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Your data matches 56 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
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Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
Description
The largest part of an integer partition.
Matching statistic: St000010
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> [1]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
Description
The length of the partition.
Matching statistic: St000734
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> [[1]]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000097
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> ([(1,5),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,3),(1,8),(2,3),(2,8),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> ([(1,5),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,3),(1,8),(2,3),(2,8),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000744
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 80% ●values known / values provided: 94%●distinct values known / distinct values provided: 80%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 80% ●values known / values provided: 94%●distinct values known / distinct values provided: 80%
Values
[.,.]
=> ([],1)
=> [1]
=> [[1]]
=> ? = 1 - 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 2 = 3 - 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 2 = 3 - 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 2 = 3 - 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 3 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 3 = 4 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 3 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 3 = 4 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 3 = 4 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> 2 = 3 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 4 = 5 - 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[[[[[.,[[.,.],.]],.],.],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[[[[.,[[[[.,.],.],.],.]],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 4 - 1
[[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[.,[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[.,[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[.,[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[.,[.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[.,[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[[[[[[[.,[.,[[.,.],.]]],.],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[[[[[.,[.,.]],.],.],.],.]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[[[[[[.,[.,.]],.],.],.],.],.]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
[.,[.,[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> ? = 10 - 1
Description
The length of the path to the largest entry in a standard Young tableau.
Matching statistic: St000676
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 85%●distinct values known / distinct values provided: 80%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 85%●distinct values known / distinct values provided: 80%
Values
[.,.]
=> ([],1)
=> [1]
=> [1,0]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001039
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 85%●distinct values known / distinct values provided: 70%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 85%●distinct values known / distinct values provided: 70%
Values
[.,.]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St001720
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[.,[.,[.,.]]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[.,[.,[.,.]]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[.,[.,[.,.]]],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[.,[[.,.],.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[.,[[.,.],.]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[.,[[.,.],.]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[.,[[.,.],.]],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[[.,[.,.]],.],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[[.,[.,.]],.],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[[.,[.,.]],.],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[[.,[.,.]],.],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[[[.,.],.],.],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[[[.,.],.],.],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[[[[.,.],.],.],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 4
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St000528
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 71%●distinct values known / distinct values provided: 70%
St000528: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 71%●distinct values known / distinct values provided: 70%
Values
[.,.]
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[.,[.,[.,[[.,.],.]]]],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[.,[.,[[[.,.],.],.]]],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[.,[[[[.,.],.],.],.]],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[[.,[.,[.,[.,.]]]],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[.,[[.,[[.,.],.]],.]]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[.,[[[.,[.,.]],.],.]]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[.,[[[[.,.],.],.],.]]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[[.,[.,[[.,.],.]]],.]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[[[.,[.,[.,.]]],.],.]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[[[.,[[.,.],.]],.],.]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[[[[.,[.,.]],.],.],.]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[.,[[[.,[.,[.,[[.,.],.]]]],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001343The dimension of the reduced incidence algebra of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000808The number of up steps of the associated bargraph. St001494The Alon-Tarsi number of a graph. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001580The acyclic chromatic number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001644The dimension of a graph. St000907The number of maximal antichains of minimal length in a poset. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001717The largest size of an interval in a poset. St000050The depth or height of a binary tree. St001330The hat guessing number of a graph. St000080The rank of the poset. St000013The height of a Dyck path. St000822The Hadwiger number of the 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. St001323The independence gap of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001820The size of the image of the pop stack sorting operator. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001626The number of maximal proper sublattices of a lattice. St000299The number of nonisomorphic vertex-induced subtrees. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000442The maximal area to the right of an up step of a Dyck path. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one.
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