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Your data matches 120 different statistics following compositions of up to 3 maps.
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Matching statistic: St001344
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
St001344: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001344: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => 1
{{1},{2}}
=> [1,2] => 1
{{1,2,3}}
=> [2,3,1] => 1
{{1,2},{3}}
=> [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => 1
{{1},{2,3}}
=> [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => 1
{{1,2,3,4}}
=> [2,3,4,1] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 1
Description
The neighbouring number of a permutation.
For a permutation $\pi$, this is
$$\min \big(\big\{|\pi(k)-\pi(k+1)|:k\in\{1,\ldots,n-1\}\big\}\cup \big\{|\pi(1) - \pi(n)|\big\}\big).$$
Matching statistic: St000069
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
Mp00209: Permutations —pattern poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => ([(0,1)],2)
=> 1
{{1,2,3}}
=> [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3}}
=> [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2}}
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,18),(1,22),(2,11),(2,14),(2,16),(2,18),(3,9),(3,14),(3,15),(3,22),(4,12),(4,13),(4,16),(4,22),(5,10),(5,13),(5,15),(5,18),(5,22),(6,8),(6,10),(6,11),(6,12),(6,22),(8,20),(8,25),(9,19),(9,25),(10,20),(10,21),(10,25),(10,26),(11,17),(11,25),(11,26),(12,17),(12,20),(12,26),(13,21),(13,26),(14,19),(14,26),(15,19),(15,21),(15,25),(16,17),(16,26),(17,24),(18,19),(18,25),(18,26),(19,23),(20,23),(20,24),(21,23),(21,24),(22,20),(22,21),(22,25),(22,26),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,11),(2,12),(2,14),(2,18),(3,10),(3,13),(3,14),(3,18),(4,7),(4,9),(4,12),(4,13),(5,8),(5,9),(5,10),(5,11),(7,16),(7,23),(8,16),(8,19),(9,16),(9,17),(9,20),(9,23),(10,19),(10,20),(11,19),(11,20),(11,23),(12,15),(12,23),(13,15),(13,20),(13,23),(14,15),(14,17),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,17),(18,19),(18,23),(19,21),(20,21),(20,22),(21,6),(22,6),(23,21),(23,22)],24)
=> ? = 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,21),(2,9),(2,16),(2,17),(3,12),(3,13),(3,16),(4,11),(4,14),(4,16),(4,17),(5,10),(5,12),(5,14),(5,17),(6,9),(6,10),(6,11),(6,13),(7,20),(9,18),(9,19),(10,1),(10,15),(10,19),(10,22),(11,18),(11,19),(11,22),(12,15),(12,22),(13,15),(13,18),(14,7),(14,22),(15,20),(15,21),(16,18),(16,22),(17,7),(17,19),(17,22),(18,21),(19,20),(19,21),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,19),(1,20),(2,15),(2,16),(2,19),(2,20),(3,9),(3,12),(3,13),(3,19),(4,8),(4,11),(4,13),(4,15),(4,20),(5,7),(5,11),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,16),(6,17),(7,21),(7,25),(7,26),(7,27),(8,21),(8,24),(8,26),(9,24),(9,25),(9,26),(11,18),(11,21),(11,25),(12,18),(12,25),(12,27),(13,18),(13,26),(13,27),(14,21),(14,27),(15,24),(15,25),(15,27),(16,24),(16,26),(17,24),(17,26),(17,27),(18,23),(19,24),(19,27),(20,21),(20,25),(20,26),(20,27),(21,22),(21,23),(22,10),(23,10),(24,22),(25,22),(25,23),(26,22),(26,23),(27,22),(27,23)],28)
=> ? = 2
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,11),(2,12),(2,14),(2,18),(3,10),(3,13),(3,14),(3,18),(4,7),(4,9),(4,12),(4,13),(5,8),(5,9),(5,10),(5,11),(7,16),(7,23),(8,16),(8,19),(9,16),(9,17),(9,20),(9,23),(10,19),(10,20),(11,19),(11,20),(11,23),(12,15),(12,23),(13,15),(13,20),(13,23),(14,15),(14,17),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,17),(18,19),(18,23),(19,21),(20,21),(20,22),(21,6),(22,6),(23,21),(23,22)],24)
=> ? = 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,17),(1,20),(2,13),(2,15),(2,17),(2,18),(2,20),(3,12),(3,14),(3,17),(3,18),(4,8),(4,10),(4,12),(4,20),(5,8),(5,11),(5,14),(5,15),(5,20),(6,9),(6,10),(6,11),(6,13),(6,18),(8,16),(8,22),(8,26),(9,21),(9,26),(10,21),(10,25),(10,26),(11,19),(11,21),(11,26),(12,16),(12,25),(13,21),(13,22),(13,26),(14,16),(14,19),(14,25),(15,19),(15,22),(15,25),(15,26),(16,23),(17,25),(17,26),(18,19),(18,22),(18,25),(18,26),(19,23),(19,24),(20,21),(20,22),(20,25),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 1
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,19),(1,20),(2,15),(2,16),(2,19),(2,20),(3,9),(3,12),(3,13),(3,19),(4,8),(4,11),(4,13),(4,15),(4,20),(5,7),(5,11),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,16),(6,17),(7,21),(7,25),(7,26),(7,27),(8,21),(8,24),(8,26),(9,24),(9,25),(9,26),(11,18),(11,21),(11,25),(12,18),(12,25),(12,27),(13,18),(13,26),(13,27),(14,21),(14,27),(15,24),(15,25),(15,27),(16,24),(16,26),(17,24),(17,26),(17,27),(18,23),(19,24),(19,27),(20,21),(20,25),(20,26),(20,27),(21,22),(21,23),(22,10),(23,10),(24,22),(25,22),(25,23),(26,22),(26,23),(27,22),(27,23)],28)
