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Your data matches 206 different statistics following compositions of up to 3 maps.
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Matching statistic: St000071
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Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 3
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 5
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> 5
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 3
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 5
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> 5
[[1,1],[3]]
=> ([(0,1)],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 2
[[1],[2],[3]]
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 4
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> 9
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> 14
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> 14
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 3
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 5
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> 5
Description
The number of maximal chains in a poset.
Matching statistic: St000909
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Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000909: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 66%●distinct values known / distinct values provided: 14%
St000909: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 66%●distinct values known / distinct values provided: 14%
Values
[[1,2]]
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 3
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 5
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 5
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 3
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 5
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 5
[[1,1],[3]]
=> ([(0,1)],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 2
[[1],[2],[3]]
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 4
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? ∊ {5,9,14,14}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {5,9,14,14}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {5,9,14,14}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 3
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 5
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {5,9,14,14}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 3
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> 5
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? ∊ {4,5,5,6,6,11,16,21,21,42,42}
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {3,5,5,9,14,14}
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? ∊ {3,5,5,9,14,14}
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {3,5,5,9,14,14}
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {3,5,5,9,14,14}
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ? ∊ {3,5,5,9,14,14}
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ? ∊ {3,5,5,9,14,14}
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? ∊ {5,9,14,14,14,28,42,42}
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? ∊ {5,9,14,14,14,28,42,42}
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? ∊ {5,9,14,14,14,28,42,42}
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> ? ∊ {5,9,14,14,14,28,42,42}
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> ? ∊ {5,9,14,14,14,28,42,42}
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? ∊ {5,9,14,14,14,28,42,42}
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {5,9,14,14,14,28,42,42}
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {5,9,14,14,14,28,42,42}
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[3,3,5]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[3,4,5]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[3,5,5]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[4,4,5]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[4,5,5]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[5,5,5]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,5],[4]]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,5],[5]]
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,2],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,3],[5]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,5],[3]]
=> ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> ? ∊ {5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St001581
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {5,5}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {5,5}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {5,5}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {5,5}
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,9,14,14}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {4,5,5,9,14,14}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {4,5,5,9,14,14}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {4,5,5,9,14,14}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,9,14,14}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,9,14,14}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 2
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 2
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(2,10),(2,11),(2,13),(2,15),(2,17),(2,19),(3,8),(3,9),(3,12),(3,14),(3,16),(3,18),(4,8),(4,9),(4,12),(4,14),(4,16),(4,18),(4,19),(5,10),(5,11),(5,13),(5,15),(5,17),(5,18),(5,19),(6,8),(6,9),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(6,19),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,18),(7,19),(8,10),(8,11),(8,13),(8,15),(8,17),(8,19),(9,10),(9,11),(9,13),(9,15),(9,17),(9,19),(10,12),(10,14),(10,16),(10,18),(11,12),(11,14),(11,16),(11,18),(12,13),(12,15),(12,17),(12,19),(13,14),(13,16),(13,18),(14,15),(14,16),(14,17),(14,19),(15,16),(15,17),(15,18),(16,17),(16,19),(17,18),(18,19)],20)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {3,4,5,5,5,9,14,14}
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {3,4,5,5,5,9,14,14}
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {3,4,5,5,5,9,14,14}
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {3,4,5,5,5,9,14,14}
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {3,4,5,5,5,9,14,14}
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {3,4,5,5,5,9,14,14}
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {3,4,5,5,5,9,14,14}
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> ? ∊ {3,4,5,5,5,9,14,14}
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ([(3,12),(4,9),(4,16),(5,6),(5,12),(5,15),(6,14),(6,17),(7,14),(7,15),(7,17),(8,13),(8,16),(8,17),(9,13),(9,17),(10,11),(10,15),(10,16),(10,17),(11,13),(11,14),(11,17),(12,14),(12,17),(13,15),(13,16),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> ([(3,14),(4,13),(5,7),(5,13),(5,17),(6,8),(6,14),(6,18),(7,15),(7,19),(8,16),(8,19),(9,15),(9,17),(9,19),(10,16),(10,18),(10,19),(11,12),(11,15),(11,16),(11,19),(12,17),(12,18),(12,19),(13,15),(13,19),(14,16),(14,19),(15,17),(15,18),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> ([(4,15),(5,14),(6,8),(6,14),(6,18),(7,9),(7,15),(7,19),(8,16),(8,20),(9,17),(9,20),(10,16),(10,18),(10,20),(11,17),(11,19),(11,20),(12,13),(12,16),(12,17),(12,20),(13,18),(13,19),(13,20),(14,16),(14,20),(15,17),(15,20),(16,18),(16,19),(17,18),(17,19),(18,20),(19,20)],21)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[2],[6]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[1,2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[3,3,5]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[3,4,5]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000228
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
Values
[[1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> [1]
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {3,5}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {3,5}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {2,3,5}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {2,3,5}
[[1,1],[3]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {2,3,5}
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {3,4,5,9,14,14}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {3,4,5,9,14,14}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {3,4,5,9,14,14}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {3,4,5,9,14,14}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {3,4,5,9,14,14}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {3,4,5,9,14,14}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [9,9,9,9,3,3]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 2
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 2
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [58,38,38,38,30]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [98,98,37,37,37,37,24,24,24,24,24,24,24,24,10,10,8,8]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[1],[2],[4]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> [26,13,7,7,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ?
