Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000383
Mapping
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => {{1}}
 => {{1}}
 => [1] => 1
[1,0,1,0]
 => {{1},{2}}
 => {{1},{2}}
 => [1,1] => 1
[1,1,0,0]
 => {{1,2}}
 => {{1,2}}
 => [2] => 2
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => 1
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,2},{3}}
 => [2,1] => 1
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => {{1},{2,3}}
 => [1,2] => 2
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1,3},{2}}
 => [2,1] => 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1,2,3}}
 => [3] => 3
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => 1
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => 1
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => [2,1,1] => 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,2,3},{4}}
 => [3,1] => 1
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => [1,1,2] => 2
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => [2,2] => 2
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => [1,2,1] => 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [2,1,1] => 1
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => {{1,3},{2,4}}
 => [2,2] => 2
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => [1,3] => 3
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1,4},{2,3}}
 => [2,2] => 2
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1,3,4},{2}}
 => [3,1] => 1
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => 4
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => 1
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => 1
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1,3},{2},{4},{5}}
 => [2,1,1,1] => 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => 1
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => {{1},{2},{3,4},{5}}
 => [1,1,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => 1
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => [1,2,1,1] => 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1,4},{2},{3},{5}}
 => [2,1,1,1] => 1
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => 1
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => 1
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,3,4},{2},{5}}
 => [3,1,1] => 1
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,2,3,4},{5}}
 => [4,1] => 1
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => [1,1,1,2] => 2
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => 2
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => [2,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => {{1,2,3},{4,5}}
 => [3,2] => 2
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => [1,1,2,1] => 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => 1
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => {{1},{2,5},{3},{4}}
 => [1,2,1,1] => 1
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => [2,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => {{1,4},{2,5},{3}}
 => [2,2,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => {{1},{2,4},{3,5}}
 => [1,2,2] => 2
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => {{1,4},{2},{3,5}}
 => [2,1,2] => 2
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => {{1,3,5},{2,4}}
 => [3,2] => 2
Description
The last part of an integer composition.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 70%
Values
[1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,1,0,0]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,5,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [6,4,3,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [6,5,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [5,2,1,3,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,1,6,5,2,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [3,1,6,4,2,5] => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,4,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [6,5,3,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [5,2,1,6,3,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => [4,2,1,6,5,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [5,3,1,6,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [6,4,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [5,3,2,1,6,4] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => [5,4,2,1,6,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [7,5,1,2,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,3,1,2,6,4,5] => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [8,1,2,3,4,7,5,6] => [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [8,1,2,3,6,4,5,7] => [6,1,2,3,4,8,5,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [8,1,2,5,3,4,6,7] => [5,1,2,3,8,4,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [8,1,2,5,3,7,4,6] => [5,1,2,3,7,4,8,6] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [8,1,4,2,3,5,6,7] => [4,1,2,8,3,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [8,1,4,2,3,7,5,6] => [4,1,2,7,3,5,8,6] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [8,1,4,2,6,3,5,7] => [4,1,2,6,3,8,5,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [8,3,1,2,4,5,6,7] => [3,1,8,2,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [8,3,1,2,4,7,5,6] => [3,1,7,2,4,5,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [8,3,1,2,6,4,5,7] => [3,1,6,2,4,8,5,7] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => [8,3,1,5,2,4,6,7] => [3,1,5,2,8,4,6,7] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [8,3,1,5,2,7,4,6] => [3,1,5,2,7,4,8,6] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => 2 = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,1] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => 8 = 7 + 1
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000877
Mapping
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1,0]
 => [1,0]
 => 10 => 11 => 0 = 1 - 1
[1,0,1,0]
 => [1,0,1,0]
 => 1010 => 1101 => 0 = 1 - 1
[1,1,0,0]
 => [1,1,0,0]
 => 1100 => 0111 => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 101010 => 110101 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 110100 => 111001 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 110010 => 011101 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 101100 => 110011 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 111000 => 001111 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 01110101 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 01111001 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 11001101 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 11010011 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 11000111 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 00111101 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 01110011 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 11100011 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 00011111 => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1101010101 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1110101001 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1110100101 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 1101101001 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 1111010001 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1110010101 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 1111001001 => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 1101100101 => ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 1101011001 => ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 1101110001 => ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 1111000101 => ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1110011001 => ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1110110001 => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 1111100001 => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 0111010101 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 0111101001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0111100101 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 0111011001 => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0111110001 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1100110101 => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 1100111001 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 1101001101 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 1101010011 => ? = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1101000111 => ? = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1100011101 => ? = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1100110011 => ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 1101100011 => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1100001111 => ? = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0011110101 => ? = 3 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0011111001 => ? = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0111001101 => ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0111010011 => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0111000111 => ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1110001101 => ? = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1110010011 => ? = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1110100011 => ? = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1110000111 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0001111101 => 3 = 4 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0011110011 => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0111100011 => ? = 2 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 1111000011 => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 0000111111 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => 110101010101 => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 110101010100 => 111010101001 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 110101010010 => 111010100101 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 101101010100 => 110110101001 => ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 111010101000 => 111101010001 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 110101001010 => 111010010101 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 111010100100 => 111101001001 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 101101010010 => 110110100101 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => 110101101001 => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 101110101000 => 110111010001 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 111010100010 => 111101000101 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 110110100100 => 111011001001 => ? = 1 - 1
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].