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(1,13),(1,16),(1,17),(2,9),(2,13),(2,15),(2,17),(3,12),(3,14),(3,15),(3,17),(4,11),(4,14),(4,16),(4,17),(5,8),(5,11),(5,12),(5,15),(5,16),(6,8),(6,9),(6,10),(6,15),(6,16),(8,19),(8,20),(9,19),(9,22),(9,23),(10,20),(10,22),(10,23),(11,19),(11,22),(11,24),(12,20),(12,22),(12,24),(13,22),(13,23),(14,22),(14,24),(15,19),(15,20),(15,23),(15,24),(16,19),(16,20),(16,23),(16,24),(17,23),(17,24),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21),(23,18),(23,21),(24,18),(24,21)],25)
=> ? = 1
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,17),(1,18),(2,12),(2,14),(2,18),(2,19),(3,11),(3,14),(3,17),(3,19),(4,10),(4,13),(4,17),(4,18),(4,19),(5,9),(5,13),(5,17),(5,18),(5,19),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,20),(9,21),(9,22),(9,25),(10,20),(10,21),(10,22),(10,25),(11,15),(11,20),(11,21),(12,15),(12,20),(12,22),(13,16),(13,25),(14,15),(14,25),(15,24),(16,23),(17,16),(17,21),(17,25),(18,16),(18,22),(18,25),(19,16),(19,20),(19,25),(20,23),(20,24),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 2
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,24),(1,25),(2,9),(2,11),(2,13),(2,15),(3,8),(3,10),(3,13),(3,14),(4,8),(4,11),(4,12),(4,16),(5,9),(5,10),(5,12),(5,17),(6,1),(6,14),(6,15),(6,16),(6,17),(8,20),(8,24),(9,20),(9,25),(10,20),(10,23),(10,25),(11,20),(11,23),(11,24),(12,19),(12,20),(13,18),(13,24),(13,25),(14,18),(14,23),(14,24),(15,18),(15,23),(15,25),(16,19),(16,23),(16,24),(16,25),(17,19),(17,23),(17,24),(17,25),(18,22),(19,21),(19,22),(20,21),(21,7),(22,7),(23,21),(23,22),(24,21),(24,22),(25,21),(25,22)],26)
=> ? = 1
{{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,21),(2,9),(2,16),(2,17),(3,12),(3,13),(3,16),(4,11),(4,14),(4,16),(4,17),(5,10),(5,12),(5,14),(5,17),(6,9),(6,10),(6,11),(6,13),(7,20),(9,18),(9,19),(10,1),(10,15),(10,19),(10,22),(11,18),(11,19),(11,22),(12,15),(12,22),(13,15),(13,18),(14,7),(14,22),(15,20),(15,21),(16,18),(16,22),(17,7),(17,19),(17,22),(18,21),(19,20),(19,21),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 1
{{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,18),(1,22),(2,11),(2,14),(2,16),(2,18),(3,9),(3,14),(3,15),(3,22),(4,12),(4,13),(4,16),(4,22),(5,10),(5,13),(5,15),(5,18),(5,22),(6,8),(6,10),(6,11),(6,12),(6,22),(8,20),(8,25),(9,19),(9,25),(10,20),(10,21),(10,25),(10,26),(11,17),(11,25),(11,26),(12,17),(12,20),(12,26),(13,21),(13,26),(14,19),(14,26),(15,19),(15,21),(15,25),(16,17),(16,26),(17,24),(18,19),(18,25),(18,26),(19,23),(20,23),(20,24),(21,23),(21,24),(22,20),(22,21),(22,25),(22,26),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 1
{{1,4},{2},{3,5},{6}}
=> [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 2
{{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,15),(1,16),(2,8),(2,11),(2,16),(2,18),(3,8),(3,10),(3,15),(3,17),(4,9),(4,13),(4,14),(4,17),(4,18),(5,10),(5,12),(5,13),(5,16),(5,18),(6,11),(6,12),(6,14),(6,15),(6,17),(8,21),(8,25),(9,23),(9,24),(10,21),(10,23),(10,25),(11,21),(11,24),(11,25),(12,23),(12,24),(12,25),(13,19),(13,23),(13,25),(14,19),(14,24),(14,25),(15,24),(15,25),(16,23),(16,25),(17,19),(17,21),(17,23),(17,24),(18,19),(18,21),(18,23),(18,24),(19,20),(19,22),(20,7),(21,20),(21,22),(22,7),(23,20),(23,22),(24,20),(24,22),(25,22)],26)
=> ? = 2
{{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(1,13),(1,16),(1,17),(2,9),(2,13),(2,15),(2,17),(3,12),(3,14),(3,15),(3,17),(4,11),(4,14),(4,16),(4,17),(5,8),(5,11),(5,12),(5,15),(5,16),(6,8),(6,9),(6,10),(6,15),(6,16),(8,19),(8,20),(9,19),(9,22),(9,23),(10,20),(10,22),(10,23),(11,19),(11,22),(11,24),(12,20),(12,22),(12,24),(13,22),(13,23),(14,22),(14,24),(15,19),(15,20),(15,23),(15,24),(16,19),(16,20),(16,23),(16,24),(17,23),(17,24),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21),(23,18),(23,21),(24,18),(24,21)],25)
=> ? = 1
{{1},{2,4},{3,6},{5}}
=> [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 2
{{1,5},{2},{3,4,6}}
=> [5,2,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,17),(1,20),(2,13),(2,15),(2,17),(2,18),(2,20),(3,12),(3,14),(3,17),(3,18),(4,8),(4,10),(4,12),(4,20),(5,8),(5,11),(5,14),(5,15),(5,20),(6,9),(6,10),(6,11),(6,13),(6,18),(8,16),(8,22),(8,26),(9,21),(9,26),(10,21),(10,25),(10,26),(11,19),(11,21),(11,26),(12,16),(12,25),(13,21),(13,22),(13,26),(14,16),(14,19),(14,25),(15,19),(15,22),(15,25),(15,26),(16,23),(17,25),(17,26),(18,19),(18,22),(18,25),(18,26),(19,23),(19,24),(20,21),(20,22),(20,25),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 2
