,9,5,10,6,2,7] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [[1,2,3,7,9],[4,5,6,8,10]]
 => [4,5,6,8,10,1,2,3,7,9] => [1,4,8,3,6,2,5,10,9,7] => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => [4,5,6,9,10,1,2,3,7,8] => [1,4,9,7,2,5,10,8,3,6] => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => [4,5,7,8,10,1,2,3,6,9] => [1,4,8,3,7,2,5,10,9,6] => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => [4,5,7,9,10,1,2,3,6,8] => [1,4,9,6,2,5,10,8,3,7] => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [[1,2,3,6,7],[4,5,8,9,10]]
 => [4,5,8,9,10,1,2,3,6,7] => [1,4,9,6,2,5,10,7,3,8] => ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => [4,6,7,8,10,1,2,3,5,9] => [1,4,8,3,7,2,6,5,10,9] => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [[1,2,3,5,8],[4,6,7,9,10]]
 => [4,6,7,9,10,1,2,3,5,8] => [1,4,9,5,10,8,3,7,2,6] => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [[1,2,3,5,7],[4,6,8,9,10]]
 => [4,6,8,9,10,1,2,3,5,7] => [1,4,9,5,10,7,2,6,3,8] => ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [[1,2,3,5,6],[4,7,8,9,10]]
 => [4,7,8,9,10,1,2,3,5,6] => [1,4,9,5,10,6,2,7,3,8] => ? = 2 - 1
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000253
Mapping
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 60%
Values
[1,0]
 => [[1],[2]]
 => {{1},{2}}
 => {{1},{2}}
 => 0 = 1 - 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 1 = 2 - 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => {{1,3,5},{2,4,6}}
 => {{1,4,5},{2,3,6}}
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => {{1,3,4},{2,5,6}}
 => {{1,5,6},{2,3,4}}
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => {{1,2,5},{3,4,6}}
 => {{1,2,4,5},{3,6}}
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => {{1,2,4},{3,5,6}}
 => {{1,2,5,6},{3,4}}
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => {{1,2,3},{4,5,6}}
 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => {{1,3,5,7},{2,4,6,8}}
 => {{1,4,5,8},{2,3,6,7}}
 => ? = 4 - 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => {{1,3,5,6},{2,4,7,8}}
 => {{1,4,5,6},{2,3,7,8}}
 => ? = 3 - 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => {{1,3,4,7},{2,5,6,8}}
 => {{1,5,8},{2,3,4,6,7}}
 => ? = 3 - 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => {{1,3,4,6},{2,5,7,8}}
 => {{1,5,6},{2,3,4,7,8}}
 => ? = 3 - 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => {{1,3,4,5},{2,6,7,8}}
 => {{1,6,7,8},{2,3,4,5}}
 => ? = 3 - 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => {{1,2,5,7},{3,4,6,8}}
 => {{1,2,4,5,8},{3,6,7}}
 => ? = 4 - 1
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => {{1,2,5,6},{3,4,7,8}}
 => {{1,2,4,5,6},{3,7,8}}
 => ? = 2 - 1
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => {{1,2,4,7},{3,5,6,8}}
 => {{1,2,5,8},{3,4,6,7}}
 => ? = 3 - 1
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => {{1,2,4,6},{3,5,7,8}}
 => {{1,2,5,6},{3,4,7,8}}
 => ? = 2 - 1
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => {{1,2,4,5},{3,6,7,8}}
 => {{1,2,6,7,8},{3,4,5}}
 => ? = 2 - 1
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => {{1,2,3,7},{4,5,6,8}}
 => {{1,2,3,5,8},{4,6,7}}
 => ? = 3 - 1
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => {{1,2,3,6},{4,5,7,8}}
 => {{1,2,3,5,6},{4,7,8}}
 => ? = 3 - 1
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => {{1,2,3,5},{4,6,7,8}}
 => {{1,2,3,6,7,8},{4,5}}
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => {{1,2,3,4},{5,6,7,8}}
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => {{1,3,5,7,9},{2,4,6,8,10}}
 => {{1,4,5,8,9},{2,3,6,7,10}}
 => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => {{1,3,5,7,8},{2,4,6,9,10}}
 => {{1,4,5,9,10},{2,3,6,7,8}}
 => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => {{1,3,5,6,9},{2,4,7,8,10}}
 => {{1,4,5,6,8,9},{2,3,7,10}}
 => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => {{1,3,5,6,8},{2,4,7,9,10}}
 => {{1,4,5,6,9,10},{2,3,7,8}}
 => ? = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => {{1,3,5,6,7},{2,4,8,9,10}}
 => {{1,4,5,6,7},{2,3,8,9,10}}
 => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => {{1,3,4,7,9},{2,5,6,8,10}}
 => {{1,5,8,9},{2,3,4,6,7,10}}
 => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => {{1,3,4,7,8},{2,5,6,9,10}}
 => {{1,5,9,10},{2,3,4,6,7,8}}
 => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => {{1,3,4,6,9},{2,5,7,8,10}}
 => {{1,5,6,8,9},{2,3,4,7,10}}
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => {{1,3,4,6,8},{2,5,7,9,10}}
 => {{1,5,6,9,10},{2,3,4,7,8}}
 => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => {{1,3,4,6,7},{2,5,8,9,10}}
 => {{1,5,6,7},{2,3,4,8,9,10}}
 => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => {{1,3,4,5,9},{2,6,7,8,10}}
 => {{1,6,8,9},{2,3,4,5,7,10}}
 => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => {{1,3,4,5,8},{2,6,7,9,10}}
 => {{1,6,9,10},{2,3,4,5,7,8}}