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [10,10,10,5,5,2]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[2],[6]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[1,2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,6,6,6,6,4,4,4,3,3]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ?
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ?
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ?
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000321
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
Values
[[1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> [1]
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {3,5}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {3,5}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {2,3,5}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {2,3,5}
[[1,1],[3]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {2,3,5}
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {3,4,5,9,14,14}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {3,4,5,9,14,14}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {3,4,5,9,14,14}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {3,4,5,9,14,14}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {3,4,5,9,14,14}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {3,4,5,9,14,14}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [9,9,9,9,3,3]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 2
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 2
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 5
[[2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [58,38,38,38,30]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [98,98,37,37,37,37,24,24,24,24,24,24,24,24,10,10,8,8]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ?
=> ? ∊ {3,3,4,5,5,6,6,11,16,21,21,42,42}
[[1],[2],[4]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> [26,13,7,7,2]
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ?
=> ? ∊ {2,3,3,4,5,5,9,14,14}
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [10,10,10,5,5,2]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[2],[6]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[1,2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,6,6,6,6,4,4,4,3,3]
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ?
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ?
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ?
=> ? ∊ {3,3,3,4,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
Description
The number of integer partitions of n that are dominated by an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.
Matching statistic: St000319
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
Values
[[1,2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1],[2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1 = 2 - 1
[[1],[3]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {5,5} - 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {5,5} - 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1 = 2 - 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1 = 2 - 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {2,5,5} - 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {2,5,5} - 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {2,5,5} - 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = 1 - 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = 1 - 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {4,5,5,9,14,14} - 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {4,5,5,9,14,14} - 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,5,9,14,14} - 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,5,9,14,14} - 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {4,5,5,9,14,14} - 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {4,5,5,9,14,14} - 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [9,9,9,9,3,3]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1 = 2 - 1
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 1 = 2 - 1
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 1 = 2 - 1
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [58,38,38,38,30]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [98,98,37,37,37,37,24,24,24,24,24,24,24,24,10,10,8,8]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[1],[2],[4]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> [26,13,7,7,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ?
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [10,10,10,5,5,2]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[2],[6]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[1,2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,6,6,6,6,4,4,4,3,3]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
Values
[[1,2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1],[2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1 = 2 - 1
[[1],[3]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {5,5} - 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {5,5} - 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1 = 2 - 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1 = 2 - 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {2,5,5} - 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {2,5,5} - 1
[[1,1],[3]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {2,5,5} - 1
[[1],[2],[3]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = 1 - 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = 1 - 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {4,5,5,9,14,14} - 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {4,5,5,9,14,14} - 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,5,9,14,14} - 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,5,9,14,14} - 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {4,5,5,9,14,14} - 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {4,5,5,9,14,14} - 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [9,9,9,9,3,3]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1 = 2 - 1
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 1 = 2 - 1
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 1 = 2 - 1
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 2 = 3 - 1
[[2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [58,38,38,38,30]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [98,98,37,37,37,37,24,24,24,24,24,24,24,24,10,10,8,8]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ?
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[1],[2],[4]]
=> ([(0,1)],2)
=> [1]
=> 0 = 1 - 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> [26,13,7,7,2]
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ?
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [10,10,10,5,5,2]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> ?
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[2],[6]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[1,2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,6,6,6,6,4,4,4,3,3]
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ?