{{1,5},{2},{3,6},{4}}
=> [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,15),(1,16),(2,8),(2,11),(2,16),(2,18),(3,8),(3,10),(3,15),(3,17),(4,9),(4,13),(4,14),(4,17),(4,18),(5,10),(5,12),(5,13),(5,16),(5,18),(6,11),(6,12),(6,14),(6,15),(6,17),(8,21),(8,25),(9,23),(9,24),(10,21),(10,23),(10,25),(11,21),(11,24),(11,25),(12,23),(12,24),(12,25),(13,19),(13,23),(13,25),(14,19),(14,24),(14,25),(15,24),(15,25),(16,23),(16,25),(17,19),(17,21),(17,23),(17,24),(18,19),(18,21),(18,23),(18,24),(19,20),(19,22),(20,7),(21,20),(21,22),(22,7),(23,20),(23,22),(24,20),(24,22),(25,22)],26)
=> ? = 2
{{1,5},{2},{3},{4,6}}
=> [5,2,3,6,1,4] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,24),(1,25),(2,9),(2,11),(2,13),(2,15),(3,8),(3,10),(3,13),(3,14),(4,8),(4,11),(4,12),(4,16),(5,9),(5,10),(5,12),(5,17),(6,1),(6,14),(6,15),(6,16),(6,17),(8,20),(8,24),(9,20),(9,25),(10,20),(10,23),(10,25),(11,20),(11,23),(11,24),(12,19),(12,20),(13,18),(13,24),(13,25),(14,18),(14,23),(14,24),(15,18),(15,23),(15,25),(16,19),(16,23),(16,24),(16,25),(17,19),(17,23),(17,24),(17,25),(18,22),(19,21),(19,22),(20,21),(21,7),(22,7),(23,21),(23,22),(24,21),(24,22),(25,21),(25,22)],26)
=> ? = 1
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,17),(1,18),(2,12),(2,14),(2,18),(2,19),(3,11),(3,14),(3,17),(3,19),(4,10),(4,13),(4,17),(4,18),(4,19),(5,9),(5,13),(5,17),(5,18),(5,19),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,20),(9,21),(9,22),(9,25),(10,20),(10,21),(10,22),(10,25),(11,15),(11,20),(11,21),(12,15),(12,20),(12,22),(13,16),(13,25),(14,15),(14,25),(15,24),(16,23),(17,16),(17,21),(17,25),(18,16),(18,22),(18,25),(19,16),(19,20),(19,25),(20,23),(20,24),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 2
{{1},{2,3,4,5,6,7}}
=> [1,3,4,5,6,7,2] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 1
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => ([(0,1),(0,4),(0,5),(0,6),(1,18),(2,8),(2,9),(2,21),(3,2),(3,11),(3,12),(3,20),(4,10),(4,14),(4,18),(5,10),(5,13),(5,18),(6,3),(6,13),(6,14),(6,18),(8,17),(8,19),(9,17),(9,19),(10,15),(10,20),(11,8),(11,16),(11,21),(12,9),(12,16),(12,21),(13,11),(13,15),(13,20),(14,12),(14,15),(14,20),(15,16),(15,21),(16,17),(16,19),(17,7),(18,20),(19,7),(20,21),(21,19)],22)
=> ? = 1
{{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => ([(0,1),(0,4),(0,5),(0,6),(1,8),(1,17),(2,10),(2,11),(2,20),(3,9),(3,12),(3,24),(4,13),(4,14),(4,17),(5,2),(5,13),(5,15),(5,17),(6,3),(6,8),(6,14),(6,15),(8,24),(9,19),(9,25),(10,18),(10,22),(11,18),(11,22),(11,25),(12,19),(12,22),(12,25),(13,10),(13,16),(13,20),(14,9),(14,16),(14,24),(15,11),(15,12),(15,16),(15,24),(16,18),(16,19),(16,25),(17,20),(17,24),(18,21),(18,23),(19,21),(19,23),(20,22),(20,25),(21,7),(22,21),(22,23),(23,7),(24,25),(25,23)],26)
=> ? = 1
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ? = 1
{{1},{2,3,4,6,7},{5}}
=> [1,3,4,6,5,7,2] => ?
=> ? = 1
{{1},{2,3,4,6},{5,7}}
=> [1,3,4,6,7,2,5] => ?
=> ? = 1
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => ?
=> ? = 1
{{1},{2,3,4},{5,6,7}}
=> [1,3,4,2,6,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(1,19),(2,13),(2,14),(2,15),(2,19),(3,11),(3,12),(3,15),(3,19),(4,9),(4,10),(4,12),(4,14),(5,8),(5,10),(5,11),(5,13),(7,20),(7,26),(8,27),(9,23),(9,27),(10,16),(10,21),(10,22),(10,27),(11,18),(11,22),(11,27),(12,17),(12,22),(12,23),(13,16),(13,18),(13,27),(14,16),(14,17),(14,23),(15,17),(15,18),(15,21),(16,24),(16,25),(16,28),(17,24),(17,25),(18,25),(18,28),(19,21),(19,23),(19,27),(20,6),(21,7),(21,24),(21,28),(22,7),(22,25),(22,28),(23,24),(23,28),(24,20),(24,26),(25,20),(25,26),(26,6),(27,28),(28,26)],29)
=> ? = 1
{{1},{2,3,4},{5,6},{7}}
=> [1,3,4,2,6,5,7] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,23),(1,28),(2,9),(2,11),(2,19),(3,13),(3,15),(3,16),(3,19),(4,12),(4,14),(4,16),(4,19),(5,10),(5,11),(5,14),(5,15),(6,9),(6,10),(6,12),(6,13),(8,20),(8,25),(9,27),(10,1),(10,21),(10,26),(10,27),(11,22),(11,27),(12,18),(12,21),(12,27),(13,18),(13,26),(13,27),(14,17),(14,21),(14,22),(15,17),(15,22),(15,26),(16,17),(16,18),(16,26),(17,23),(17,24),(18,24),(18,28),(19,22),(19,26),(19,27),(20,7),(21,8),(21,24),(21,28),(22,23),(22,28),(23,20),(23,25),(24,20),(24,25),(25,7),(26,23),(26,24),(26,28),(27,28),(28,25)],29)
=> ? = 1
{{1},{2,3,4,7},{5},{6}}
=> [1,3,4,7,5,6,2] => ?