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => {{1,3,4,5,7},{2,6,8,9,10}}
 => {{1,6,7},{2,3,4,5,8,9,10}}
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => {{1,3,4,5,6},{2,7,8,9,10}}
 => {{1,7,8,9,10},{2,3,4,5,6}}
 => ? = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => {{1,2,5,7,9},{3,4,6,8,10}}
 => {{1,2,4,5,8,9},{3,6,7,10}}
 => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => {{1,2,5,7,8},{3,4,6,9,10}}
 => {{1,2,4,5,9,10},{3,6,7,8}}
 => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => {{1,2,5,6,9},{3,4,7,8,10}}
 => {{1,2,4,5,6,8,9},{3,7,10}}
 => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => {{1,2,5,6,8},{3,4,7,9,10}}
 => {{1,2,4,5,6,9,10},{3,7,8}}
 => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => {{1,2,5,6,7},{3,4,8,9,10}}
 => {{1,2,4,5,6,7},{3,8,9,10}}
 => ? = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => {{1,2,4,7,9},{3,5,6,8,10}}
 => {{1,2,5,8,9},{3,4,6,7,10}}
 => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => {{1,2,4,7,8},{3,5,6,9,10}}
 => {{1,2,5,9,10},{3,4,6,7,8}}
 => ? = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => {{1,2,4,6,9},{3,5,7,8,10}}
 => {{1,2,5,6,8,9},{3,4,7,10}}
 => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => {{1,2,4,6,8},{3,5,7,9,10}}
 => {{1,2,5,6,9,10},{3,4,7,8}}
 => ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => {{1,2,4,6,7},{3,5,8,9,10}}
 => {{1,2,5,6,7},{3,4,8,9,10}}
 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => {{1,2,4,5,9},{3,6,7,8,10}}
 => {{1,2,6,8,9},{3,4,5,7,10}}
 => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => {{1,2,4,5,8},{3,6,7,9,10}}
 => {{1,2,6,9,10},{3,4,5,7,8}}
 => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => {{1,2,4,5,7},{3,6,8,9,10}}
 => {{1,2,6,7},{3,4,5,8,9,10}}
 => ? = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => {{1,2,4,5,6},{3,7,8,9,10}}
 => {{1,2,7,8,9,10},{3,4,5,6}}
 => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [[1,2,3,7,9],[4,5,6,8,10]]
 => {{1,2,3,7,9},{4,5,6,8,10}}
 => {{1,2,3,5,8,9},{4,6,7,10}}
 => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [[1,2,3,7,8],[4,5,6,9,10]]
 => {{1,2,3,7,8},{4,5,6,9,10}}
 => {{1,2,3,5,9,10},{4,6,7,8}}
 => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => {{1,2,3,6,9},{4,5,7,8,10}}
 => {{1,2,3,5,6,8,9},{4,7,10}}
 => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => {{1,2,3,6,8},{4,5,7,9,10}}
 => {{1,2,3,5,6,9,10},{4,7,8}}
 => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [[1,2,3,6,7],[4,5,8,9,10]]
 => {{1,2,3,6,7},{4,5,8,9,10}}
 => {{1,2,3,5,6,7},{4,8,9,10}}
 => ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => {{1,2,3,5,9},{4,6,7,8,10}}
 => {{1,2,3,6,8,9},{4,5,7,10}}
 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [[1,2,3,5,8],[4,6,7,9,10]]
 => {{1,2,3,5,8},{4,6,7,9,10}}
 => {{1,2,3,6,9,10},{4,5,7,8}}
 => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [[1,2,3,5,7],[4,6,8,9,10]]
 => {{1,2,3,5,7},{4,6,8,9,10}}
 => {{1,2,3,6,7},{4,5,8,9,10}}
 => ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [[1,2,3,5,6],[4,7,8,9,10]]
 => {{1,2,3,5,6},{4,7,8,9,10}}
 => {{1,2,3,7,8,9,10},{4,5,6}}
 => ? = 2 - 1
Description
The crossing number of a set partition. This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000718
(load all 2 compositions to match this statistic)
Mapping
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [.,.]
 => [[]]
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0]
 => [.,[.,.]]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,0,0]
 => [[.,.],.]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => [[[.,.],.],.]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 1
[1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 1
[1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 1
[1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],.]]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,[.,[.,.]]],.]]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[[.,[.,.]],.],.]]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[.,[[.,[.,.]],.]],.]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => [[[[],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => [[[[[],[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => [[[[[[]]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => [[[[[]],[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => [[[[]],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [[[.,.],[[.,.],.]],.]
 => [[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [[[[.,.],.],[.,.]],.]
 => [[[[[]]],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => [[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 3 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [[.,.],[[.,.],[.,.]]]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2 + 1
Description
The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]