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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000668
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [2,2]
=> [2]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [2,2]
=> [2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [2,2]
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [2,2]
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [2,2]
=> [2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [3,2]
=> [2]
=> 2
([(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [3,2,1]
=> [2,1]
=> 2
([(4,5)],6)
=> [2,1,1,1,1]
=> [4,2]
=> [2]
=> 2
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [4,2]
=> [2]
=> 2
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [4,2]
=> [2]
=> 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [4,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St001845
Values
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ([(0,9),(2,14),(3,13),(4,11),(5,10),(6,5),(6,14),(7,4),(7,14),(8,6),(8,7),(9,2),(9,8),(10,12),(11,12),(12,13),(13,1),(14,3),(14,10),(14,11)],15)
=> ? = 2 - 1
([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(1,9),(2,9),(4,8),(5,6),(6,4),(6,7),(7,1),(7,2),(7,8),(8,9),(9,3)],10)
=> ? = 2 - 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(1,7),(2,7),(3,7),(5,6),(6,1),(6,2),(6,3),(7,4)],8)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ([(0,7),(2,9),(3,9),(4,8),(5,4),(5,9),(6,1),(7,2),(7,3),(7,5),(8,6),(9,8)],10)
=> ? = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ([(0,6),(2,7),(3,7),(4,7),(5,1),(6,2),(6,3),(6,4),(7,5)],8)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
([],5)
=> ?
=> ?
=> ?
=> ? = 2 - 1
([(3,4)],5)
=> ?
=> ?
=> ?
=> ? = 1 - 1
([(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
=> ?
=> ? = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ?
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ?
=> ? = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ?
=> ? = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ?
=> ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ?
=> ? = 1 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ?
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
=> ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
=> ([(0,10),(2,14),(3,14),(4,15),(5,12),(6,11),(7,6),(7,15),(8,5),(8,15),(9,2),(9,3),(10,4),(10,7),(10,8),(11,13),(12,13),(13,9),(14,1),(15,11),(15,12)],16)
=> ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
=> ([(0,9),(2,11),(3,11),(4,10),(5,6),(6,1),(7,4),(7,11),(8,5),(9,2),(9,3),(9,7),(10,8),(11,10)],12)
=> ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ([(0,7),(2,8),(3,8),(4,8),(5,1),(6,5),(7,2),(7,3),(7,4),(8,6)],9)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
=> ([(0,11),(1,15),(3,17),(4,13),(5,12),(6,5),(6,17),(7,4),(7,17),(8,1),(8,14),(9,2),(10,6),(10,7),(11,3),(11,10),(12,16),(13,16),(14,15),(15,9),(16,14),(17,8),(17,12),(17,13)],18)
=> ? = 2 - 1
([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ?
=> ? = 2 - 1
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
=> ?
=> ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
=> ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
=> ([(0,10),(2,15),(3,15),(4,14),(5,14),(6,13),(7,6),(7,11),(8,7),(8,14),(9,2),(9,3),(9,12),(10,4),(10,5),(10,8),(11,9),(11,13),(12,15),(13,12),(14,11),(15,1)],16)
=> ? = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,2),(0,3),(1,11),(2,1),(2,12),(3,4),(3,5),(3,12),(4,8),(4,10),(5,8),(5,9),(6,14),(7,14),(8,13),(9,6),(9,13),(10,7),(10,13),(11,6),(11,7),(12,9),(12,10),(12,11),(13,14)],15)
=> ([(0,2),(0,3),(1,11),(2,1),(2,12),(3,4),(3,5),(3,12),(4,8),(4,10),(5,8),(5,9),(6,14),(7,14),(8,13),(9,6),(9,13),(10,7),(10,13),(11,6),(11,7),(12,9),(12,10),(12,11),(13,14)],15)
=> ?
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
=> ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
=> ([(0,9),(2,16),(3,12),(4,14),(5,13),(6,5),(6,17),(7,4),(7,17),(8,11),(8,12),(9,10),(10,3),(10,8),(11,6),(11,7),(12,17),(13,15),(14,15),(15,16),(16,1),(17,2),(17,13),(17,14)],18)
=> ? = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9)
=> 0 = 1 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> ([(0,10),(1,15),(3,11),(4,11),(5,13),(6,12),(7,6),(7,14),(8,5),(8,14),(9,7),(9,8),(10,3),(10,4),(11,9),(12,15),(13,15),(14,1),(14,12),(14,13),(15,2)],16)
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
=> ([(0,7),(1,11),(2,11),(4,10),(5,6),(6,8),(7,5),(8,4),(8,9),(9,1),(9,2),(9,10),(10,11),(11,3)],12)
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ([(0,6),(1,8),(2,8),(3,8),(5,7),(6,5),(7,1),(7,2),(7,3),(8,4)],9)
=> 1 = 2 - 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,7),(2,10),(3,11),(4,9),(5,4),(5,11),(6,1),(7,8),(8,3),(8,5),(9,10),(10,6),(11,2),(11,9)],12)
=> ? = 1 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,6),(2,8),(3,8),(4,1),(5,4),(6,7),(7,2),(7,3),(8,5)],9)
=> ? = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
([],6)
=> ?