=> ? = 1
{{1},{2,3,4},{5,7},{6}}
=> [1,3,4,2,7,6,5] => ([(0,3),(0,4),(0,5),(0,7),(1,22),(2,12),(2,24),(2,25),(3,13),(3,20),(4,15),(4,17),(4,20),(5,16),(5,17),(5,20),(6,2),(6,9),(6,10),(6,14),(7,6),(7,13),(7,15),(7,16),(8,1),(8,25),(9,24),(10,19),(10,24),(10,25),(12,22),(12,23),(13,9),(13,21),(14,12),(14,19),(14,24),(15,10),(15,18),(15,21),(16,14),(16,18),(16,21),(17,8),(17,18),(18,19),(18,25),(19,22),(19,23),(20,8),(20,21),(21,24),(21,25),(22,11),(23,11),(24,23),(25,22),(25,23)],26)
=> ? = 1
{{1},{2,3,4},{5},{6,7}}
=> [1,3,4,2,5,7,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,22),(1,27),(2,9),(2,24),(3,11),(3,13),(3,14),(3,24),(4,12),(4,14),(4,15),(4,24),(5,10),(5,13),(5,15),(5,24),(6,9),(6,10),(6,11),(6,12),(8,19),(8,25),(9,26),(10,18),(10,20),(10,26),(11,1),(11,17),(11,20),(11,26),(12,17),(12,18),(12,26),(13,16),(13,20),(13,21),(14,16),(14,17),(14,21),(15,16),(15,18),(15,21),(16,22),(16,23),(17,22),(17,23),(17,27),(18,23),(18,27),(19,7),(20,8),(20,23),(20,27),(21,22),(21,27),(22,19),(22,25),(23,19),(23,25),(24,21),(24,26),(25,7),(26,27),(27,25)],28)
=> ? = 1
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,18),(2,8),(2,9),(2,21),(3,2),(3,11),(3,12),(3,20),(4,10),(4,14),(4,18),(5,10),(5,13),(5,18),(6,3),(6,13),(6,14),(6,18),(8,17),(8,19),(9,17),(9,19),(10,15),(10,20),(11,8),(11,16),(11,21),(12,9),(12,16),(12,21),(13,11),(13,15),(13,20),(14,12),(14,15),(14,20),(15,16),(15,21),(16,17),(16,19),(17,7),(18,20),(19,7),(20,21),(21,19)],22)
=> ? = 1
{{1},{2,3,5,6,7},{4}}
=> [1,3,5,4,6,7,2] => ?
=> ? = 1
{{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => ?
=> ? = 1
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => ?
=> ? = 1
{{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,12),(1,16),(1,27),(2,11),(2,13),(2,27),(3,13),(3,15),(3,27),(4,8),(4,9),(4,10),(4,26),(5,11),(5,12),(5,14),(5,27),(6,4),(6,14),(6,15),(6,16),(6,27),(8,18),(8,23),(9,18),(9,23),(9,25),(10,23),(10,24),(11,21),(11,22),(12,17),(12,21),(13,22),(14,9),(14,17),(14,21),(14,26),(15,10),(15,22),(15,26),(16,8),(16,17),(16,26),(17,18),(17,25),(18,19),(19,7),(20,7),(21,24),(21,25),(22,24),(23,19),(23,20),(24,20),(25,19),(25,20),(26,23),(26,24),(26,25),(27,21),(27,22),(27,26)],28)
=> ? = 1
{{1},{2,3,5},{4,6,7}}
=> [1,3,5,6,2,7,4] => ?
=> ? = 1
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => ?
=> ? = 1
{{1},{2,3,5,7},{4},{6}}
=> [1,3,5,4,7,6,2] => ?
=> ? = 1
{{1},{2,3,5},{4,7},{6}}
=> [1,3,5,7,2,6,4] => ?
=> ? = 2
{{1},{2,3,5},{4},{6,7}}
=> [1,3,5,4,2,7,6] => ?
=> ? = 1
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => ?
=> ? = 1
{{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => ?
=> ? = 1
{{1},{2,3,6},{4,5,7}}
=> [1,3,6,5,7,2,4] => ?
=> ? = 1
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => ?
=> ? = 1
{{1},{2,3,7},{4,5,6}}
=> [1,3,7,5,6,4,2] => ?
