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Your data matches 84 different statistics following compositions of up to 3 maps.
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Matching statistic: St001537
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Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001537: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001537: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 0
[[.,.],.]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => 0
[[.,.],[.,.]]
=> [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 4
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
Description
The number of cyclic crossings of a permutation.
The pair $(i,j)$ is a cyclic crossing of a permutation $\pi$ if $i, \pi(j), \pi(i), j$ are cyclically ordered and all distinct, see Section 5 of [1].
Matching statistic: St000175
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 0
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> 0
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000225
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 0
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> 0
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000749
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,4}
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> 0
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
Description
The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St001175
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,4}
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> 0
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001586
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 0
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,4}
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> 0
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
Description
The number of odd parts smaller than the largest even part in an integer partition.
Matching statistic: St001714
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0}
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 1
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 0
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,4}
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> 0
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 2
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,3,3,4,4,4,4,4,4,4,4,4,5,5}
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St001912
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001912: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001912: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 0
[.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,0}
[[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,0}
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,0,0,0}
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {0,0,0,0}
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 0
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 0
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 0
[[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,4}
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> [1]
=> 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [2]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> [1]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> 0
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 2
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 0
[.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[[.,[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
[[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[5,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,4,5,5}
Description
The length of the preperiod in Bulgarian solitaire corresponding to an integer partition.
Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.
Matching statistic: St000455
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 67%●distinct values known / distinct values provided: 33%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 67%●distinct values known / distinct values provided: 33%
Values
[.,.]
=> [1] => [1] => ([],1)
=> ? = 0
[.,[.,.]]
=> [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0}
[[.,.],.]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0}
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0}
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0}
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0}
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,1}
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,1}
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,1}
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,1}
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2,4}
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,.],[.,.]]]]]
=> [4,6,5,3,2,1] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [4,3,6,5,2,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0
[.,[.,[[[.,.],.],[.,.]]]]
=> [3,4,6,5,2,1] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> [3,4,5,6,2,1] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,.],[[.,.],[.,.]]]]
=> [2,4,6,5,3,1] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[.,.]],.],[.,.]]]
=> [3,2,4,6,5,1] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,.],[.,.]]],.]]
=> [3,5,4,2,6,1] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,[.,.]],.]],.]]
=> [4,3,5,2,6,1] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,.],[.,[.,.]]],.]]
=> [2,5,4,3,6,1] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[[[.,.],.],.],.],.]]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[.,[[.,.],[.,.]]]]
=> [1,4,6,5,3,2] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[.,.],[.,[.,.]]]]
=> [1,3,6,5,4,2] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[.,.],[[.,.],.]]]
=> [1,3,5,6,4,2] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[[.,.],.],[.,.]]]
=> [1,3,4,6,5,2] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[.,[.,[.,.]]],.]]
=> [1,5,4,3,6,2] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[[.,.],[.,.]],.]]
=> [1,3,5,4,6,2] => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[.,.]],[[.,.],[.,.]]]
=> [2,1,4,6,5,3] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,.],.],[[.,.],[.,.]]]
=> [1,2,4,6,5,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,[.,.]],.],[.,[.,.]]]
=> [2,1,3,6,5,4] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,[.,.]],.],[[.,.],.]]
=> [2,1,3,5,6,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[[.,.],[.,.]]],[.,.]]
=> [2,4,3,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[[.,[.,.]],.]],[.,.]]
=> [3,2,4,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,[.,[.,.]]],.],[.,.]]
=> [3,2,1,4,6,5] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[.,[[.,.],.]],.],[.,.]]
=> [2,3,1,4,6,5] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[[[.,[.,.]],.],.],[.,.]]
=> [2,1,3,4,6,5] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[.,[[.,.],[.,.]]]],.]
=> [3,5,4,2,1,6] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[.,[[.,[.,.]],.]]],.]
=> [4,3,5,2,1,6] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[[.,.],[.,[.,.]]]],.]
=> [2,5,4,3,1,6] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[[.,.],[[.,.],.]]],.]
=> [2,4,5,3,1,6] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,[[.,[.,.]],[.,.]]],.]
=> [3,2,5,4,1,6] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000929
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 62%●distinct values known / distinct values provided: 17%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 62%●distinct values known / distinct values provided: 17%
Values
[.,.]
=> ([],1)
=> [1]
=> ? = 0
[.,[.,.]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0}
[[.,.],.]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0}
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0}
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0}
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 0
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0}
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0}
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1}
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1}
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 0
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1}
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1}
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 0
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,4}
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> 0
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0
[.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,5,5}
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000655The length of the minimal rise of a Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001964The interval resolution global dimension of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St000302The determinant of the distance matrix of a connected 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. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001498The normalised height of a Nakayama algebra with magnitude 1. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000938The number of zeros of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001845The number of join irreducibles minus the rank of a lattice. St000741The Colin de Verdière graph invariant. St000068The number of minimal elements in a poset. St001490The number of connected components of a skew partition. St000842The breadth of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000629The defect of a binary word. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000383The last part of an integer composition. St001867The number of alignments of type EN of a signed permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001549The number of restricted non-inversions between exceedances. St001570The minimal number of edges to add to make a graph Hamiltonian. St001811The Castelnuovo-Mumford regularity of a permutation. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001162The minimum jump of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001344The neighbouring number of a permutation. St000445The number of rises of length 1 of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001584The area statistic between a Dyck path and its bounce path. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices.
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