=> ?
=> ?
=> ? = 2 - 1
([(4,5)],6)
=> ?
=> ?
=> ?
=> ? = 2 - 1
([(3,4),(3,5)],6)
=> ?
=> ?
=> ?
=> ? = 2 - 1
([(2,3),(2,4),(2,5)],6)
=> ?
=> ?
=> ?
=> ? = 2 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> ?
=> ?
=> ?
=> ? = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ?
=> ?
=> ?
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(1,3),(1,4),(1,5),(1,6),(2,7),(3,8),(3,9),(3,10),(4,10),(4,12),(4,13),(5,9),(5,11),(5,13),(6,8),(6,11),(6,12),(8,14),(8,15),(9,14),(9,16),(10,15),(10,16),(11,14),(11,17),(12,15),(12,17),(13,16),(13,17),(14,18),(15,18),(16,18),(17,2),(17,18),(18,7)],19)
=> ?
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
=> ?
=> ? = 3 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,7),(2,14),(3,4),(3,5),(3,6),(3,14),(4,10),(4,12),(4,13),(5,9),(5,11),(5,13),(6,8),(6,11),(6,12),(8,15),(8,16),(9,15),(9,17),(10,16),(10,17),(11,15),(11,18),(12,16),(12,18),(13,17),(13,18),(14,8),(14,9),(14,10),(15,19),(16,19),(17,19),(18,1),(18,19),(19,7)],20)
=> ?
=> ?
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ([(0,6),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,5)],11)
=> ?
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ([(0,6),(2,10),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(6,3),(6,4),(6,5),(7,11),(8,11),(9,2),(9,11),(10,1),(11,10)],12)
=> ?
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> ([(0,6),(1,11),(2,7),(2,8),(3,8),(3,9),(4,7),(4,9),(5,1),(5,10),(6,2),(6,3),(6,4),(7,12),(8,12),(9,5),(9,12),(10,11),(12,10)],13)
=> ?
=> ?
=> ? = 1 - 1
([(2,3),(2,4),(4,5)],6)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ([(0,1),(1,2),(1,3),(2,13),(3,4),(3,5),(3,6),(3,13),(4,9),(4,11),(4,12),(5,8),(5,10),(5,12),(6,7),(6,10),(6,11),(7,14),(7,15),(8,14),(8,16),(9,15),(9,16),(10,14),(10,17),(11,15),(11,17),(12,16),(12,17),(13,7),(13,8),(13,9),(14,18),(15,18),(16,18),(17,18)],19)
=> ([(0,1),(1,2),(1,3),(2,13),(3,4),(3,5),(3,6),(3,13),(4,9),(4,11),(4,12),(5,8),(5,10),(5,12),(6,7),(6,10),(6,11),(7,14),(7,15),(8,14),(8,16),(9,15),(9,16),(10,14),(10,17),(11,15),(11,17),(12,16),(12,17),(13,7),(13,8),(13,9),(14,18),(15,18),(16,18),(17,18)],19)
=> ?
=> ? = 3 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> ([(0,4),(0,6),(1,10),(2,8),(2,12),(3,8),(3,11),(4,7),(5,1),(5,9),(6,2),(6,3),(6,7),(7,11),(7,12),(8,5),(8,13),(9,10),(11,13),(12,13),(13,9)],14)
=> ?
=> ?
=> ? = 1 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> ?
=> ? = 2 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ?
=> ?
=> ? = 2 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ?
=> ? = 2 - 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ?
=> ?
=> ? = 2 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 0 = 1 - 1
Description
The number of join irreducibles minus the rank of a lattice.
A lattice is join-extremal, if this statistic is $0$.
Matching statistic: St001804
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> 1
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> ? = 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 2
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 1
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> ? = 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]]
=> ? = 2
([(3,4)],5)
=> [6,6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
=> ? = 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
=> ? = 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
=> ? = 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 2
([(1,4),(2,3)],5)
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 2
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16]]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 2
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> ? = 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 2
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [[1,2,3,4,5,6]]
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> ? = 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ? = 2
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ? = 3
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ? = 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> ? = 1
([(2,3),(2,4),(4,5)],6)
=> [14,14]
=> ?
=> ? = 3
([(1,4),(1,5),(5,2),(5,3)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3
([(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,4,4,2,2,2,2]
=> ?
=> ? = 3
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 1
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St001695
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 1 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]]
=> ? = 2 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
=> ? = 1 - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
=> ? = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
=> ? = 1 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 2 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 2 - 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16]]
=> ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> ? = 1 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 2 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 1 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> ? = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ? = 2 - 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> ? = 1 - 1
([(2,3),(2,4),(4,5)],6)
=> [14,14]
=> ?