=> ? = 1
{{1},{2,3},{4,5,6,7}}
=> [1,3,2,5,6,7,4] => ([(0,3),(0,4),(0,5),(0,6),(1,8),(1,25),(2,9),(2,10),(2,11),(3,12),(3,13),(3,15),(4,12),(4,14),(4,17),(5,13),(5,14),(5,16),(6,2),(6,15),(6,16),(6,17),(8,21),(8,23),(9,19),(9,22),(10,22),(10,25),(11,19),(11,22),(11,25),(12,24),(13,1),(13,20),(13,24),(14,18),(14,24),(15,10),(15,20),(15,24),(16,9),(16,18),(16,20),(17,11),(17,18),(17,24),(18,19),(18,25),(19,21),(19,23),(20,8),(20,22),(20,25),(21,7),(22,21),(22,23),(23,7),(24,25),(25,23)],26)
=> ? = 1
{{1},{2,3},{4,5,6},{7}}
=> [1,3,2,5,6,4,7] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,23),(1,28),(2,9),(2,11),(2,19),(3,13),(3,15),(3,16),(3,19),(4,12),(4,14),(4,16),(4,19),(5,10),(5,11),(5,14),(5,15),(6,9),(6,10),(6,12),(6,13),(8,20),(8,25),(9,27),(10,1),(10,21),(10,26),(10,27),(11,22),(11,27),(12,18),(12,21),(12,27),(13,18),(13,26),(13,27),(14,17),(14,21),(14,22),(15,17),(15,22),(15,26),(16,17),(16,18),(16,26),(17,23),(17,24),(18,24),(18,28),(19,22),(19,26),(19,27),(20,7),(21,8),(21,24),(21,28),(22,23),(22,28),(23,20),(23,25),(24,20),(24,25),(25,7),(26,23),(26,24),(26,28),(27,28),(28,25)],29)
=> ? = 1
{{1},{2,3,7},{4,5},{6}}
=> [1,3,7,5,4,6,2] => ?
=> ? = 1
Description
The number of maximal elements of a poset.
Matching statistic: St000065
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [2,1] => [[0,1],[1,0]]
=> 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => [[1,0],[0,1]]
=> 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,4,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,4,3] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,5,3,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,2,4,3,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,4,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,3,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,3,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [6,3,2,1,5,4] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [5,2,1,6,4,3] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [6,5,2,1,4,3] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [5,2,1,4,3,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [5,6,2,1,4,3] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [6,4,2,1,5,3] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [4,6,2,1,5,3] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [3,6,2,1,5,4] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2 - 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,6,3,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [5,4,1,6,3,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [6,4,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [4,1,6,5,3,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [4,1,5,3,2,6] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [4,1,5,6,3,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [5,2,6,4,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [5,4,1,3,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 - 1
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [4,1,3,2,6,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [4,1,3,2,5,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [4,5,1,6,3,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => [2,6,3,1,5,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [4,6,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [6,1,3,2,5,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [4,5,1,3,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 2 - 1
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [5,4,6,1,3,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [4,5,6,1,3,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [5,3,1,6,4,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [6,5,3,1,4,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,4},{2,3,5},{6}}
=> [4,3,5,1,2,6] => [5,3,1,4,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 - 1
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [5,6,3,1,4,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => [6,4,3,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => [4,6,3,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => [3,2,6,1,5,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [1,6,3,2,5,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => [3,5,1,6,4,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [2,5,1,6,4,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => [3,6,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [3,6,5,1,4,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,4},{2,5},{3},{6}}
=> [4,5,3,1,2,6] => [3,5,1,4,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 - 1
{{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => [5,1,6,3,4,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => [2,6,5,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,4},{2},{3,5},{6}}
=> [4,2,5,1,3,6] => [2,5,1,4,3,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 2 - 1
{{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [3,5,6,1,4,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 2 - 1
{{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => [2,5,6,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => [3,6,4,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => [6,2,4,1,5,3] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1},{2,4,5},{3,6}}
=> [1,4,6,5,2,3] => [1,5,2,6,4,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1},{2,4},{3,5,6}}
=> [1,4,5,2,6,3] => [1,6,5,2,4,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => [1,5,2,4,3,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 - 1
{{1},{2,4},{3,6},{5}}
=> [1,4,6,2,5,3] => [1,5,6,2,4,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 2 - 1
Description
The number of entries equal to -1 in an alternating sign matrix.
The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St001947
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001947: Parking functions ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001947: Parking functions ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [2,3,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,3,2,1] => [4,5,3,2,1] => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,4,2,1] => [3,5,4,2,1] => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,4,3] => [5,2,1,4,3] => 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,2,1,5] => [3,4,2,1,5] => 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,2,1] => [4,3,5,2,1] => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,2,1] => [3,4,5,2,1] => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,4,3,1] => [2,5,4,3,1] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,5,3,2] => [4,1,5,3,2] => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,2,4,3,1] => [5,2,4,3,1] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,4,1,3,2] => [5,4,1,3,2] => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,3,1] => [2,4,5,3,1] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,1,3,2] => [4,5,1,3,2] => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,4,1] => [3,2,5,4,1] => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,3,1,4,2] => [5,3,1,4,2] => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => 0 = 1 - 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [6,3,2,1,5,4] => [6,3,2,1,5,4] => ? = 1 - 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [5,2,1,6,4,3] => [5,2,1,6,4,3] => ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [6,5,2,1,4,3] => [6,5,2,1,4,3] => ? = 1 - 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [5,2,1,4,3,6] => [5,2,1,4,3,6] => ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [5,6,2,1,4,3] => [5,6,2,1,4,3] => ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [6,4,2,1,5,3] => [6,4,2,1,5,3] => ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [4,6,2,1,5,3] => [4,6,2,1,5,3] => ? = 1 - 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [3,6,2,1,5,4] => [3,6,2,1,5,4] => ? = 2 - 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,6,3,5,4] => [2,1,6,3,5,4] => ? = 1 - 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [5,4,1,6,3,2] => [5,4,1,6,3,2] => ? = 1 - 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [6,4,2,5,3,1] => [6,4,2,5,3,1] => ? = 1 - 1
