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Your data matches 59 different statistics following compositions of up to 3 maps.
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Matching statistic: St001438
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Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [[1],[]]
=> 0
{{1,2}}
=> [2] => [2] => [[2],[]]
=> 0
{{1},{2}}
=> [1,1] => [1,1] => [[1,1],[]]
=> 0
{{1,2,3}}
=> [3] => [3] => [[3],[]]
=> 0
{{1,2},{3}}
=> [2,1] => [1,2] => [[2,1],[]]
=> 0
{{1,3},{2}}
=> [2,1] => [1,2] => [[2,1],[]]
=> 0
{{1},{2,3}}
=> [1,2] => [2,1] => [[2,2],[1]]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => [[1,1,1],[]]
=> 0
{{1,2,3,4}}
=> [4] => [4] => [[4],[]]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,3] => [[3,1],[]]
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,3] => [[3,1],[]]
=> 0
{{1,2},{3,4}}
=> [2,2] => [2,2] => [[3,2],[1]]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,3] => [[3,1],[]]
=> 0
{{1,3},{2,4}}
=> [2,2] => [2,2] => [[3,2],[1]]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> 1
{{1,4},{2,3}}
=> [2,2] => [2,2] => [[3,2],[1]]
=> 1
{{1},{2,3,4}}
=> [1,3] => [3,1] => [[3,3],[2]]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,1,2] => [[2,1,1],[]]
=> 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,2,1] => [[2,2,1],[1]]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,1,2] => [[2,1,1],[]]
=> 0
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> 0
{{1,2,3,4,5}}
=> [5] => [5] => [[5],[]]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => [[4,1],[]]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => [[4,1],[]]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => [[4,1],[]]
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => [[4,3],[2]]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [[3,2,1],[1]]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [[3,2,1],[1]]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> 3
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 2
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => [[4,1],[]]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => [[4,3],[2]]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [[3,2,1],[1]]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,3,1] => [[3,3,1],[2]]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [[3,2,1],[1]]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> 3
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> 2
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => [[4,2],[1]]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => [[4,3],[2]]
=> 2
Description
The number of missing boxes of a skew partition.
Matching statistic: St000612
St000612: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 0
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> 2
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 0
{{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> 1
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 2
{{1,3,4},{2,5}}
=> 2
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 2
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 2
{{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.
Matching statistic: St001964
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Mp00080: Set partitions —to permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 80%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [2,1] => ([],2)
=> 0
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1)],2)
=> 0
{{1,2,3}}
=> [2,3,1] => [3,2,1] => ([],3)
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,1,2] => ([(1,2)],3)
=> ? = 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {1,1,2}
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,3,2,1] => ([],4)
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {1,1,2}
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {1,1,2}
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 2
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,4}
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001384
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1]
=> [1]
=> []
=> ? = 0
{{1,2}}
=> [2]
=> [2]
=> []
=> ? = 0
{{1},{2}}
=> [1,1]
=> [1,1]
=> [1]
=> 0
{{1,2,3}}
=> [3]
=> [2,1]
=> [1]
=> 0
{{1,2},{3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1,3},{2}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2,3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> 1
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,1,2}
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,1,2}
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,1,2}
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2
{{1,2,3,4,5}}
=> [5]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,2,5},{3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,3,5},{2,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,5},{2,3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [3,2]
=> [2]
=> 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 1
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St000771
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 80%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> ? = 0 + 1
{{1},{2}}
=> [1,2] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0} + 1
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0} + 1
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,2} + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,2} + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,2} + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,2} + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,2} + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,2} + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4} + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000510
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Values
{{1}}
=> [1] => [[1],[]]
=> []
=> ? = 0
{{1,2}}
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1}
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1}
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1,2,3,4}}
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,5},{2,3,4}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2,3,4,5}}
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2,3,4},{5}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1},{2,3,5},{4}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,4},{2},{3,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1},{2,4,5},{3}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
{{1,5},{2},{3,4}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2},{3,4,5}}
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000566
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 80%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1] => [[1],[]]
=> []
=> ? = 0
{{1,2}}
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1}
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1}
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1,2,3,4}}
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2}
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2}
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2}
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2}
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2}
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2}
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2,2}
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,2,2}
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 3
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,5},{2,3,4}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1},{2,3,4,5}}
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1},{2,3,4},{5}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1},{2,3,5},{4}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,4},{2},{3,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1},{2,4,5},{3}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1},{2},{3,4,5}}
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {2,2,2,2,2,2,2,2,2,2,2,3,3,4}
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000681
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Values
{{1}}
=> [1] => [[1],[]]
=> []
=> ? = 0
{{1,2}}
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1}
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,1}
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1}
{{1,2,3,4}}
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 2
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,5},{2,3,4}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2,3,4,5}}
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2,3,4},{5}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1},{2,3,5},{4}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1,4},{2},{3,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
{{1},{2,4,5},{3}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
{{1,5},{2},{3,4}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2},{3,4,5}}
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,3,3,3,3,3,3,4}
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St001435
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 80%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 80%
Values
{{1}}
=> [1] => [1,0]
=> [[1],[]]
=> 0
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [[2],[]]
=> 0
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [[1,1],[]]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {0,1,2}
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {0,1,2}
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {0,1,2}
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 0
Description
The number of missing boxes in the first row.
Matching statistic: St001487
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 60%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 60%
Values
{{1}}
=> [1] => [1,0]
=> [[1],[]]
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [[2],[]]
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [[1,1],[]]
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {0,2,2} + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {0,2,2} + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {0,2,2} + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1 = 0 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 2 = 1 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 2 = 1 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 2 = 1 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 1 = 0 + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 2 = 1 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4} + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
Description
The number of inner corners of a skew partition.
The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000937The number of positive values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000454The largest eigenvalue of a graph if it is integral. St001862The number of crossings of a signed permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001875The number of simple modules with projective dimension at most 1. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. 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. St001100The 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 trees. St001101The 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 increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001857The number of edges in the reduced word graph of a signed permutation.
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