=> ? = 3 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,4,4,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
Description
The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
Matching statistic: St001698
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 1 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]]
=> ? = 2 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
=> ? = 1 - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
=> ? = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
=> ? = 1 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 2 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 2 - 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16]]
=> ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> ? = 1 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 2 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 1 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> ? = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ? = 2 - 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> ? = 1 - 1
([(2,3),(2,4),(4,5)],6)
=> [14,14]
=> ?
=> ? = 3 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,4,4,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001699
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> ? = 2 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> ? = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> ? = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> ? = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]]
=> ? = 2 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
=> ? = 1 - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
=> ? = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,15,16,17,18],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,17],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
=> ? = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,15,16,17,18],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,17],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> ? = 1 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [[1,2,5,9,10,11,12,13],[3,4,8],[6,7]]
=> ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> ? = 2 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 2 - 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [[1,2,3,4,5,6,13,14,15,16],[7,8,9,10,11,12]]
=> ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4],[5,6],[7,8],[9,10]]
=> ? = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> ? = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> ? = 1 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [[1,2,5,9,10,11,12,13],[3,4,8],[6,7]]
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? = 2 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 1 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 0 = 1 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> ? = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 0 = 1 - 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ? = 2 - 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> ? = 1 - 1
([(2,3),(2,4),(4,5)],6)
=> [14,14]
=> ?
=> ? = 3 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,4,4,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> 0 = 1 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> 0 = 1 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> 0 = 1 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> 0 = 1 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> 0 = 1 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
Description
The major index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001712
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> ? = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 1 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 1 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> ? = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]]
=> ? = 2 - 1
([(3,4)],5)
=> [6,6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
=> ? = 1 - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1 - 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
=> ? = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1 - 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
=> ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
=> ? = 1 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 2 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 2 - 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16]]
=> ? = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> ? = 1 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 2 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 1 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> ? = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 0 = 1 - 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ? = 2 - 1
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3 - 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
=> ? = 1 - 1
([(2,3),(2,4),(4,5)],6)
=> [14,14]
=> ?
=> ? = 3 - 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [14,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [7,2,2,2,2,2,2]
=> ?
=> ? = 3 - 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,4,4,2,2,2,2]
=> ?
=> ? = 3 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> 0 = 1 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> 0 = 1 - 1
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St001462
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> ? = 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> ? = 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> ? = 2
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 1
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> ? = 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> ? = 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [16,16]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]]
=> ? = 2
([(3,4)],5)
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20],[21,22,23,24]]
=> ? = 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,1,1,1,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14],[15],[16],[17],[18]]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16],[17]]
=> ? = 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [6,6,2,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,1,1,1,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14],[15],[16],[17],[18]]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16],[17]]
=> ? = 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13],[14],[15]]
=> ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,3,2,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12],[13]]
=> ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14]]
=> ? = 2
([(1,4),(2,3)],5)
=> [6,6,6]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 2
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [2,2,2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13],[14],[15],[16]]
=> ? = 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [5,5,1,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12],[13],[14]]
=> ? = 2
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13],[14],[15]]
=> ? = 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14]]
=> ? = 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [3,3,2,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11],[12],[13]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 2
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 1
([],6)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 2
([(4,5)],6)
=> [6,6,6,6,6,6,6,6]
=> ?
=> ?
=> ? = 2
([(3,4),(3,5)],6)
=> [6,6,6,6,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 2
([(2,3),(2,4),(2,5)],6)
=> [6,6,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 2
([(1,2),(1,3),(1,4),(1,5)],6)
=> [6,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [16,16,1]
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [7,6,6,6]
=> [4,4,4,4,4,4,1]
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [7,6,2,2,2,2]
=> [6,6,2,2,2,2,1]
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [7,2,2,2,2,2,2]
=> [7,7,1,1,1,1,1]
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,2,2,2,2,2,2,2]
=> [8,8,1,1]
=> ?
=> ? = 3
([(1,3),(1,4),(1,5),(5,2)],6)
=> [14,6,6]
=> ?
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [7,6,2,2,2,2]
=> [6,6,2,2,2,2,1]
=> ?
=> ? = 3
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [14,2,2,2,2]
=> ?
=> ?
=> ? = 3
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [4,4,2,2,2,2,2,2]
=> ?
=> ?
=> ? = 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> ? = 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11],[12]]
=> ? = 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> ? = 1
([(2,3),(2,4),(4,5)],6)
=> [14,14]
=> ?
=> ?
=> ? = 3
([(1,4),(1,5),(5,2),(5,3)],6)
=> [14,2,2,2,2]
=> ?
=> ?
=> ? = 3
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [7,2,2,2,2,2,2]
=> [7,7,1,1,1,1,1]
=> ?
=> ? = 3
([(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,4,4,2,2,2,2]
=> ?
=> ?
=> ? = 3
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [6,2]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
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