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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000737
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000737: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000737: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> [[1]]
=> 1
[[1,2]]
=> [[2,0],[1]]
=> [[1,2]]
=> 1
[[1],[2]]
=> [[1,1],[1]]
=> [[1],[2]]
=> 1
[[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> 1
[[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> 1
[[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> 1
[[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> 1
[[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> 1
[[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2]]
=> 1
[[1,2,4],[3]]
=> [[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3]]
=> 1
[[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4]]
=> 1
[[1,3],[2,4]]
=> [[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4]]
=> 4
[[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4]]
=> 4
[[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2],[3]]
=> 1
[[1,3],[2],[4]]
=> [[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2],[4]]
=> 1
[[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4]]
=> 1
[[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> 1
[[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5]]
=> 1
[[1,3,4,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5],[2]]
=> 1
[[1,2,4,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5],[3]]
=> 1
[[1,2,3,5],[4]]
=> [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5],[4]]
=> 1
[[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4],[5]]
=> 1
[[1,3,5],[2,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2,4]]
=> 4
[[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3,4]]
=> 4
[[1,3,4],[2,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2,5]]
=> 5
[[1,2,4],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3,5]]
=> 5
[[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4,5]]
=> 5
[[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4,5],[2],[3]]
=> 1
[[1,3,5],[2],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3,5],[2],[4]]
=> 1
[[1,2,5],[3],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2,5],[3],[4]]
=> 1
[[1,3,4],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4],[2],[5]]
=> 1
[[1,2,4],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4],[3],[5]]
=> 1
[[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3],[4],[5]]
=> 1
[[1,4],[2,5],[3]]
=> [[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2,5],[3]]
=> 5
[[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,5],[4]]
=> 5
[[1,2],[3,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,5],[4]]
=> 5
[[1,3],[2,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2,4],[5]]
=> 4
[[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3,4],[5]]
=> 4
[[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1,5],[2],[3],[4]]
=> 1
[[1,4],[2],[3],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> [[1,4],[2],[3],[5]]
=> 1
[[1,3],[2],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2],[4],[5]]
=> 1
[[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3],[4],[5]]
=> 1
[[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> 1
[[1,2,3,4,5,6]]
=> [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5,6]]
=> 1
[[1,3,4,5,6],[2]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,4,5,6],[2]]
=> 1
[[1,2,4,5,6],[3]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2,4,5,6],[3]]
=> 1
[[1,2,3,5,6],[4]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,5,6],[4]]
=> 1
[[1,2,3,4,6],[5]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,6],[5]]
=> 1
[[1,2,3,4,5],[6]]
=> [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4,5],[6]]
=> 1
[[1,3,5,6],[2,4]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> [[1,3,5,6],[2,4]]
=> 4
Description
The last entry on the main diagonal of a semistandard tableau.
Matching statistic: St000735
St000735: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> ? = 1
[[1,2]]
=> 1
[[1],[2]]
=> 1
[[1,2,3]]
=> 1
[[1,3],[2]]
=> 1
[[1,2],[3]]
=> 1
[[1],[2],[3]]
=> 1
[[1,2,3,4]]
=> 1
[[1,3,4],[2]]
=> 1
[[1,2,4],[3]]
=> 1
[[1,2,3],[4]]
=> 1
[[1,3],[2,4]]
=> 4
[[1,2],[3,4]]
=> 4
[[1,4],[2],[3]]
=> 1
[[1,3],[2],[4]]
=> 1
[[1,2],[3],[4]]
=> 1
[[1],[2],[3],[4]]
=> 1
[[1,2,3,4,5]]
=> 1
[[1,3,4,5],[2]]
=> 1
[[1,2,4,5],[3]]
=> 1
[[1,2,3,5],[4]]
=> 1
[[1,2,3,4],[5]]
=> 1
[[1,3,5],[2,4]]
=> 4
[[1,2,5],[3,4]]
=> 4
[[1,3,4],[2,5]]
=> 5
[[1,2,4],[3,5]]
=> 5
[[1,2,3],[4,5]]
=> 5
[[1,4,5],[2],[3]]
=> 1
[[1,3,5],[2],[4]]
=> 1
[[1,2,5],[3],[4]]
=> 1
[[1,3,4],[2],[5]]
=> 1
[[1,2,4],[3],[5]]
=> 1
[[1,2,3],[4],[5]]
=> 1
[[1,4],[2,5],[3]]
=> 5
[[1,3],[2,5],[4]]
=> 5
[[1,2],[3,5],[4]]
=> 5
[[1,3],[2,4],[5]]
=> 4
[[1,2],[3,4],[5]]
=> 4
[[1,5],[2],[3],[4]]
=> 1
[[1,4],[2],[3],[5]]
=> 1
[[1,3],[2],[4],[5]]
=> 1
[[1,2],[3],[4],[5]]
=> 1
[[1],[2],[3],[4],[5]]
=> 1
[[1,2,3,4,5,6]]
=> 1
[[1,3,4,5,6],[2]]
=> 1
[[1,2,4,5,6],[3]]
=> 1
[[1,2,3,5,6],[4]]
=> 1
[[1,2,3,4,6],[5]]
=> 1
[[1,2,3,4,5],[6]]
=> 1
[[1,3,5,6],[2,4]]
=> 4
[[1,2,5,6],[3,4]]
=> 4
Description
The last entry on the main diagonal of a standard tableau.