=> ? ∊ {4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001633
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001633: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
St001633: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 58%●distinct values known / distinct values provided: 8%
Values
[[1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
[[1],[3]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2 = 3 - 1
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {5,5} - 1
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {5,5} - 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2 = 3 - 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {2,5,5} - 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {2,5,5} - 1
[[1,1],[3]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {2,5,5} - 1
[[1],[2],[3]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1],[2,2]]
=> ([],1)
=> 0 = 1 - 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {4,5,5,9,14,14} - 1
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? ∊ {4,5,5,9,14,14} - 1
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {4,5,5,9,14,14} - 1
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {4,5,5,9,14,14} - 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2 = 3 - 1
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {4,5,5,9,14,14} - 1
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {4,5,5,9,14,14} - 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2 = 3 - 1
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 2 - 1
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 2 - 1
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2 = 3 - 1
[[2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42} - 1
[[1],[2],[4]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ? ∊ {2,3,4,5,5,5,9,14,14} - 1
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[2],[6]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {4,5,9,14,14,14,28,42,42} - 1
[[1,2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462} - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000172
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {5,5}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {5,5}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {2,5,5}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,5,5}
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {2,5,5}
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,9,14,14}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {4,5,5,9,14,14}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {4,5,5,9,14,14}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {4,5,5,9,14,14}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,9,14,14}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,9,14,14}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 2
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 2
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(2,10),(2,11),(2,13),(2,15),(2,17),(2,19),(3,8),(3,9),(3,12),(3,14),(3,16),(3,18),(4,8),(4,9),(4,12),(4,14),(4,16),(4,18),(4,19),(5,10),(5,11),(5,13),(5,15),(5,17),(5,18),(5,19),(6,8),(6,9),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(6,19),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,18),(7,19),(8,10),(8,11),(8,13),(8,15),(8,17),(8,19),(9,10),(9,11),(9,13),(9,15),(9,17),(9,19),(10,12),(10,14),(10,16),(10,18),(11,12),(11,14),(11,16),(11,18),(12,13),(12,15),(12,17),(12,19),(13,14),(13,16),(13,18),(14,15),(14,16),(14,17),(14,19),(15,16),(15,17),(15,18),(16,17),(16,19),(17,18),(18,19)],20)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ([(3,12),(4,9),(4,16),(5,6),(5,12),(5,15),(6,14),(6,17),(7,14),(7,15),(7,17),(8,13),(8,16),(8,17),(9,13),(9,17),(10,11),(10,15),(10,16),(10,17),(11,13),(11,14),(11,17),(12,14),(12,17),(13,15),(13,16),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> ([(3,14),(4,13),(5,7),(5,13),(5,17),(6,8),(6,14),(6,18),(7,15),(7,19),(8,16),(8,19),(9,15),(9,17),(9,19),(10,16),(10,18),(10,19),(11,12),(11,15),(11,16),(11,19),(12,17),(12,18),(12,19),(13,15),(13,19),(14,16),(14,19),(15,17),(15,18),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> ([(4,15),(5,14),(6,8),(6,14),(6,18),(7,9),(7,15),(7,19),(8,16),(8,20),(9,17),(9,20),(10,16),(10,18),(10,20),(11,17),(11,19),(11,20),(12,13),(12,16),(12,17),(12,20),(13,18),(13,19),(13,20),(14,16),(14,20),(15,17),(15,20),(16,18),(16,19),(17,18),(17,19),(18,20),(19,20)],21)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[2],[6]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[1,2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
Description
The Grundy number of a graph.
