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Your data matches 121 different statistics following compositions of up to 3 maps.
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Matching statistic: St000964
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(load all 2 compositions to match this statistic)
St000964: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 1
[1,1,0,0]
=> 3
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 6
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 6
[1,1,1,1,0,0,0,0]
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 3
Description
Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra.
Matching statistic: St000208
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 68%●distinct values known / distinct values provided: 60%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 68%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,3,6}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,3,6}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[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]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[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]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[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]
=> 1
[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]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[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]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,7,7,10,10,15,21}
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St001389
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 68%●distinct values known / distinct values provided: 60%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 68%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,3,6}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,3,6}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[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]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[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]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 1
[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]
=> 1
[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]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[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]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000063
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [[1],[]]
=> []
=> ?
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ?
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> ?
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> []
=> 1
[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],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> []
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> []
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
Description
The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
Matching statistic: St000108
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 68%●distinct values known / distinct values provided: 60%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 68%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [[1],[]]
=> []
=> ?
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ?
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> ?
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> []
=> 1
[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],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> []
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> []
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,7,7,10,10,15,21}
Description
The number of partitions contained in the given partition.
Matching statistic: St000532
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 68%●distinct values known / distinct values provided: 60%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 68%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [[1],[]]
=> []
=> ?
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ?
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> ?
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> []
=> 1
[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],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> []
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> []
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,7,10,10,15,21}
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St000738
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [[1,2,3]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [[1]]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[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],[2]]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [[1,2,3]]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [[1,2]]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [[1]]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,5,6,6,6,6,6,6,7,7,10,10,15,21}
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St001291
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 68%●distinct values known / distinct values provided: 40%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 68%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St001400
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001400: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001400: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [[1],[]]
=> []
=> ?
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> ?
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> ?
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,3,6}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> []
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> []
=> 1
[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],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> []
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> []
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> []
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> []
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> []
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> []
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> []
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> []
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> []
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,7,7,10,10,15,21}
Description
The total number of Littlewood-Richardson tableaux of given shape.
This is the multiplicity of the Schur function $s_\lambda$ in $\sum_{\mu, \nu} s_\mu s_\nu$, where the sum is over all partitions $\mu$ and $\nu$.
Matching statistic: St001733
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 68%●distinct values known / distinct values provided: 40%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 68%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {1,3}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {1,1,3,6}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {1,1,2,2,3,3,6,10}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {1,2,2,2,3,3,3,3,3,4,4,4,6,6,10,15}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {1,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,6,7,7,10,10,15,21}
Description
The number of weak left to right maxima of a Dyck path.
A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its
left.
The following 111 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001809The index of the step at the first peak of maximal height in a Dyck path. St001924The number of cells in an integer partition whose arm and leg length coincide. St000993The multiplicity of the largest part of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000456The monochromatic index 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. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001933The largest multiplicity of a part in an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001568The smallest positive integer that does not appear twice in the partition. St000327The number of cover relations in a poset. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000260The radius of a connected graph. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001118The acyclic chromatic index of a graph. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000474Dyson's crank of a partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000681The Grundy value of Chomp on Ferrers diagrams. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001890The maximum magnitude of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000741The Colin de Verdière graph invariant. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000264The girth of a graph, which is not a tree. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000770The major index of an integer partition when read from bottom to top. St001875The number of simple modules with projective dimension at most 1. St000454The largest eigenvalue of a graph if it is integral.
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