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [4,1,6,5,3,2] => [4,1,6,5,3,2] => ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [4,1,5,3,2,6] => [4,1,5,3,2,6] => ? = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [4,1,5,6,3,2] => [4,1,5,6,3,2] => ? = 1 - 1
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [6,5,2,4,3,1] => [6,5,2,4,3,1] => ? = 1 - 1
{{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => [6,1,5,4,3,2] => [6,1,5,4,3,2] => ? = 1 - 1
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [5,2,6,4,3,1] => [5,2,6,4,3,1] => ? = 1 - 1
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [6,5,4,1,3,2] => [6,5,4,1,3,2] => ? = 1 - 1
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [5,4,1,3,2,6] => [5,4,1,3,2,6] => ? = 1 - 1
{{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => [5,6,2,4,3,1] => [5,6,2,4,3,1] => ? = 1 - 1
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [5,6,4,1,3,2] => [5,6,4,1,3,2] => ? = 1 - 1
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [4,1,3,2,6,5] => [4,1,3,2,6,5] => ? = 1 - 1
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [4,5,1,6,3,2] => [4,5,1,6,3,2] => ? = 1 - 1
{{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => [2,6,3,1,5,4] => [2,6,3,1,5,4] => ? = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [4,6,2,5,3,1] => [4,6,2,5,3,1] => ? = 1 - 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [6,1,3,2,5,4] => [6,1,3,2,5,4] => ? = 1 - 1
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [4,5,1,3,2,6] => [4,5,1,3,2,6] => ? = 2 - 1
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [5,4,6,1,3,2] => [5,4,6,1,3,2] => ? = 1 - 1
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [5,3,1,6,4,2] => [5,3,1,6,4,2] => ? = 1 - 1
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 1 - 1
{{1,4},{2,3,5},{6}}
=> [4,3,5,1,2,6] => [5,3,1,4,2,6] => [5,3,1,4,2,6] => ? = 1 - 1
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [5,6,3,1,4,2] => [5,6,3,1,4,2] => ? = 1 - 1
{{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => [6,4,3,1,5,2] => [6,4,3,1,5,2] => ? = 1 - 1
{{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => [4,6,3,1,5,2] => [4,6,3,1,5,2] => ? = 1 - 1
{{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => [3,2,6,1,5,4] => [3,2,6,1,5,4] => ? = 1 - 1
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,6,3,2,5,4] => ? = 1 - 1
{{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [2,5,1,6,4,3] => [2,5,1,6,4,3] => ? = 1 - 1
{{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => [3,6,2,5,4,1] => [3,6,2,5,4,1] => ? = 1 - 1
{{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ? = 1 - 1
{{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => [6,1,5,2,4,3] => ? = 1 - 1
{{1,4},{2,5},{3},{6}}
=> [4,5,3,1,2,6] => [3,5,1,4,2,6] => [3,5,1,4,2,6] => ? = 1 - 1
{{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => [5,1,6,3,4,2] => [5,1,6,3,4,2] => ? = 1 - 1
{{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => [2,6,5,1,4,3] => [2,6,5,1,4,3] => ? = 1 - 1
{{1,4},{2},{3,5},{6}}
=> [4,2,5,1,3,6] => [2,5,1,4,3,6] => [2,5,1,4,3,6] => ? = 2 - 1
{{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [3,5,6,1,4,2] => [3,5,6,1,4,2] => ? = 2 - 1
{{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => [2,5,6,1,4,3] => [2,5,6,1,4,3] => ? = 1 - 1
{{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => [3,6,4,1,5,2] => [3,6,4,1,5,2] => ? = 1 - 1
{{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => [6,2,4,1,5,3] => [6,2,4,1,5,3] => ? = 1 - 1
{{1},{2,4,5},{3,6}}
=> [1,4,6,5,2,3] => [1,5,2,6,4,3] => [1,5,2,6,4,3] => ? = 1 - 1
Description
The number of ties in a parking function.
This is the number of indices $i$ such that $p_i=p_{i+1}$.
Matching statistic: St001434
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001434: Signed permutations ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001434: Signed permutations ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [2,3,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => [3,4,2,1] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,3,2,1] => [4,5,3,2,1] => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,4,2,1] => [3,5,4,2,1] => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,4,3] => [5,2,1,4,3] => 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,2,1,5] => [3,4,2,1,5] => 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,2,1] => [4,3,5,2,1] => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,2,1] => [3,4,5,2,1] => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,5,3] => 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,4,3,1] => [2,5,4,3,1] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,5,3,2] => [4,1,5,3,2] => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,3,1,5] => [2,4,3,1,5] => 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,2,4,3,1] => [5,2,4,3,1] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,4,1,3,2] => [5,4,1,3,2] => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,3,1] => [2,4,5,3,1] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,1,3,2] => [4,5,1,3,2] => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,5,4] => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,4,1] => [3,2,5,4,1] => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,3,1,4,2] => [5,3,1,4,2] => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => [3,2,4,1,5] => 0 = 1 - 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [6,3,2,1,5,4] => [6,3,2,1,5,4] => ? = 1 - 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [5,2,1,6,4,3] => [5,2,1,6,4,3] => ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [6,5,2,1,4,3] => [6,5,2,1,4,3] => ? = 1 - 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [5,2,1,4,3,6] => [5,2,1,4,3,6] => ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [5,6,2,1,4,3] => [5,6,2,1,4,3] => ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [6,4,2,1,5,3] => [6,4,2,1,5,3] => ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [4,6,2,1,5,3] => [4,6,2,1,5,3] => ? = 1 - 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [3,6,2,1,5,4] => [3,6,2,1,5,4] => ? = 2 - 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,6,3,5,4] => [2,1,6,3,5,4] => ? = 1 - 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [5,4,1,6,3,2] => [5,4,1,6,3,2] => ? = 1 - 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [6,4,2,5,3,1] => [6,4,2,5,3,1] => ? = 1 - 1