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St000363
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {5,5}
[[4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {5,5}
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {2,5,5}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,5,5}
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {2,5,5}
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1
[[2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,9,14,14}
[[3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {4,5,5,9,14,14}
[[4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {4,5,5,9,14,14}
[[5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {4,5,5,9,14,14}
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[2],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[3],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,9,14,14}
[[4],[5]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,9,14,14}
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 2
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 2
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 2
[[1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 2
[[2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 3
[[2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(2,10),(2,11),(2,13),(2,15),(2,17),(2,19),(3,8),(3,9),(3,12),(3,14),(3,16),(3,18),(4,8),(4,9),(4,12),(4,14),(4,16),(4,18),(4,19),(5,10),(5,11),(5,13),(5,15),(5,17),(5,18),(5,19),(6,8),(6,9),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(6,19),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,18),(7,19),(8,10),(8,11),(8,13),(8,15),(8,17),(8,19),(9,10),(9,11),(9,13),(9,15),(9,17),(9,19),(10,12),(10,14),(10,16),(10,18),(11,12),(11,14),(11,16),(11,18),(12,13),(12,15),(12,17),(12,19),(13,14),(13,16),(13,18),(14,15),(14,16),(14,17),(14,19),(15,16),(15,17),(15,18),(16,17),(16,19),(17,18),(18,19)],20)
=> ? ∊ {4,5,5,5,5,6,6,11,16,21,21,42,42}
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> ? ∊ {2,3,4,5,5,5,9,14,14}
[[2,6]]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> ([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3,6]]
=> ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> ([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4,6]]
=> ([(0,1),(1,4),(1,5),(2,13),(3,12),(4,14),(5,7),(5,14),(6,10),(7,8),(7,15),(8,6),(8,17),(10,11),(11,9),(12,9),(13,3),(13,16),(14,2),(14,15),(15,13),(15,17),(16,11),(16,12),(17,10),(17,16)],18)
=> ([(3,12),(4,9),(4,16),(5,6),(5,12),(5,15),(6,14),(6,17),(7,14),(7,15),(7,17),(8,13),(8,16),(8,17),(9,13),(9,17),(10,11),(10,15),(10,16),(10,17),(11,13),(11,14),(11,17),(12,14),(12,17),(13,15),(13,16),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5,6]]
=> ([(0,1),(1,5),(1,6),(2,15),(3,14),(4,10),(5,16),(6,8),(6,16),(7,12),(8,9),(8,17),(9,7),(9,19),(11,13),(12,11),(13,10),(14,4),(14,13),(15,3),(15,18),(16,2),(16,17),(17,15),(17,19),(18,11),(18,14),(19,12),(19,18)],20)
=> ([(3,14),(4,13),(5,7),(5,13),(5,17),(6,8),(6,14),(6,18),(7,15),(7,19),(8,16),(8,19),(9,15),(9,17),(9,19),(10,16),(10,18),(10,19),(11,12),(11,15),(11,16),(11,19),(12,17),(12,18),(12,19),(13,15),(13,19),(14,16),(14,19),(15,17),(15,18),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[6,6]]
=> ([(0,10),(1,20),(2,19),(4,18),(5,17),(6,13),(7,8),(7,17),(8,9),(8,11),(9,6),(9,15),(10,5),(10,7),(11,15),(11,18),(12,16),(12,20),(13,16),(14,19),(15,12),(15,13),(16,14),(17,4),(17,11),(18,1),(18,12),(19,3),(20,2),(20,14)],21)
=> ([(4,15),(5,14),(6,8),(6,14),(6,18),(7,9),(7,15),(7,19),(8,16),(8,20),(9,17),(9,20),(10,16),(10,18),(10,20),(11,17),(11,19),(11,20),(12,13),(12,16),(12,17),(12,20),(13,18),(13,19),(13,20),(14,16),(14,20),(15,17),(15,20),(16,18),(16,19),(17,18),(17,19),(18,20),(19,20)],21)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[2],[6]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[3],[6]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[4],[6]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[5],[6]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {4,5,9,14,14,14,28,42,42}
[[1,2,5]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,3,5]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,4,5]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[1,5,5]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,2,5]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,3,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,4,5]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
[[2,5,5]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? ∊ {3,4,5,5,5,5,5,5,6,7,9,10,14,14,16,18,21,21,23,29,35,42,42,52,56,70,84,94,168,210,252,462,462}
Description
The number of minimal vertex covers of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices.
This is also the leading coefficient of the clique polynomial of the complement of $G$.
This is also the number of independent sets of maximal cardinality of $G$.
The following 196 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000388The number of orbits of vertices of a graph under automorphisms. St000468The Hosoya index of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001963The tree-depth of a graph. St000171The degree of the graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St000081The number of edges of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000298The order dimension or Dushnik-Miller dimension of a poset. St000632The jump number of the poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001397Number of pairs of incomparable elements in a finite poset. St000456The monochromatic index of a connected graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St000343The number of spanning subgraphs of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001734The lettericity of a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001649The length of a longest trail in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St000087The number of induced subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000364The exponent of the automorphism group of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000972The composition number of a graph. St001029The size of the core of a graph. St001093The detour number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001116The game chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000271The chromatic index of a graph. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000361The second Zagreb index of a graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001323The independence gap of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001362The normalized Knill dimension of a graph. St001391The disjunction number of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001743The discrepancy of a graph. St001783The number of odd automorphisms of a graph. St001792The arboricity of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001973The Gromov width of a graph. St000527The width of the poset. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001624The breadth of a lattice. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001964The interval resolution global dimension of a poset. 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 $eA$. St001877Number of indecomposable injective modules with projective dimension 2. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001118The acyclic chromatic index of a graph. St000464The Schultz index of a connected graph. St001281The normalized isoperimetric number of a graph. St001545The second Elser number of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
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