Matching statistic: St001195
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {4,4}
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {4,4}
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,3,4,5],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,4,5],[3]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,3,5],[4]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,3,4],[5]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,3,5],[2,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,5],[4,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4],[5,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5],[2,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5],[3,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4],[2,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4],[3,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3],[4,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [3,1,2] => 1
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? ∊ {4,4}
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {4,4}
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,3,4,5],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,4,5],[3]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,3,5],[4]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,3,4],[5]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,3,5],[2,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,5],[4,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4],[5,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5],[2,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5],[3,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4],[2,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4],[3,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3],[4,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001490
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {4,4}
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 1
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 1
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {4,4}
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,3,4,5],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,4,5],[3]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,3,5],[4]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,3,4],[5]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,3,5],[2,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5}
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,5],[4,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4],[5,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5],[2,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5],[3,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4],[2,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,4],[3,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,3],[4,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
Description
The number of connected components of a skew partition.
Matching statistic: St001001
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {4,4} - 1
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {4,4} - 1
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3,4,5],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,4,5],[3]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,5],[4]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,4],[5]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3,5],[2,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,6],[2,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,6],[3,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,6],[4,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5],[2,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5],[3,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5],[4,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4],[5,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5],[2,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5],[3,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4],[2,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4],[3,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3],[4,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001371
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 0 = 1 - 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 0 = 1 - 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 0 = 1 - 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0 = 1 - 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0 = 1 - 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {4,4} - 1
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0 = 1 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {4,4} - 1
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3,4,5],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,4,5],[3]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,5],[4]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,4],[5]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3,5],[2,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,6],[2,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,6],[3,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,6],[4,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5],[2,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5],[3,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5],[4,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4],[5,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5],[2,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5],[3,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4],[2,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4],[3,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3],[4,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 0 = 1 - 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 0 = 1 - 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 0 = 1 - 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0 = 1 - 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0 = 1 - 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {4,4} - 1
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0 = 1 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {4,4} - 1
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3,4,5],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,4,5],[3]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,5],[4]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,4],[5]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3,5],[2,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,6],[2,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,6],[3,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,6],[4,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5],[2,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5],[3,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5],[4,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4],[5,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5],[2,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5],[3,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4],[2,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4],[3,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3],[4,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001803
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0 = 1 - 1
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 0 = 1 - 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 1 - 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 0 = 1 - 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 1 - 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 1 - 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? ∊ {4,4} - 1
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0 = 1 - 1
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0 = 1 - 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0 = 1 - 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0 = 1 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? ∊ {4,4} - 1
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3,4,5],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,4,5],[3]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,5],[4]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,4],[5]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3,5],[2,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0 = 1 - 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0 = 1 - 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0 = 1 - 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0 = 1 - 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0 = 1 - 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} - 1
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,6],[2,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,6],[3,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,6],[4,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5],[2,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5],[3,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5],[4,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4],[5,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5],[2,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5],[3,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4],[2,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,4],[3,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,3],[4,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0 = 1 - 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001804
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 39%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 2 = 1 + 1
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 2 = 1 + 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 2 = 1 + 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 2 = 1 + 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 1 + 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? ∊ {4,4} + 1
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 1 + 1
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 2 = 1 + 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? ∊ {4,4} + 1
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1,3,4,5],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1,2,4,5],[3]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1,2,3,5],[4]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1,2,3,4],[5]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1,3,5],[2,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 1 + 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 1 + 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 1 + 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 1 + 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 1 + 1
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? ∊ {4,4,4,4,5,5,5,5,5,5} + 1
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,4,6],[2,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,4,6],[3,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3,6],[4,5]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,4,5],[2,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,4,5],[3,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3,5],[4,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3,4],[5,6]]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,5],[2,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,5],[3,4,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,4],[2,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,4],[3,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,3],[4,5,6]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 2 = 1 + 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6} + 1
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000422The energy of a graph, if it is integral. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000456The monochromatic index of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001964The interval resolution global dimension of a poset. St000068The number of minimal elements in a poset. St001845The number of join irreducibles minus the rank of a lattice. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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