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [4,1,6,5,3,2] => [4,1,6,5,3,2] => ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [4,1,5,3,2,6] => [4,1,5,3,2,6] => ? = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [4,1,5,6,3,2] => [4,1,5,6,3,2] => ? = 1 - 1
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [6,5,2,4,3,1] => [6,5,2,4,3,1] => ? = 1 - 1
{{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => [6,1,5,4,3,2] => [6,1,5,4,3,2] => ? = 1 - 1
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [5,2,6,4,3,1] => [5,2,6,4,3,1] => ? = 1 - 1
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [6,5,4,1,3,2] => [6,5,4,1,3,2] => ? = 1 - 1
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [5,4,1,3,2,6] => [5,4,1,3,2,6] => ? = 1 - 1
{{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => [5,6,2,4,3,1] => [5,6,2,4,3,1] => ? = 1 - 1
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [5,6,4,1,3,2] => [5,6,4,1,3,2] => ? = 1 - 1
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [4,1,3,2,6,5] => [4,1,3,2,6,5] => ? = 1 - 1
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [4,5,1,6,3,2] => [4,5,1,6,3,2] => ? = 1 - 1
{{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => [2,6,3,1,5,4] => [2,6,3,1,5,4] => ? = 1 - 1
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [4,6,2,5,3,1] => [4,6,2,5,3,1] => ? = 1 - 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [6,1,3,2,5,4] => [6,1,3,2,5,4] => ? = 1 - 1
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [4,5,1,3,2,6] => [4,5,1,3,2,6] => ? = 2 - 1
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [5,4,6,1,3,2] => [5,4,6,1,3,2] => ? = 1 - 1
{{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [4,5,6,1,3,2] => [4,5,6,1,3,2] => ? = 1 - 1
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [5,3,1,6,4,2] => [5,3,1,6,4,2] => ? = 1 - 1
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? = 1 - 1
{{1,4},{2,3,5},{6}}
=> [4,3,5,1,2,6] => [5,3,1,4,2,6] => [5,3,1,4,2,6] => ? = 1 - 1
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [5,6,3,1,4,2] => [5,6,3,1,4,2] => ? = 1 - 1
{{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => [6,4,3,1,5,2] => [6,4,3,1,5,2] => ? = 1 - 1
{{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => [4,6,3,1,5,2] => [4,6,3,1,5,2] => ? = 1 - 1
{{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => [3,2,6,1,5,4] => [3,2,6,1,5,4] => ? = 1 - 1
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [1,6,3,2,5,4] => [1,6,3,2,5,4] => ? = 1 - 1
{{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => [3,5,1,6,4,2] => [3,5,1,6,4,2] => ? = 1 - 1
{{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [2,5,1,6,4,3] => [2,5,1,6,4,3] => ? = 1 - 1
{{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => [3,6,2,5,4,1] => [3,6,2,5,4,1] => ? = 1 - 1
{{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ? = 1 - 1
{{1,4},{2,5},{3,6}}
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => [6,1,5,2,4,3] => ? = 1 - 1
{{1,4},{2,5},{3},{6}}
=> [4,5,3,1,2,6] => [3,5,1,4,2,6] => [3,5,1,4,2,6] => ? = 1 - 1
{{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => [5,1,6,3,4,2] => [5,1,6,3,4,2] => ? = 1 - 1
{{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => [2,6,5,1,4,3] => [2,6,5,1,4,3] => ? = 1 - 1
{{1,4},{2},{3,5},{6}}
=> [4,2,5,1,3,6] => [2,5,1,4,3,6] => [2,5,1,4,3,6] => ? = 2 - 1
{{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [3,5,6,1,4,2] => [3,5,6,1,4,2] => ? = 2 - 1
{{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => [2,5,6,1,4,3] => [2,5,6,1,4,3] => ? = 1 - 1
{{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => [3,6,4,1,5,2] => [3,6,4,1,5,2] => ? = 1 - 1
Description
The number of negative sum pairs of a signed permutation.
The number of negative sum pairs of a signed permutation $\sigma$ is:
$$\operatorname{nsp}(\sigma)=\big|\{1\leq i < j\leq n \mid \sigma(i)+\sigma(j) < 0\}\big|,$$
see [1, Eq.(8.1)].
Matching statistic: St001260
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001260: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 50%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001260: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [1,2] => [[1,0],[0,1]]
=> 1
{{1},{2}}
=> [1,2] => [2,1] => [[0,1],[1,0]]
=> 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,6,3,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,2,6,4,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,2,5,4,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,2,6,5,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,6,2,4,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,6,2,3,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,5,2,3,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,6,2,5,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,3,2,6,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [1,6,3,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 1
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [1,6,3,4,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [1,5,3,4,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [1,6,3,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [1,4,3,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [1,4,3,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [1,6,4,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [1,3,6,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 2
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [1,6,4,3,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [1,5,6,3,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [1,5,4,3,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [1,6,5,4,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1
{{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => [1,3,6,4,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [1,3,5,4,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [1,6,4,5,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [1,3,6,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [1,3,4,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [6,1,3,4,2,5] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 1
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [6,1,3,2,5,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [5,1,3,2,6,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [6,1,3,5,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [2,1,3,6,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => [6,1,2,3,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 1
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [6,1,2,4,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [5,1,2,4,6,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [6,1,2,5,4,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [4,1,2,6,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [4,1,2,5,6,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [6,1,4,3,2,5] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 1
{{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => [2,1,6,3,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [6,1,4,2,5,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [5,1,6,2,4,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [5,1,4,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 2
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [6,1,5,4,2,3] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 1
Description
The permanent of an alternating sign matrix.
Matching statistic: St001975
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001975: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 50%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001975: Alternating sign matrices ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [1,2] => [[1,0],[0,1]]
=> 0 = 1 - 1
{{1},{2}}
=> [1,2] => [2,1] => [[0,1],[1,0]]
=> 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0 = 1 - 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,6,3,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,2,6,4,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,2,5,4,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,2,6,5,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,6,2,4,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,6,2,3,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,5,2,3,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,6,2,5,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,3,2,6,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [1,6,3,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [1,6,3,4,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [1,5,3,4,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [1,6,3,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [1,4,3,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [1,4,3,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [1,6,4,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [1,3,6,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 2 - 1
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [1,6,4,3,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [1,5,6,3,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [1,5,4,3,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [1,6,5,4,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => [1,3,6,4,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [1,3,5,4,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [1,6,4,5,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1 - 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [1,3,6,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [1,3,4,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [6,1,3,4,2,5] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [6,1,3,2,5,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [5,1,3,2,6,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [6,1,3,5,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [2,1,3,6,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => [6,1,2,3,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [6,1,2,4,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => [5,1,2,4,6,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [6,1,2,5,4,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [4,1,2,6,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1 - 1
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => [4,1,2,5,6,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 - 1
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [6,1,4,3,2,5] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => [2,1,6,3,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [6,1,4,2,5,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [5,1,6,2,4,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 - 1
{{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [5,1,4,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 2 - 1
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [6,1,5,4,2,3] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 - 1
Description
The corank of the alternating sign matrix.
Matching statistic: St000022
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [3,1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [4,3,1,2] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,4,1,3] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0 = 1 - 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ? = 1 - 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 1 - 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? = 1 - 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => ? = 1 - 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ? = 1 - 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 1 - 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ? = 1 - 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => ? = 1 - 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => ? = 1 - 1
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => ? = 1 - 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 1 - 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? = 1 - 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ? = 1 - 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 1 - 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ? = 1 - 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? = 1 - 1
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 1 - 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 1 - 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? = 1 - 1
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 1 - 1
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => ? = 1 - 1
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => ? = 1 - 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => ? = 1 - 1
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 - 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 - 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? = 1 - 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? = 2 - 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 1 - 1
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => ? = 1 - 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 1 - 1
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 1 - 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 1 - 1
{{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => ? = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 1 - 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 1 - 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 1 - 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 1 - 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => ? = 1 - 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? = 1 - 1
Description
The number of fixed points of a permutation.
Matching statistic: St001465
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [3,1,2] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [4,3,1,2] => 0 = 1 - 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [2,4,1,3] => 0 = 1 - 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0 = 1 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0 = 1 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 0 = 1 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 0 = 1 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 0 = 1 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0 = 1 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 0 = 1 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 0 = 1 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 0 = 1 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 0 = 1 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 0 = 1 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 0 = 1 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 0 = 1 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 0 = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 0 = 1 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 0 = 1 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 0 = 1 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 0 = 1 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 0 = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 0 = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 0 = 1 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0 = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 0 = 1 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 0 = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => 0 = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 0 = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => 0 = 1 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => 0 = 1 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => 0 = 1 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 0 = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 0 = 1 - 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ? = 1 - 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 1 - 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? = 1 - 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => ? = 1 - 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ? = 1 - 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 1 - 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ? = 1 - 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => ? = 1 - 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 1 - 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => ? = 1 - 1
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => ? = 1 - 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 1 - 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? = 1 - 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ? = 1 - 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ? = 1 - 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 1 - 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => ? = 1 - 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 1 - 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ? = 1 - 1
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 1 - 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 1 - 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 1 - 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? = 1 - 1
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 1 - 1
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => ? = 1 - 1
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => ? = 1 - 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => ? = 1 - 1
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 1 - 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1 - 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 1 - 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ? = 1 - 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => ? = 2 - 1
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 1 - 1
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,6,3,5] => ? = 1 - 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 1 - 1
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 1 - 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 1 - 1
{{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => ? = 1 - 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 1 - 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ? = 1 - 1
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 1 - 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 1 - 1
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ? = 1 - 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 1 - 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => ? = 1 - 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? = 1 - 1
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000788
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> ? = 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> ? = 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]
=> ? = 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> ? = 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> ? = 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)]
=> ? = 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
=> ? = 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> ? = 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9)]
=> ? = 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> ? = 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9)]
=> ? = 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> ? = 1
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,7),(5,6),(8,9),(10,11)]
=> ? = 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> ? = 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)]
=> ? = 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,1,0,1,1,0,1,0,0,1,0,0]
=> [(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)]
=> ? = 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8),(11,12)]
=> ? = 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [(1,12),(2,3),(4,7),(5,6),(8,11),(9,10)]
=> ? = 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> ? = 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]
=> ? = 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> ? = 1
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [1,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
=> ? = 1
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [1,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
=> ? = 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 1
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 1
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]
=> ? = 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [1,1,0,1,1,1,0,1,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [1,1,0,1,1,1,0,0,1,0,0,0]
=> [(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)]
=> ? = 2
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> ? = 1
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> ? = 1
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1
{{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => [1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> ? = 1
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1
{{1,2},{3,6},{4},{5}}
=> [2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> ? = 1
Description
The number of nesting-similar perfect matchings of a perfect matching.
Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$.
Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of nestings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$.
[thm 1.2, 1] shows that to find out whether $M$ and $N$ are nesting-similar, it is enough to check that $M$ and $N$ have the same number of nestings, and that the distribution of nestings agrees for their direct children.
[thm 3.5, 1], see also [2], gives the number of equivalence classes of nesting-similar matchings with $n$ arcs as $$2\cdot 4^{n-1} - \frac{3n-1}{2n+2}\binom{2n}{n}.$$ [prop 3.6, 1] has further interpretations of this number.
The following 110 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000787The number of flips required to make a perfect matching noncrossing. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001979The size of the permutation set corresponding to the alternating sign matrix variety. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001429The number of negative entries in a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001890The maximum magnitude of the Möbius function of a poset. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. 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. St000488The number of cycles of a permutation of length at most 2. St000542The number of left-to-right-minima of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000879The number of long braid edges in the graph of braid moves of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001845The number of join irreducibles minus the rank of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000068The number of minimal elements in a poset. St001862The number of crossings of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001768The number of reduced words of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset.
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